Электродинамика СВЧ

Œˆˆ‘’…‘’‚Ž Ž™…ƒŽ ˆ Ž”…‘‘ˆŽ€‹œŽƒŽ Ž€‡Ž‚€ˆŸ Ž‘‘ˆ‰‘ŠŽ‰ ”…„…€–ˆˆ Ž‚Ž‘ˆˆ‘Šˆ‰ ƒŽ‘“„€‘’‚…›‰ “ˆ‚…‘ˆ’…’ ”¨§¨ç¥áª¨© ä ªã«ìâ¥â Š ä¥¤à  à ¤¨®ä¨§¨ª¨

Œ.Œ. Š€‹ˆ…

‹…Š’Ž„ˆ€ŒˆŠ€ ‘‚— Šãàá «¥ªæ¨©

Ž‚Ž‘ˆˆ‘Š 1999

à¥¤« £ ¥¬ë© ªãàá «¥ªæ¨© \«¥ªâத¨­ ¬¨ª  ‘‚—" ç¨â «áï ¢ â¥ç¥­¨¥ à鸞 «¥â ¤«ï áâ㤥­â®¢ 䨧¨ç¥áª®£® ä ªã«ìâ¥â  ƒ“, ᯥ樠«¨§¨àãîé¨åáï ­  ª ä¥¤à¥ à ¤¨®ä¨§¨ª¨. Šãàá ᮤ¥à¦¨â ⥮à¨î ¤«¨­­ëå «¨­¨©, ¢®«­®¢®¤®¢, १®­ â®à®¢, ­¥¢§ ¨¬­ëå ãáâனáâ¢: ¢¥­â¨«¥© ¨ æ¨àªã«ïâ®à®¢. ‚ ¤ ­­®© à ¡®â¥ ¯à¥¤áâ ¢«¥­ë ­¥ ⮫쪮 ®á­®¢­ë¥ ᢮©á⢠ í«¥¬¥­â®¢ ‘‚—-â¥å­¨ª¨, ­® ¨ ¬¥â®¤ë ¨å à áç¥â®¢: ª¢ §¨áâ â¨ç¥áª¨©, ¬¥â®¤ ç áâ¨ç­ëå ®¡« á⥩, ¢ à¨ æ¨®­­ë¥ ¬¥â®¤ë ¨ ¤à. à¨¬¥­¥­¨¥ íâ¨å ¬¥â®¤®¢ ¨««îáâà¨à®¢ ­® ᮮ⢥âáâ¢ãî騬¨ ¯à¨¬¥à ¬¨. Šãàá «¥ªæ¨© ¯à¥¤­ §­ ç¥­ ¤«ï áâ㤥­â®¢ à §«¨ç­ëå ᯥ樠«ì­®á⥩,   â ª¦¥ ¤«ï  á¯¨à ­â®¢ ¨ ¤à㣨å ᯥ樠«¨á⮢, á¢ï§ ­­ëå á ‘‚—â¥å­¨ª®©. ¥ç â ¥âáï ¯® à¥è¥­¨î ¬¥â®¤¨ç¥áª®© ª®¬¨áᨨ 䨧¨ç¥áª®£® ä ªã«ìâ¥â 

¥æ¥­§¥­â: ª ­¤. â¥å­. ­ ãª ‚.„. ˜¥¬¥«¨­

c ®¢®á¨¡¨à᪨© £®á㤠àá⢥­­ë© ã­¨¢¥àá¨â¥â, 1999

Ž£« ¢«¥­¨¥

1 Ž¡é¨¥ ᢥ¤¥­¨ï ¨§ ⥮ਨ í«¥ªâ஬ £­¨â­®£® ¯®«ï 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

“à ¢­¥­¨ï Œ ªá¢¥««  . . . . . . . . ‘¨á⥬  ¥¤¨­¨æ . . . . . . . . . . . . ƒà ­¨ç­ë¥ ãá«®¢¨ï . . . . . . . . . . ­¥à£¨ï í«¥ªâ஬ £­¨â­®£® ¯®«ï . . ’¥®à¥¬  ¥¤¨­á⢥­­®á⨠. . . . . . . ¥è¥­¨ï ãà ¢­¥­¨© Œ ªá¢¥««  . . . ‘ª¨­-íä䥪⠨ £à ­¨ç­ë¥ ãá«®¢¨ï . «¥ªâத¨­ ¬¨ç¥áª®¥ ¯®¤®¡¨¥ . . . .

. . . . . . . .

. . . . . . . .

®¯¥à¥ç­ë¥ ¢®«­ë ¢ «¨­¨¨ ¯¥à¥¤ ç¨ . . . . . . . . . . . .  ¯à殮­¨¥, ⮪ ¨ ¬®é­®áâì ¢ TEM-«¨­¨¨ . . . . . . . . Š® ªá¨ «ì­ ï «¨­¨ï . . . . . . . . . . . . . . . . . . . . . „¢ã寫 á⨭ç â ï «¨­¨ï . . . . . . . . . . . . . . . . . . . „¢ãå¯à®¢®¤­ ï «¨­¨ï . . . . . . . . . . . . . . . . . . . . . ’¥«¥£à ä­ë¥ ãà ¢­¥­¨ï . . . . . . . . . . . . . . . . . . . Žâà ¦¥­¨¥ ®â ­ £à㧪¨ . . . . . . . . . . . . . . . . . . . . ’à ­áä®à¬ æ¨ï ᮯà®â¨¢«¥­¨© ¨ ¯à®¢®¤¨¬®á⥩ . . . . . 2.8.1 Š®à®âª®§ ¬ª­ãâ ï «¨­¨ï . . . . . . . . . . . . . . 2.8.2  §®¬ª­ãâ ï «¨­¨ï . . . . . . . . . . . . . . . . . . 2.8.3 ‹¨­¨ï, § ¬ª­ãâ ï ­   ªâ¨¢­®¥ ᮯà®â¨¢«¥­¨¥ . . 2.8.4 ‹¨­¨ï, § ¬ª­ãâ ï ­  ॠªâ¨¢­®¥ ᮯà®â¨¢«¥­¨¥ . ’à ­áä®à¬ æ¨ï ᮯà®â¨¢«¥­¨© ®â१ª ¬¨ «¨­¨¨ . . . . 2.9.1 ®«ã¢®«­®¢ë© ®â१®ª . . . . . . . . . . . . . . . . 2.9.2 —¥â¢¥àâ좮«­®¢ë© ®â१®ª . . . . . . . . . . . . . . à¥¤áâ ¢«¥­¨¥ ®â१ª  «¨­¨¨ T-®¡à §­ë¬ ç¥âëà¥å¯®«îá­¨ª®¬ . . . . . . . . . . . . . . . . . . . . . . ®¬®£à ¬¬  ¯®«­ëå ᮯà®â¨¢«¥­¨© (¤¨ £à ¬¬  ‘¬¨â ) 2.11.1 à¥®¡à §®¢ ­¨¥ ª®íää¨æ¨¥­â  ®âà ¦¥­¨ï ®â१ª®¬ «¨­¨¨ . . . . . . . . . . . . . . . . . . . . 2.11.2 ®¬®£à ¬¬  ¯®«­ëå ᮯà®â¨¢«¥­¨© . . . . . . . . Œ­®£®¯à®¢®¤­ë¥ «¨­¨¨ TEM . . . . . . . . . . . . . . . . ‹¨­¨ï á ¬ «ë¬¨ ¯®â¥àﬨ . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

2 ’¥®à¨ï ¤«¨­­ëå «¨­¨© 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

2.9 2.10 2.11 2.12 2.13

3

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

7

7 9 9 11 13 14 20 25

27

27 30 34 35 36 36 39 41 42 43 43 44 45 45 45

. 47 . 48 . . . .

48 48 50 55

3 ‚®«­®¢®¤ë

3.1 ‚ëà ¦¥­¨¥ ¢¥ªâ®à®¢ ¯®«ï ç¥à¥§ ¯®â¥­æ¨ «ì­ë¥ ä㭪樨. E- ¨ H-¬®¤ë . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 àאַ㣮«ì­ë¥ ¢®«­®¢®¤ë . . . . . . . . . . . . . . . . . . . 3.2.1 ‚®«­ë H-⨯  . . . . . . . . . . . . . . . . . . . . . . 3.2.2 ‚®«­ë E-⨯  . . . . . . . . . . . . . . . . . . . . . . 3.3 Šàã£«ë¥ ¢®«­®¢®¤ë . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Œ £­¨â­ë¥ ¬®¤ë . . . . . . . . . . . . . . . . . . . . 3.3.2 «¥ªâà¨ç¥áª¨¥ ¬®¤ë . . . . . . . . . . . . . . . . . . 3.3.3 ‚®«­®¢®¤­ë¥ ¬®¤ë ¢ ª® ªá¨ «ì­®© «¨­¨¨ . . . . . . 3.4 ‚®«­®¢®¤ë á á¥ç¥­¨¥¬ á«®¦­®© ä®à¬ë ¨ ª¢ §¨áâ â¨ç¥áª¨¥ ¢®«­®¢®¤ë . . . . . . . . . . . . . . . . . . 3.5 ­¥à£¥â¨ç¥áª¨¥ ᮮ⭮襭¨ï ¤«ï ¢®«­®¢®¤®¢ . . . . . . . . 3.6 ” §®¢ ï ¨ £à㯯®¢ ï ᪮à®á⨠. . . . . . . . . . . . . . . . . 3.7 ‚®«­ë à¨««îí­  . . . . . . . . . . . . . . . . . . . . . . . . 3.8 ‡ âãå ­¨¥ ¢ ¢®«­®¢®¤ å . . . . . . . . . . . . . . . . . . . . . 3.8.1 àאַ㣮«ì­ë© ¢®«­®¢®¤. H10 {¬®¤  . . . . . . . . . 3.8.2 Šàã£«ë© ¢®«­®¢®¤. Œ®¤  H01 . . . . . . . . . . . . .

4 ’¥®à¨ï 楯¥© ¢ à áç¥â å ¢®«­®¢®¤®¢

57 57 68 68 74 78 78 83 86 88 92 94 97 98 104 105

107

4.1 ‚®«­®¢®¤­ ï «¨­¨ï ¯¥à¥¤ ç¨ . . . . . . . . . . . . . . . . . 107 4.2 ’¥«¥£à ä­ë¥ ãà ¢­¥­¨ï ¤«ï H-¢®«­ ¢ ¢®«­®¢®¤¥ . . . . . . 109 4.3 Žª®­¥ç­®¥ ãáâனá⢮ (¤¢ã寮«îá­¨ª) . . . . . . . . . . . . 113 4.3.1 ®«­®¥ ᮯà®â¨¢«¥­¨¥ ¨ ¯à®¢®¤¨¬®áâì . . . . . . . . 114 4.3.2 ‚®«­ë ¢ ®ª®­¥ç­®¬ ãáâனá⢥ . . . . . . . . . . . . 115 4.4 ‘®¥¤¨­¥­¨¥ ­¥áª®«ìª¨å ¢®«­®¢®¤®¢ . . . . . . . . . . . . . . 116 4.4.1 Œ âà¨æë ᮯà®â¨¢«¥­¨ï ¨ ¯à®¢®¤¨¬®á⨠. . . . . . 117 4.4.2 ‘¨¬¬¥âà¨ï ¬ âà¨æë ᮯà®â¨¢«¥­¨ï (¯à®¢®¤¨¬®áâ¨) 118 4.4.3 Œ­®£®¯®«îá­®¥ á®ç«¥­¥­¨¥ ¡¥§ ¯®â¥àì . . . . . . . . 120 4.4.4 ‚ à¨ æ¨ï ¬ âà¨æë ᮯà®â¨¢«¥­¨ï (¯à®¢®¤¨¬®áâ¨) á®ç«¥­¥­¨ï ¡¥§ ¯®â¥àì . . . . . . . . . . . . . . . . . 121 4.4.5 Ž¡®¡é¥­¨¥ ­  á«ãç © ¢®«­®¢®¤®¢, ¯® ª®â®àë¬ ¬®£ãâ à á¯à®áâà ­ïâìáï ¢®«­ë ­¥áª®«ìª¨å ¬®¤ . . 124 4.4.6 Œ âà¨æ  à áá¥ï­¨ï . . . . . . . . . . . . . . . . . . . 125 4.4.7 ‘¢®©á⢠ ¬ âà¨æë à áá¥ï­¨ï . . . . . . . . . . . . . 127 4.4.8 ­¥à£¥â¨ç¥áª¨¥ ᮮ⭮襭¨ï . . . . . . . . . . . . . 128 4.4.9 à¥®¡à §®¢ ­¨¥ ¬ âà¨æë à áá¥ï­¨ï ¯à¨ ¯¥à¥­®á¥ ®âáç¥â­ëå ¯«®áª®á⥩ . . . . . . . . . . . . . . . . . 130 4.4.10 „¢®©­®© âன­¨ª . . . . . . . . . . . . . . . . . . . . 131 4

4.5 — áâ®â­ë¥ ᢮©á⢠ . . . . . . . . . . . . . . . . . . . . . . 4.5.1 — áâ®â­ ï § ¢¨á¨¬®áâì ¬ âà¨æë ᮯà®â¨¢«¥­¨ï (¯à®¢®¤¨¬®áâ¨) á®ç«¥­¥­¨ï ¡¥§ ¯®â¥àì . . . . . . . 4.5.2 — áâ®â­ ï § ¢¨á¨¬®áâì ¬ âà¨æë à áá¥ï­¨ï . . . . 4.5.3 Žª®­¥ç­®¥ ãáâனá⢮ . . . . . . . . . . . . . . . . 4.6 ¥®¤­®à®¤­®á⨠¢ ¢®«­®¢®¤ å . . . . . . . . . . . . . . . . 4.6.1 à¥¤áâ ¢«¥­¨¥ ­¥®¤­®à®¤­®á⨠¢ ¢¨¤¥ ç¥âëà¥å¯®«îá­¨ª  . . . . . . . . . . . . . . . . . . 4.6.2 ‘ª çª®®¡à §­®¥ ¨§¬¥­¥­¨¥ ¯ à ¬¥â஢ ¢¥é¥á⢠, § ¯®«­ïî饣® ¢®«­®¢®¤ . . . . . . . . . . . . . . . 4.6.3 „¨ äà £¬ë ¢ ¢®«­®¢®¤¥ . . . . . . . . . . . . . . . 4.6.4 ‚ à¨ æ¨®­­ë¥ ¬¥â®¤ë . . . . . . . . . . . . . . . . 4.6.5 à¨¡«¨¦¥­­®¥ ¢ëç¨á«¥­¨¥ ¯à®¢®¤¨¬®á⨠. . . . . 4.6.6 ‚ à¨ æ¨®­­ë¥ ¬¥â®¤ë. Œ¥â®¤ ¨âæ  . . . . . . . 4.6.7 ‚ à¨ æ¨®­­ë¥ ¬¥â®¤ë. Œ¥â®¤ ƒ «¥àª¨­  . . . . . 4.6.8 „¨ äà £¬ë ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥ . . . . . . 4.6.9 ‘®¥¤¨­¥­¨¥ ¢®«­®¢®¤®¢ à §«¨ç­®£® á¥ç¥­¨ï . . . . 4.6.10 ‘®£« á®¢ ­¨¥ ¢®«­®¢®¤®¢ ¬­®£®áâ㯥­ç â묨 ¯¥à¥å®¤ ¬¨ . . . . . . . . . . . . . . . . . . . . . . .

. 134 . . . .

134 137 137 139

. 139 . . . . . . . .

140 142 146 148 149 152 154 156

. 165

5 ‚®§¡ã¦¤¥­¨¥ ¢®«­®¢®¤®¢ § ¤ ­­ë¬¨ ⮪ ¬¨

171

6 ¥§®­ â®àë

185

5.1 ‹¥¬¬  ‹®à¥­æ . «¥ªâà¨ç¥áª¨¥ ¨ ¬ £­¨â­ë¥ ⮪¨ . . . . 172 5.2 ‚®§¡ã¦¤¥­¨¥ ¢®«­®¢®¤®¢ í«¥ªâà¨ç¥áª¨¬¨ ¨ ¬ £­¨â­ë¬¨ ⮪ ¬¨ . . . . . . . . . . . . . . . . . . . . . . 175 5.3 „à㣮© ¢ë¢®¤ ä®à¬ã«ë ¢®§¡ã¦¤¥­¨ï í«¥ªâà¨ç¥áª¨¬¨ ¨ ¬ £­¨â­ë¬¨ ⮪ ¬¨ . . . . . . . . . . . . . . . . . . . . . . 181

6.1 ‘¢®¡®¤­ë¥ ª®«¥¡ ­¨ï ¢ १®­ â®à å . . . . . . . . . . . . . 6.2 ‘®¡á⢥­­ë¥ ä㭪樨 ¨ ᮡá⢥­­ë¥ §­ ç¥­¨ï. Žà⮣®­ «ì­®áâì ᮡá⢥­­ëå ä㭪権 . . . . . . . . . . . 6.3 ¥§®­ â®àë, ®¡à §®¢ ­­ë¥ ¨§ ®â१ª®¢ «¨­¨¨ ¯¥à¥¤ ç¨ . . 6.4 à¨¬¥àë १®­ â®à®¢, ®¡à §®¢ ­­ëå ¨§ ®â१ª®¢ «¨­¨¨ ¯¥à¥¤ ç¨ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 à¨¡«¨¦¥­­ë¥ ¬¥â®¤ë à áç¥â  ç áâ®âë ᢮¡®¤­ëå ª®«¥¡ ­¨© १®­ â®à®¢ . . . . . . . . . . . . . . . . . . . . . 6.5.1 Š¢ §¨áâ â¨ç¥áª¨© ¬¥â®¤ . . . . . . . . . . . . . . . . 6.5.2 à¨¬¥àë: â®à®¨¤ «ì­ë© १®­ â®à, १®­ â®à \饫ì { ®â¢¥àá⨥" . . . . . . . . . . . . . . . . . . . 5

185 185 190 192

196 196 197

6.5.3 Œ¥â®¤ ç áâ¨ç­ëå ®¡« á⥩ . . . . . . . . . . . 6.5.4 à¨¬¥àë . . . . . . . . . . . . . . . . . . . . . . 6.5.5 ‚ à¨ æ¨®­­ë© ¬¥â®¤. ”®à¬ã«  ¢®§¬ã饭¨© . 6.6 ®â¥à¨ ¢ १®­ â®à å . . . . . . . . . . . . . . . . . . . 6.6.1 à¨¬¥àë à áç¥â  ¤®¡à®â­®á⨠. . . . . . . . . 6.6.2 ®â¥à¨ ¢ á।¥ . . . . . . . . . . . . . . . . . . . 6.6.3 ®â¥à¨ ­  ¨§«ã祭¨¥ . . . . . . . . . . . . . . . 6.7 ‚ë­ã¦¤¥­­ë¥ ª®«¥¡ ­¨ï ¢ १®­ â®à å . . . . . . . . 6.7.1 ‚®§¡ã¦¤¥­¨¥ १®­ â®à  § ¤ ­­ë¬¨ ⮪ ¬¨ . 6.7.2 ‚®§¡ã¦¤¥­¨¥ १®­ â®à  ¯®â®ª®¬ § à殮­­ëå ç áâ¨æ . . . . . . . . . . . . . . . . . . . . . . . 6.7.3 „à㣨¥ á¯®á®¡ë ¢®§¡ã¦¤¥­¨ï १®­ â®à®¢ . . 6.8 ¥§®­ â®à ª ª í«¥¬¥­â «¨­¨¨ ¯¥à¥¤ ç¨ . . . . . . . . 6.8.1 ‚室­®¥ ᮯà®â¨¢«¥­¨¥ १®­ â®à  . . . . . . . 6.8.2 à¨¬¥­¥­¨¥ १®­ â®à®¢ . . . . . . . . . . . . .

7 «¥ªâ஬ £­¨â­ë¥ ¢®«­ë ¢ £¨à®¬ £­¨â­®© á।¥

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

199 200 204 208 213 216 217 218 218

. . . . .

. . . . .

. . . . .

223 228 229 230 237

7.1 ‘¢®©á⢠ ä¥àà¨â®¢ . . . . . . . . . . . . . . . . . . . . . . . 7.2  á¯à®áâà ­¥­¨¥ ¯«®áª¨å ¢®«­ ¢ ­ ¬ £­¨ç¥­­®¬ ä¥àà¨â¥ 7.2.1 à®¤®«ì­® ­ ¬ £­¨ç¥­­ë© ä¥àà¨â . . . . . . . . . . 7.2.2 ®¯¥à¥ç­® ­ ¬ £­¨ç¥­­ë© ä¥àà¨â . . . . . . . . . . 7.3  á¯à®áâà ­¥­¨¥ ¢®«­ ¢ ¢®«­®¢®¤¥, ç áâ¨ç­® § ¯®«­¥­­®¬ ä¥àà¨â®¬ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 ‚¥­â¨«¨ ¨ æ¨àªã«ïâ®àë . . . . . . . . . . . . . . . . . . . . 7.4.1 ”¥àà¨â ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥ . . . . . . . . . 7.4.2 Š® ªá¨ «ì­ë© ¢¥­â¨«ì . . . . . . . . . . . . . . . . . 7.4.3 –¨àªã«ïâ®àë . . . . . . . . . . . . . . . . . . . . . .

6

239

239 246 246 250

252 259 260 262 262

1.

Ž™ˆ… ‘‚…„…ˆŸ ˆ‡ ’…Žˆˆ ‹…Š’ŽŒ€ƒˆ’ŽƒŽ Ž‹Ÿ

1.1.

“à ¢­¥­¨ï Œ ªá¢¥«« 

«¥ªâ஬ £­¨â­®¥ ¯®«¥ ¢ «î¡®© á।¥ ®¯¨á뢠¥âáï ç¥âëàì¬ï ¢¥ªâ®à ¬¨ ~ H; ~ B~ , ¨§ ª®â®àëå ¯¥à¢ë¥ ¤¢  å à ªâ¥à¨§ãîâ í«¥ªâà¨ç¥áª®¥ ¯®«¥, E~ ; D;   ¢â®àë¥ { ¬ £­¨â­®¥. ‚ ãà ¢­¥­¨ïå ¯®«ï ᮤ¥à¦ âáï â ª¦¥ ¢¥«¨ç¨­ë:  { ¯«®â­®áâì § à鸞 ¨ ¢¥ªâ®à­ ï ¢¥«¨ç¨­  J~ { ¯«®â­®áâì ⮪ . “à ¢­¥­¨ï Œ ªá¢¥«« , á¢ï§ë¢ î騥 ¬¥¦¤ã ᮡ®© ¢¥ªâ®àë ¯®«ï, ®¯à¥¤¥«ïîâ í«¥ªâ஬ £­¨â­®¥ ¯®«¥:

~

~

~ rot E~ = ; @@tB ; rot H~ = @@tD + J: «®â­®áâì § à鸞 ¨ ¯«®â­®áâì ⮪  á¢ï§ ­ë ãà ¢­¥­¨¥¬ ­¥¯à¥à뢭®áâ¨: @ + div J~ = 0; @t

ª®â®à®¥ ¬®¦¥â ¡ëâì ¯®«ã祭® ¨§ § ª®­  á®åà ­¥­¨ï § à鸞. Š ª á«¥¤á⢨¥ ¨§ ãà ¢­¥­¨© Œ ªá¢¥««  ¨ ãà ¢­¥­¨ï ­¥¯à¥à뢭®á⨠¬®£ãâ ¡ëâì ¯®«ãç¥­ë ¥é¥ ¤¢  ãà ¢­¥­¨ï, ª®â®àë¥ ¨­®£¤  ®â­®áïâ ª á¨á⥬¥ ãà ¢­¥­¨© Œ ªá¢¥«« : div B~ = 0; div D~ = : à¨¢¥¤¥­­ëå ãà ¢­¥­¨© ­¥¤®áâ â®ç­® ¤«ï ®¯à¥¤¥«¥­¨ï ¢¥ªâ®à®¢ ¯®«ï, ¥á«¨ § ¤ ­ë ¨áâ®ç­¨ª¨ ¯®«ï  ¨ J~. „ ­­ ï á¨á⥬  ¤®«¦­  ¡ëâì ¤®¯®«­¥­  â ª ­ §ë¢ ¥¬ë¬¨ ¬ â¥à¨ «ì­ë¬¨ ãà ¢­¥­¨ï¬¨, ®¯¨á뢠î騬¨ ¢«¨ï­¨¥ á।ë (¤«ï «¨­¥©­®© á।ë):

~ B~ = H; ~ J~¯à = E: ~ D~ = "E; ‚¥«¨ç¨­ë ",  ¨  ¬®£ãâ ¡ëâì ⥭§®à­ë¬¨,   ¨­®£¤  ¤¨ää¥à¥­æ¨ «ì­ë¬¨ ¨«¨ ¨­â¥£à «ì­ë¬¨ ®¯¥à â®à ¬¨. ‚ ¤ «ì­¥©è¥¬ í⨠¢¥«¨ç¨­ë ¡ã¤ãâ à áᬠâਢ âìáï ª ª ᪠«ïà­ë¥, ªà®¬¥ § à ­¥¥ ®£®¢®à¥­­ëå á«ãç ¥¢. ‘ ãç¥â®¬ ¬ â¥à¨ «ì­ëå ãà ¢­¥­¨© á¨á⥬  ãà ¢­¥­¨© Œ ªá¢¥««  áâ ­®¢¨âáï ¯®«­®©. 7

‚ ¤ «ì­¥©è¥¬ ­ á ¡ã¤ãâ ¨­â¥à¥á®¢ âì ¯à¥¦¤¥ ¢á¥£® £ à¬®­¨ç¥áª¨¥ ¯à®æ¥ááë, ª®£¤  § ¢¨á¨¬®áâì ¯®«¥© ®â ¢à¥¬¥­¨ ¨¬¥¥â ¢¨¤ E~ = E~ (r)
e j!t ; £¤¥ E~ (r) { ª®¬¯«¥ªá­ ï  ¬¯«¨â㤠, ¢ ®¡é¥¬ á«ãç ¥ § ¢¨áïé ï ®â ª®®à¤¨­ â. “à ¢­¥­¨ï Œ ªá¢¥««  «¨­¥©­ë, ¥á«¨ «¨­¥©­ë ¬ â¥à¨ «ì­ë¥ ãà ¢­¥­¨ï. …᫨ § ¢¨á¨¬®áâì ®â ¢à¥¬¥­¨ ¯à¥¤áâ ¢«¥­  ¬­®¦¨â¥«¥¬ e j!t , â® ãà ¢­¥­¨ï Œ ªá¢¥««  ¬®£ãâ ¡ëâì § ¯¨á ­ë ¢ ª®¬¯«¥ªá­®© ä®à¬¥: ~ rot E~ = ; j!B~ ; rot H~ = j!D~ + J; £¤¥ ¢á¥ ¢¥ªâ®àë ¯®«ï ¯à¥¤áâ ¢«ïîâ ᮡ®© ª®¬¯«¥ªá­ë¥  ¬¯«¨âã¤ë. ˆá¯®«ì§ãï ¬ â¥à¨ «ì­ë¥ ãà ¢­¥­¨ï, í⨠ãà ¢­¥­¨ï ¬®¦­® § ¯¨á âì ¢ ¢¨¤¥ ~ rot H~ = j!"E~ + J: ~ rot E~ = ; j!H; ’ ª ª ª á¢ï§ì ¬¥¦¤ã D~ ¨ E~ , B~ ¨ H~ ¬®¦¥â ®¯¨á뢠âìáï ¤¨ää¥à¥­æ¨ «ì­ë¬¨ ãà ¢­¥­¨ï¬¨ (¢à¥¬¥­­묨), â® ¢ ®¡é¥¬ á«ãç ¥ " ¨  á«¥¤ã¥â áç¨â âì ª®¬¯«¥ªá­ë¬¨ ç¨á« ¬¨ (" = "0 ; j"00 ;  = 0 ; j00 ). ’®ª J~ ᪫ ¤ë¢ ¥âáï ¨§ ⮪  ¯à®¢®¤¨¬®á⨠¨ áâ®à®­­¥£® ⮪ , ¢ë§¢ ­­®£® ¨áâ®ç­¨ª ¬¨: J~ = J~¯à + J~áâ®à = E~ + J~áâ®à : ®¤áâ ¢«ïï J~ ¢ ãà ¢­¥­¨ï Œ ªá¢¥«« , ¯®«ã稬 ~ rot H~ = j!"E~ + E~ + J~áâ®à: rot E~ = ; j!H; ®á«¥¤­¥¥ ãà ¢­¥­¨¥ ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ á«¥¤ãî饬 ¢¨¤¥: rot H~ = j!"k E~ + J~áâ®à; £¤¥ "k = " ; j ! : ‘«¥¤ã¥â ¨¬¥âì ¢ ¢¨¤ã, çâ® ¢ ⥮ਨ £ à¬®­¨ç¥áª¨å ¯®«¥© ãà ¢­¥­¨ï, ¢ ª®â®àë¥ ¢å®¤¨â ¯«®â­®áâì § à鸞, ­¥ ¨¬¥îâ á ¬®áâ®ï⥫쭮£® §­ ç¥­¨ï, â ª ª ª ¯«®â­®áâì § à鸞 ®¤­®§­ ç­® ®¯à¥¤¥«ï¥âáï ¯«®â­®áâìî ⮪ : 1 div J~ :  = ; j! 8

1.2.

‘¨á⥬  ¥¤¨­¨æ

à¨¢¥¤¥­­ ï ¢ëè¥ á¨á⥬  ãà ¢­¥­¨© Œ ªá¢¥««  § ¯¨á ­  ¢ á¨á⥬¥ ¥¤¨­¨æ ‘ˆ, ª®â®à ï ¡ë«  ¯à¨­ïâ  ¢ 1960 £. ­  II ƒ¥­¥à «ì­®© ª®­ä¥à¥­æ¨¨ ¯® ¬¥à ¬ ¨ ¢¥á ¬. ‚ ‘‘‘ á¨á⥬  ‘ˆ ¢¢¥¤¥­  ¢ 1963 £. ‚ á¨á⥬¥ ‘ˆ ¢ ª ç¥á⢥ ®á­®¢­ëå ¥¤¨­¨æ ¯à¨­ïâë á«¥¤ãî騥: ¬¥âà { ¤«¨­ , ª¨«®£à ¬¬ { ¬ áá , ᥪ㭤  { ¢à¥¬ï; ¢ ª ç¥á⢥ ¥¤¨­¨æë ᨫë ⮪  ¯à¨­ïâ  ¬¯¥à. ‚ á¢ï§¨ á í⨬ ¢¥«¨ç¨­ë " ¨  ¨¬¥îâ à §¬¥à­®áâì €2
á4 = ” ; [] = ª£
¬ = ƒ­ : ["] = ª£
¬3 ¬ €2
á2 ¬ “¤¥«ì­ ï ¯à®¢®¤¨¬®áâì ¢ í⮩ á¨á⥬¥ ¨¬¥¥â à §¬¥à­®áâì €2
á3 = ‘¨¬ : [] = ª£
¬3 ¬ ‘«¥¤ã¥â ®â¬¥â¨âì, çâ® ¤«ï ¢ ªã㬠 " ¨  ¨¬¥îâ ®¯à¥¤¥«¥­­ë¥ §­ ç¥­¨ï,   ¨¬¥­­®: 10;9  0:884
10;11 ” : 0 = 4
10;7 ƒ­ ; " = 0 ¬ 36 ¬ ‚¥«¨ç¨­  r  = " 0 = 120 Ž¬  377 Ž¬ 0 ¨¬¥¥â à §¬¥à­®áâì ᮯà®â¨¢«¥­¨ï. …¥ ­ §ë¢ îâ ¢®«­®¢ë¬ ᮯà®â¨¢«¥­¨¥¬ ¢ ªã㬠; 䨧¨ç¥áª¨© á¬ëá« í⮣® ­ §¢ ­¨ï ¡ã¤¥â ïᥭ ¨§ ¤ «ì­¥©è¥£®. „à㣨¥ ¥¤¨­¨æë ¢ á¨á⥬¥ ‘ˆ (¯à®¨§¢®¤­ë¥) á«¥¤ãî騥: à §­®áâì ¯®â¥­æ¨ «®¢ { ¢®«ìâ, § àï¤ { ªã«®­, ᮯà®â¨¢«¥­¨¥ { ®¬, ᨫ  { ­ìîâ®­, í­¥à£¨ï { ¤¦®ã«ì, ¬®é­®áâì { ¢ ââ. …¤¨­¨æë ¬ £­¨â­®£® ¯®«ï: ¯®â®ª { ¢¥¡¥à, à ¢­ë© 108 ¬ ªá¢¥««, ¨­¤ãªæ¨ï { â¥á« , à ¢­ë© 104 £ ãáá.  ¯à殮­­®áâì í«¥ªâà¨ç¥áª®£® ¯®«ï { ‚/¬, ¬ £­¨â­®£® ¯®«ï { €/¬ (0:4
10;2 íàá⥤). 1.3.

ƒà ­¨ç­ë¥ ãá«®¢¨ï

ƒà ­¨ç­ë¥ ãá«®¢¨ï, â.¥. ãá«®¢¨ï ¤«ï ¢¥ªâ®à®¢ ¯®«ï ­  £à ­¨æ¥ ¤¢ãå á। á à §«¨ç­ë¬¨ å à ªâ¥à¨á⨪ ¬¨, ¨¬¥îâ á«¥¤ãî騩 ¢¨¤: 9

1: (B~ 2 ; B~ 1 )
~n = 0; £¤¥ B~ 1 ¨ B~ 2 { ¢¥ªâ®àë ¨­¤ãªæ¨¨ ¬ £­¨â­®£® ¯®«ï ¢ á। å 1 ¨ 2, ~n { ¥¤¨­¨ç­ë© ¢¥ªâ®à ­®à¬ «¨ ª £à ­¨æ¥ á।. â® ãá«®¢¨¥ ¬®¦­® áä®à¬ã«¨à®¢ âì á«¥¤ãî騬 ®¡à §®¬: ­®à¬ «ì­ ï ª®¬¯®­¥­â  B~ ­¥¯à¥à뢭  ¯à¨ ¯¥à¥å®¤¥ ç¥à¥§ £à ­¨æã. 2: (D~ 2 ; D~ 1 )
~n = ¯®¢ ; £¤¥ ¯®¢ { ¯®¢¥àå­®áâ­ ï ¯«®â­®áâì § à鸞, â.¥. § àï¤, ®â­¥á¥­­ë© ª ¥¤¨­¨æ¥ ¯«®é ¤¨ ¯®¢¥àå­®áâ¨, ¯à¨ç¥¬ ¢¥ªâ®à ­®à¬ «¨ ­ ¯à ¢«¥­ ¨§ á।ë 1 ¢ á।ã 2. …᫨ ­  ¯®¢¥àå­®á⨠­¥â § à鸞, â® ­®à¬ «ì­ ï ª®¬¯®­¥­â  D~ ­¥¯à¥à뢭 . ®¢¥¤¥­¨¥ â ­£¥­æ¨ «ì­ëå ª®¬¯®­¥­â ®¯à¥¤¥«ï¥âáï ãá«®¢¨ï¬¨: 3: ~n  (E~ 2 ; E~ 1 ) = 0; â.¥. â ­£¥­æ¨ «ì­ ï ª®¬¯®­¥­â  E~ ­¥¯à¥à뢭  ¯à¨ ¯¥à¥å®¤¥ ç¥à¥§ £à ­¨æã.

~ 4: ~n  (H~ 2 ; H~ 1 ) = K; £¤¥ K~ { ¯®¢¥àå­®áâ­ ï ¯«®â­®áâì ⮪ , ~n { ­®à¬ «ì ¨§ á।ë 1 ¢ á।ã 2. ‚¥«¨ç¨­  K~ ®¯à¥¤¥«ï¥âáï á«¥¤ãî騬 ®¡à §®¬. ãáâì ⮪¨ ¯à®¢®¤¨¬®á⨠á®á।®â®ç¥­ë ¢ â®­ª®¬ á«®¥ ⮫騭®©
l ®ª®«® ¯®¢¥àå­®á⨠ࠧà뢠. ’®£¤  ⮪ ­  ¥¤¨­¨æã ¤«¨­ë «¨­¨¨, ®à⮣®­ «ì­®© «¨­¨ï¬ ¢¥ªâ®à  K~ , à ¢¥­ J~

l. ¥à¥©¤¥¬ ª ¯à¥¤¥«ã
l ! 0. …᫨ J~ ! 1, â® ¯à®¨§¢¥¤¥­¨¥ J~

l ¬®¦¥â áâ६¨âìáï ª ª®­¥ç­®¬ã ¯à¥¤¥«ã

K~ =
lim J~

l: l!0

Žç¥¢¨¤­®, çâ® ¤«ï í⮣® ¯à®¢®¤¨¬®áâì ®¤­®© ¨§ á। ¤®«¦­  ¡ëâì ¡¥áª®­¥ç­® ¡®«ì让. ‚ á«ãç ¥ ª®­¥ç­®© ¯à®¢®¤¨¬®á⨠®¡¥¨å á।

~n  (H~ 2 ; H~ 1 ) = 0; â.¥. â ­£¥­æ¨ «ì­ ï ª®¬¯®­¥­â  H~ ­¥¯à¥à뢭 . 10

1.4.

­¥à£¨ï í«¥ªâ஬ £­¨â­®£® ¯®«ï

Š ª ¨§¢¥áâ­®, í­¥à£¨ï í«¥ªâ஬ £­¨â­®£® ¯®«ï ᪫ ¤ë¢ ¥âáï ¨§ í­¥à£¨¨ í«¥ªâà¨ç¥áª®£® ¯®«ï ¨ í­¥à£¨¨ ¬ £­¨â­®£® ¯®«ï Z  "E 2 H 2  W = 2 + 2 dV ¯à¨ ãá«®¢¨¨, çâ®  ¨ " { ¢¥é¥á⢥­­ë¥ ¢¥«¨ç¨­ë, ­¥ § ¢¨áï騥 ®â ç áâ®âë. ˆ§ ãà ¢­¥­¨© Œ ªá¢¥««  ¬®¦­® ¯®«ãç¨âì ⥮६㠏®©­â¨­£ :   @ Z "E 2 + H 2 dV + Z E~
J~ dV = ; I (E~  H~ )
~n dS; @t 2 2 V

V

S

£¤¥ V { ­¥ª®â®àë© ®¡ê¥¬ í«¥ªâ஬ £­¨â­®£® ¯®«ï, ~n { ¥¤¨­¨ç­ë© ¢¥ªâ®à ­®à¬ «¨, ­ ¯à ¢«¥­­ë© ­ àã¦ã § ¬ª­ã⮩ ¯®¢¥àå­®á⨠S , ®£à ­¨ç¨¢ î饩 ®¡ê¥¬. ‚¥ªâ®à S~ = E~  H~ ­ §ë¢ îâ ¢¥ªâ®à®¬ ®©­â¨­£ . …£® á¬ëá« á«¥¤ã¥â ¨§ â¥®à¥¬ë ®©­â¨­£ : íâ® ¯«®â­®áâì ¯®â®ª  í­¥à£¨¨ ç¥à¥§ ¯®¢¥àå­®áâì. €­ «®£¨ç­®¥ ᮮ⭮襭¨¥ ¬®¦¥â ¡ëâì ¯®«ã祭® ¤«ï ª®¬¯«¥ªá­ëå  ¬¯«¨âã¤. „«ï í⮣® ¡ã¤¥¬ ¨á室¨âì ¨§ ãà ¢­¥­¨© Œ ªá¢¥««  ¢ ª®¬¯«¥ªá­®© ä®à¬¥

~ rot H~ = j!"E~ + J: ~ rot E~ = ; j!H; „«ï ¢ë¢®¤  ¨á¯®«ì§ã¥¬ ¨§¢¥áâ­®¥ ¢¥ªâ®à­®¥ ⮦¤¥á⢮ div (E~  H~  ) = H~ 
rot E~ ; E~
rot H~  ; £¤¥ H~  { ª®¬¯«¥ªá­®-ᮯà殮­­ ï  ¬¯«¨â㤠. ®¤áâ ¢«ïï á rot E~ ¨ rot H~  ¨§ ãà ¢­¥­¨© Œ ªá¢¥«« , ¯®«ã稬 div (E~  H~  ) = H~ 
(;j!H~ ) ; E~
(;j!"E~  + J~ ) = = j!"jE~ j2 ; j!jH~ j2 ; E~
J~

¢ ¯à¥¤¯®«®¦¥­¨¨, çâ® " ¨  { ¢¥é¥á⢥­­ë¥ ¢¥«¨ç¨­ë. à®¨­â¥£à¨à㥬 ¯®«ã祭­®¥ à ¢¥­á⢮ ¯® ®¡ê¥¬ã V , ®£à ­¨ç¥­­®¬ã ¯®¢¥àå­®áâìî S : 11

I S

(E~  H~  )
~n dS = 2j!

Z jH~ j2 "jE~ j2 ! 2 ; 2 dV

V

+

Z V

E~
J~ dV;

£¤¥ ~n { ¥¤¨­¨ç­ë© ¢¥ªâ®à ¢­ãâ७­¥© ­®à¬ «¨. à¥¤¯®«®¦¨¬, çâ® ¢­ãâਠ®¡ê¥¬  V ®âáãâáâ¢ãîâ áâ®à®­­¨¥ ⮪¨. ’®£¤  J~ = E~ . ®¤áâ ¢«ïï ¢ ¯à¨¢¥¤¥­­®¥ ¢ëè¥ à ¢¥­á⢮, ¯®á«¥ ¤¥«¥­¨ï ­  2 ¯®«ã稬 ! 1 I (E~  H~  )
~n dS = 1 Z jE~ j2 dV + j!Z jH~ j2 ; "jE~ j2 dV: 2 2 2 2 S

V

V

‚ëïá­¨¬ á¬ëá« á« £ ¥¬ëå ¢ ¯à ¢®© ç áâ¨. Žç¥¢¨¤­®, çâ® ¨­â¥£à « Z P = 12 jE~ j2 dV V

¯à¥¤áâ ¢«ï¥â ᮡ®© á।­îî ¬®é­®áâì, à áᥨ¢ ¥¬ãî ¢­ãâਠ®¡ê¥¬  V ¢á«¥¤á⢨¥ ­ «¨ç¨ï ⮪®¢ ¯à®¢®¤¨¬®áâ¨. ‚â®à®© ¨­â¥£à « ¢ëà ¦ ¥âáï ç¥à¥§ á।­¨¥ §­ ç¥­¨ï § ¯ á¥­­®© í­¥à£¨¨ í«¥ªâà¨ç¥áª®£® ¨ ¬ £­¨â­®£® ¯®«¥© Z jH~ j2 Z "jE~ j2 dV = 2 W ; H 2 2 dV = 2WE ; V

V

£¤¥ WH ¨ WE { á।­¨¥ §­ ç¥­¨ï § ¯ á®¢ ¬ £­¨â­®© ¨ í«¥ªâà¨ç¥áª®© í­¥à£¨¨. ˆâ ª, 1 I (E~  H~  )
~n dS = P + 2j! (W ; W ): H E 2 ‚¥ªâ®à S~k = 12 (E~  H~  ) ­ §ë¢ îâ ª®¬¯«¥ªá­ë¬ ¢¥ªâ®à®¬ ®©­â¨­£ . …£® ¢¥é¥á⢥­­ ï ç áâì ¯à¥¤áâ ¢«ï¥â ᮡ®© á।­îî ¯«®â­®áâì ¯®â®ª  í­¥à£¨¨ ç¥à¥§ ¯®¢¥àå­®áâì; ¬­¨¬ãî ç áâì í⮣® ¢¥ªâ®à  ­ §ë¢ îâ ¯«®â­®áâìî ॠªâ¨¢­®© ¬®é­®áâ¨.

12

1.5.

’¥®à¥¬  ¥¤¨­á⢥­­®áâ¨

‡ ¤ ç¨, à¥è ¥¬ë¥ ¢ í«¥ªâத¨­ ¬¨ª¥ ‘‚—, ¤¥«ïâáï ­  ¤¢  ®á­®¢­ëå ª« áá : ¢­ãâ७­¨¥ ¨ ¢­¥è­¨¥. ‚® ¢­ãâ७­¨å § ¤ ç å à áᬠâਢ ¥âáï ¯®«¥ ¢ ­¥ª®â®à®© ®¡« á⨠¯à®áâà ­á⢠, ®£à ­¨ç¥­­®© § ¤ ­­®© ¯®¢¥àå­®áâìî S . ‚­ãâਠS § ¤ ­ë áâ®à®­­¨¥ ⮪¨,   ­  á ¬®© ¯®¢¥àå­®á⨠{ «¨¡® â ­£¥­æ¨ «ì­ ï ª®¬¯®­¥­â  í«¥ªâà¨ç¥áª®£® ¯®«ï Et (­ ¯à¨¬¥à, ­  S1 ), «¨¡® â ­£¥­æ¨ «ì­ ï ª®¬¯®­¥­â  ¬ £­¨â­®£® ¯®«ï Ht (­  S2 ). „®ª ¦¥¬ ⥮६㠥¤¨­á⢥­­®á⨠¤«ï ¢­ãâ७­¥© § ¤ ç¨. à¥¤¯®«®¦¨¬ ®¡à â­®¥: ¯ãáâì ¯à¨ ®¤¨­ ª®¢ëå ⮪ å ¨ ®¤¨­ ª®¢ëå §­ ç¥­¨ïå â ­£¥­æ¨ «ì­ëå ª®¬¯®­¥­â ¯®«¥© ­  ¯®¢¥àå­®á⨠S E~ 1 ; H~ 1 { ®¤­® à¥è¥­¨¥,   E~ 2 ; H~ 2 { ¤à㣮¥. ’®£¤  E~ = E~ 2 ; E~ 1 ¨ H~ = H~ 2 ; H~ 1 㤮¢«¥â¢®àïîâ ®¤­®à®¤­®© á¨á⥬¥ ãà ¢­¥­¨© Œ ªá¢¥«« .  ¯®¬­¨¬, çâ® ­  S1 § ¤ ­® Et ,   ­  S2 { Ht . ‘«¥¤®¢ â¥«ì­®, E~ 1 ¨ E~ 2 ¨¬¥îâ ᮢ¯ ¤ î騥 â ­£¥­æ¨ «ì­ë¥ ª®¬¯®­¥­âë ­  S1 ,   H~ 1 ¨ H~ 2 { ­  S2 . à¨ íâ¨å ãá«®¢¨ïå ­  S1 à ¢­  ­ã«î â ­£¥­æ¨ «ì­ ï ª®¬¯®­¥­â  E~ ,   ­  S2 { â ­£¥­æ¨ «ì­ ï ª®¬¯®­¥­â  H~ . ®í⮬㠤«ï ®¡ê¥¬  V ¨¬¥¥â ¬¥áâ® à ¢¥­á⢮

P + 2j! (WH ; WE ) = 0; â ª ª ª ­  ¢á¥© ¯®¢¥àå­®á⨠S «¨¡® Et = 0, «¨¡® Ht = 0. ˆ§ ¯®«ã祭­®£® à ¢¥­á⢠ á«¥¤ã¥â, çâ®

P = 0; WH ; WE = 0: ¥à¢®¥ à ¢¥­á⢮ ¤ ¥â

Z V

jE~ 2 ; E~ 1 j2 dV = 0:

’ ª ª ª ¯®¤ë­â¥£à «ì­®¥ ¢ëà ¦¥­¨¥ ­¥®âà¨æ â¥«ì­®, ¨§ í⮣® à ¢¥­á⢠ á«¥¤ã¥â, çâ® ¢ «î¡®© â®çª¥ ¢­ãâਠV 
jE~ 2 ; E~ 1 j2 = 0: …᫨  6= 0, â® ¨§ í⮣® á«¥¤ã¥â

E~ 2 ; E~ 1 = 0: „ «¥¥ 13

WH ; WE = 0: ® â ª ª ª WE = 0, â® ¨ WH = 0, ®âªã¤  á«¥¤ã¥â, çâ® H~ 2 ; H~ 1 = 0: ’¥®à¥¬  ¥¤¨­á⢥­­®á⨠á¯à ¢¥¤«¨¢  â ª¦¥ ¢ ¯à¥¤¯®«®¦¥­¨¨, çâ® ¯®â¥à¨ ®â«¨ç­ë ®â ­ã«ï «¨èì ¢ ­¥ª®â®à®© ç á⨠®¡ê¥¬  V . 1.6.

¥è¥­¨ï ãà ¢­¥­¨© Œ ªá¢¥«« 

“à ¢­¥­¨ï Œ ªá¢¥««  rot E~ = ; j!H~ ; rot H~ = j!"E~ + J~

à¥è îâáï ¯ã⥬ ¨áª«î祭¨ï E~ ¨«¨ H~ .  ¯à¨¬¥à, ¨§ ¯¥à¢®£® ãà ¢­¥­¨ï ­ ©¤¥¬ H~ : H~ = ; 1 rot E~

j! ¨ ¯®¤áâ ¢¨¬ ¢® ¢â®à®¥ ãà ¢­¥­¨¥ (¯à¨  = const): ;rot rot E~ = ; !2 "E~ + j!J:~ Ž¡ëç­® ®¡®§­ ç îâ

k2 = !2 " : „«ï á®áâ ¢«ïîé¨å E~ ¢ ¤¥ª à⮢ëå ª®®à¤¨­ â å ¨¬¥¥â ¬¥á⮠⮦¤¥á⢮

~ rot rot E~ = grad div E~ ;
E; £¤¥
E~ { ¢¥ªâ®à, ¤¥ª à⮢ë á®áâ ¢«ïî騥 ª®â®à®£® à ¢­ë
Ex ;
Ey ;
Ez . ®¤áâ ¢«ïï ¨ ãç¨â뢠ï, çâ® ¯à¨ " = const; div E~ = " , ¯®«ã稬

~
E~ + k2 E~ = grad " + j!J: 14

ˆ§ ãà ¢­¥­¨ï ­¥¯à¥à뢭®áâ¨

j! + div J~ = 0 á«¥¤ã¥â 1 div J: ~  = ; j! ®¤áâ ¢«ïï ¢ ¯à¨¢¥¤¥­­®¥ ¢ëè¥ ãà ¢­¥­¨¥ ¤«ï E~ , ¯®«ã稬 ãà ¢­¥­¨¥ ~
E~ + k2 E~ = ; 1 (grad div + k2 )J:

j!

¥è¥­¨¥ ãà ¢­¥­¨ï á ¯à ¢®© ç áâìî § âà㤭¨â¥«ì­®, ¯®í⮬ã â ª¨¬ ãà ¢­¥­¨¥¬ ¯®«ì§ãîâáï «¨èì ¤«ï ®¡« á⥩, £¤¥ J~ = 0. ‚ í⮩ ®¡« á⨠ãà ¢­¥­¨¥ ¯à¨®¡à¥â ¥â ¢¨¤
E~ + k2 E~ = 0: „ ­­®¥ ãà ¢­¥­¨¥ ­ §ë¢ îâ ãà ¢­¥­¨¥¬ ƒ¥«ì¬£®«ìæ . ®á«¥ ⮣® ª ª ­ ©¤¥­® ¯®«¥ E~ , ¬ £­¨â­®¥ ¯®«¥ H~ ¬®¦¥â ¡ëâì ­ ©¤¥­® á ¯®¬®éìî ãà ¢­¥­¨© Œ ªá¢¥«« . €­ «®£¨ç­®, ­ å®¤ï E~ ¨§ ¢â®à®£® ãà ¢­¥­¨ï Œ ªá¢¥««  ¨ ¯®¤áâ ¢«ïï ¢ ¯¥à¢®¥, ¯®«ã稬 (¯à¨ " = const) E~ = 1 rot H~ ; 1 J~

j!"

¨

j!"

~ rot rot H~ = !2 "H~ + rot J:

“ç¨â뢠ï, çâ® rot rot H~ = grad div H~ ;
H~ = ;
H~ (div H~ = 0); ¯®«ã稬

~
H~ + k2 H~ = ; rot J:

‚ ®¡« áâ¨, £¤¥ J~  0, ãà ¢­¥­¨¥ ¨¬¥¥â â®â ¦¥ ¢¨¤, çâ® ¨ ãà ¢­¥­¨¥ ¤«ï E~ : 15


H~ + k2 H~ = 0: ˆ­®£¤  ¤«ï à¥è¥­¨ï ãà ¢­¥­¨© Œ ªá¢¥««  ¨á¯®«ì§ãî⠢ᯮ¬®£ â¥«ì­ë¥ ä㭪樨 { ¢¥ªâ®à­ë© ¨ ᪠«ïà­ë© ¯®â¥­æ¨ «ë.  ¯à¨¬¥à, ¢¢¥¤¥¬ ¢¥ªâ®à A~ â ª¨¬ ®¡à §®¬, ç⮡ë ~ B~ = rot A~ ¨«¨ H~ = 1 rot A:



®¤áâ ¢¨¬ íâ® ¢ ¯¥à¢®¥ ãà ¢­¥­¨¥ Œ ªá¢¥«« : rot E~ = ; j! rot A~ ¨«¨ rot (E~ + j!
A~ ) = 0; ®âªã¤ 

E~ + j!
A~ = ; grad '; £¤¥ ' { ᪠«ïà­ë© ¯®â¥­æ¨ «. ˆ§ í⮣® à ¢¥­á⢠ ¨¬¥¥¬

~ E~ = ; grad ' ; j!A: „«ï ­ å®¦¤¥­¨ï A~ ¯®¤áâ ¢¨¬ H~ ¨ E~ ¢® ¢â®à®¥ ãà ¢­¥­¨¥: ~ rot rot A~ = ; j!"
grad ' + !2 "A~ + J: ’ ª ª ª

~ rot rot A~ = grad div A~ ;
A; â® ¨§ ¯à¥¤ë¤ã饣® ãà ¢­¥­¨ï ¯®«ãç ¥¬
A~ + k2 A~ = ; J~ + grad div A~ + j!" grad ': ‘®®â­®è¥­¨¥¬ H~ = 1 rot A~ ¢¥ªâ®à­ë© ¯®â¥­æ¨ « A~ ¥é¥ ­¥ ®¯à¥¤¥-



«¥­ ®¤­®§­ ç­®, â ª ª ª ª ­¥¬ã ¬®¦­® ¯à¨¡ ¢¨âì £à ¤¨¥­â ¯à®¨§¢®«ì­®£® ᪠«ïà . ‚ ç áâ­®áâ¨, ¬®¦­® ¢ë¡à âì A~ â ª (ª «¨¡à®¢ª ), ç⮡ë div A~ + j!"' = 0: 16

’®£¤  ãà ¢­¥­¨¥ ¤«ï A ¯à¨¬¥â ¢¨¤ ~
A~ + k2 A~ = ; J: ’ ª®¥ ã¯à®é¥­¨¥ ¨¬¥¥â ¬¥á⮠⮫쪮 ¢ ¤¥ª à⮢ëå ª®®à¤¨­ â å. Žá®¡¥­­® ã¯à®é ¥âáï à¥è¥­¨¥ § ¤ ç¨ ¢ ⮬ á«ãç ¥, ª®£¤  ¯® ª ª¨¬-«¨¡® á®®¡à ¦¥­¨ï¬ § à ­¥¥ ïá­®, çâ® A~ ¨¬¥¥â «¨èì ®¤­ã ª®¬¯®­¥­âã (¤¥ª à⮢ã). ‘®®â­®è¥­¨¥ ª «¨¡à®¢ª¨ ¯®§¢®«ï¥â ¢ëà §¨âì ' ç¥à¥§ A~ :

j div A: ~ ' = !" ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï E~ , ¯®«ã稬

1 (grad div + k2 ) A; ~ E~ = j!" ¨­ë¬¨ á«®¢ ¬¨, ª ª E~ , â ª ¨ H~ ¬®£ãâ ¡ëâì ¢ëà ¦¥­ë ç¥à¥§ ¢¥ªâ®à­ë© ¯®â¥­æ¨ « A~ . ‹¥£ª® ¯®ª § âì, ç⮠᪠«ïà­ë© ¯®â¥­æ¨ « ' 㤮¢«¥â¢®àï¥â ãà ¢­¥-

­¨î


' + k2 ' = ; " : ‚ ­¥ª®â®àëå á«ãç ïå ¯à¨¬¥­ï¥âáï ªã«®­®¢áª ï ª «¨¡à®¢ª  div A~ = 0:

’®£¤  A~ 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î
A~ + k2 A~ = ; J~ + j!"
grad ';   ' 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î ã áá®­ , â.¥. íâ® { áâ â¨ç¥áª¨© ¯®â¥­æ¨ «. ®«®¦¨¬

A~ = j!" ~ ;

~ { ¢¥ªâ®à ƒ¥àæ , äã­ªæ¨ï, ¤«ï ᨭãᮨ¤ «ì­ëå ¯®«¥© ®â«¨ç îé £¤¥  ïáï ®â ¢¥ªâ®à­®£® ¯®â¥­æ¨ «  A~ «¨èì ç¨á«¥­­ë¬ ¬­®¦¨â¥«¥¬. ’®£¤  ¨§ ᮮ⭮襭¨ï ª «¨¡à®¢ª¨ ¯®«ã稬 17

' = ; div ~ :

~ 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î ‚ â®çª å, £¤¥ J~ 6= 0,  ~ + k2 ~ = ; J~ :
 j!"

J~ ¨¬¥¥â à §¬¥à­®áâì ¯®«ïਧ æ¨¨. „¥©á⢨⥫쭮, ‡ ¬¥â¨¬, çâ® j! D~ = "
E~ + P~áâ®à ; ®âªã¤ , ¯®¤áâ ¢«ïï íâ® ¢® ¢â®à®¥ ãà ¢­¥­¨¥ Œ ªá¢¥«« , ¯®«ã稬

~

J ); rot H~ = j!"
E~ + j! (P~áâ®à + j! £¤¥ P~áâ®à { áâ®à®­­ïï ¯®«ïਧ æ¨ï. ~ , ¯®«ã稬 ®¤áâ ¢«ïï ¢ëà ¦¥­¨¥ ¢ ᪮¡ª å ¢ ãà ¢­¥­¨¥ ¤«ï 

~ + k2 ~ = ; P~áâ®à + J~
 "

!

j!" :

~ ï¥âáï ¯®«ïਧ æ¨ï. Žâá ¤à㣮¥ ’ ª¨¬ ®¡à §®¬, ¨áâ®ç­¨ª®¬  ~ { ¯®«ïਧ æ¨®­­ë© ¯®â¥­æ¨ «. ­ §¢ ­¨¥ ¢¥ªâ®à   «¥ªâà¨ç¥áª®¥ ¨ ¬ £­¨â­®¥ ¯®«ï ¬®£ãâ ¡ëâì ¢ëà ¦¥­ë ç¥à¥§ ¢¥ªâ®à ~ (í«¥ªâà¨ç¥áª¨© ¢¥ªâ®à ƒ¥àæ ):  E~ = !2 " ~ + grad div ~ ; H~ = j!"
rot ~ : Ž¤­®à®¤­ ï á¨á⥬  ãà ¢­¥­¨© Œ ªá¢¥««  ᨬ¬¥âà¨ç­  á â®ç­®áâìî ¤® §­ ª  ®â­®á¨â¥«ì­® E~ ¨ H~ . ®í⮬㠬®¦­® ¯®«ãç¨âì à¥è¥­¨¥ ®¤­®à®¤­®© á¨á⥬ë, ¯®« £ ï (â ª ª ª div D~ = 0 ¯à¨ J~ = 0)

D~ = ; rot A~ 0 : €­ «®£¨ç­® ¯à¥¤ë¤ã饬㠬®¦­® ¯®«ãç¨âì 18

H~ = ; j!A~ 0 ; grad '0 ; E~ = ; 1" rot A~ 0 ; ¯à¨ç¥¬ A~ 0 ¨ '0 㤮¢«¥â¢®àïîâ ãà ¢­¥­¨ï¬
A~ 0 + k2 A~ 0 = 0;
'0 + k2 '0 = 0; div A~ 0 + j!"
'0 = 0:

~ ¬®¦­® ¢¢¥á⨠¬ £­¨â­ë© ¢¥ªâ®à ƒ¥àæ  á ¯®¬®éìî ®  ­ «®£¨¨ á  ᮮ⭮襭¨© A~ 0 = j!" ~ 0 ; '0 = div ~ 0 ; ~ 0 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î ¯à¨ç¥¬  ~ 0 + k2 ~ 0 = 0 :
 —¥à¥§ ¢¥ªâ®àë ƒ¥àæ  ®¡é¥¥ à¥è¥­¨¥ ãà ¢­¥­¨© Œ ªá¢¥««  § ¯¨á뢠¥âáï ¢ ¢¨¤¥ á㬬ë

E~ = ; j!
rot ~ 0 + (grad div + k2 ) ~ ; H~ = j!"
rot ~ + (grad div + k2 ) ~ 0 : ‡ ¬¥â¨¬, çâ® ¯®â¥­æ¨ «ë ¨ ¢¥ªâ®àë ƒ¥àæ  ã¤®¢«¥â¢®àïîâ ãà ¢­¥­¨ï¬, ªã¤  ¢å®¤¨â « ¯« á¨ ­. ‚믨襬 ¥£® ¢ëà ¦¥­¨ï ¢ ¤¥ª à⮢ëå ¨ 樫¨­¤à¨ç¥áª¨å ª®®à¤¨­ â å: @2 + @2 + @2 ;
= @x 2 @y2 @z 2 @ r @  + 1 @ 2 + @ 2 :
= 1r @r @r r2 @2 @z 2

19

1.7.

‘ª¨­-íä䥪⠨ £à ­¨ç­ë¥ ãá«®¢¨ï

x

à®á⥩訬 à¥è¥­¨¥¬ ãà ¢­¥­¨© Œ ªá¢¥««  ¢ ®¤­®à®¤­®© ¨§®âய­®© á।¥ ï¥âáï ¯«®áª ï ¢®«­ .  ¯à¨¬¥à, ¯«®áª ï ¢®«­ , à á¯à®áâà ­ïîé ïáï ¢ ¯®«®¦¨â¥«ì­®¬ ­ ¯à ¢«¥­¨¨ ®á¨ z , ¯®«ïਧ®¢ ­­ ï ¢ ¯«®áª®á⨠x0z (á¬. à¨á.), ¬®¦¥â ¡ëâì § ¯¨á ­  ¢ ¢¨¤¥

E

z y

H

Ex Ey Ez Hx Hy Hz

= = = = = =

A
e;jkz + B
e jkz ; 0; 0; 0; r r" " ; jkz A 
e ; B 
e jkz ; 0;

1 @Ex . £¤¥ k = !p"; Hy = ; j! @z ¥è¥­¨¥ §¤¥áì § ¯¨á ­® ¢ ¢¨¤¥ áã¬¬ë ¤¢ãå ¢®«­, à á¯à®áâà ­ïîé¨åáï ¢ ¯à®â¨¢®¯®«®¦­ëå ­ ¯à ¢«¥­¨ïå ᮠ᪮à®áâìî v = 1=p".  áᬮâਬ ¢®«­ã, à á¯à®áâà ­ïîéãîáï ¢ ®¤­®¬ ­ ¯à ¢«¥­¨¨. „«ï ­¥¥

Ex = A
re;jkz ; Hy = A "
e;jkz : Žâ­®è¥­¨¥

Ex =  = H y

r "

§ ¢¨á¨â ⮫쪮 ®â ¯ à ¬¥â஢ á।ë. âã ¢¥«¨ç¨­ã ­ §ë¢ îâ ¢®«­®¢ë¬ ᮯà®â¨¢«¥­¨¥¬ á।ë. „«ï ¢¥é¥á⢥­­ëå " ¨   ¬¯«¨â㤠 ¯®«ï 20

¯à¨ à á¯à®áâà ­¥­¨¨ ®áâ ¥âáï ­¥¨§¬¥­­®©. …᫨ " ¨«¨  ¨¬¥îâ ¬­¨¬ãî á®áâ ¢«ïîéãî, â® ¢á«¥¤á⢨¥ ¯®â¥àì ¢®«­  § âã堥⠢ ­ ¯à ¢«¥­¨¨ à á¯à®áâà ­¥­¨ï.  áᬮâਬ á«ãç © ­®à¬ «ì­®£® ¯ ¤¥­¨ï ¢®«­ë ­  ¯«®áªãî £à ­¨æã ¬¥¦¤ã ¤¢ã¬ï á। ¬¨. 1

2

0

z

‘«¥¢  ­  £à ­¨æã ¯ ¤ ¥â ¢®«­  ¥¤¨­¨ç­®©  ¬¯«¨âã¤ë (á¬. à¨á.): Ex¯ ¤ = er;jk1 z ; Hy¯ ¤ = "1
e;jk1 z = 1
e;jk1 z : 1 1

‚®«­  ç áâ¨ç­® ®âà ¦ ¥âáï, â ª çâ® ®âà ¦¥­­ ï ¢®«­  à ¢­ 

r Ex®âà = R
e jk1 z ; Hy®âà = ; R "1
e jk1 z = ; R
e jk1 z : 1 1

à®è¥¤è ï ¢®«­  à ¢­ 

r Ex = T
e;jk2 z ; Hy = T "2
e;jk2 z = T
e;jk2 z : 2 2 ‚ í⨠ᮮ⭮襭¨ï ¢å®¤ïâ ­¥¨§¢¥áâ­ë¥ ¢¥«¨ç¨­ë R ¨ T . ƒà ­¨ç­ë¥ ãá«®¢¨ï { ­¥¯à¥à뢭®áâì Ex ¨ Hy { ¤ îâ ¢®§¬®¦­®áâì ­ ©â¨ ¨å. ‘®¢¬¥é ï £à ­¨æã á ¯«®áª®áâìî z = 0 ¨ ¯à¨à ¢­¨¢ ï Ex ¨ Hy á¯à ¢  ¨ á«¥¢ , ¯®«ã稬

1 + R = T; 1 (1 ; R) = 1 T: 1 2 ˆ§ íâ¨å ¤¢ãå ãà ¢­¥­¨© ¬®¦­® ­ ©â¨ R ¨ T :

1 ; R = 2 ; + 2 1 T =  2+2  : 2 1 21

…᫨ 2 ! 0, â® jRj ! 1,   jT j ! 0, â.¥. ¢®«­  ¯®«­®áâìî ®âà ¦ ¥âáï. â®â á«ãç © ï¥âáï ¯à¥¤¥«ì­ë¬, ¢ ç áâ­®áâ¨, ¥á«¨ ¢â®à ï á।  ¯à¥¤áâ ¢«ï¥â ᮡ®© ¬¥â «« á ®ç¥­ì ¡®«ì让 ¯à®¢®¤¨¬®áâìî. ‚ ¯®á«¥¤­¥¬ á«ãç ¥ ¬®¦­® áç¨â âì, çâ® 1 = 0 ; "1 = "0 ,   "2  ;j ! , â ª ª ª ¯à®¢®¤¨¬®áâì ¬¥â ««  ¬­®£® ¡®«ìè¥ "0 . ’®£¤ 

2 =

r 2

"2 =

r

r

j!2 = (1 + j ) !2 :  2

®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï à ¢­ 

r ! p p 2 = (1 ; j ) : k2 = ! 2 "2 = ;j2 ! = (1 ; j ) 2 

‚®«­  ¯à¨ íâ¨å ãá«®¢¨ïå § ¯¨á뢠¥âáï ¢ ¢¨¤¥

r

z z z e;j  (1;j) = e; 
e;j  :

2 å à ªâ¥à¨§ã¥â ᪮à®áâì ã¡ë¢ ­¨ï  ¬¯«¨âã¤ë ‚¥«¨ç¨­   = ! ¢®«­ë ¯® ¬¥à¥ à á¯à®áâà ­¥­¨ï ¥¥ ¢ á।¥ 2 (⮫騭  ᪨­-á«®ï). ‚­ãâਠ¬¥â ««  ¨¬¥¥â ¬¥á⮠ᮮ⭮襭¨¥ ª®¬¯®­¥­â ¯®«ï

Ex = 2
Hx : „ ­­®¥ ᮮ⭮襭¨¥ á¯à ¢¥¤«¨¢® â ª¦¥ ­  £à ­¨æ¥ à §¤¥« , ¯® ®¡¥ áâ®à®­ë ®â ­¥¥, ¢á«¥¤á⢨¥ ­¥¯à¥à뢭®á⨠⠭£¥­æ¨ «ì­ëå ª®¬¯®­¥­â ¯®«ï. ’¥¯¥àì ¯à¥¤¯®«®¦¨¬, çâ® ¢®«­  ¢­¥ ¯à®¢®¤­¨ª  ¯ ¤ ¥â ­ ª«®­­® ­  ¯«®áª®áâì à §¤¥« . ‚ í⮬ á«ãç ¥ ¯à¨ ¯¥à¥å®¤¥ ¢ ®¯â¨ç¥áª¨ ¡®«¥¥ ¯«®â­ãî á।㠭 ¯à ¢«¥­¨¥ à á¯à®áâà ­¥­¨ï ¯à¨¡«¨¦ ¥âáï ª ­®à¬ «¨. …᫨ j"j  "0 0 , â® ­ ¯à ¢«¥­¨¥ à á¯à®áâà ­¥­¨ï ®ç¥­ì ¡«¨§ª® ª ­®à¬ «¨. ®í⮬ã Ex ¨ Hy ¢­ãâਠ¬¥â ««  ®ç¥­ì ¡«¨§ª¨ ª â ­£¥­æ¨ «ì­ë¬ ª®¬¯®­¥­â ¬. ®í⮬㠨 ¢ í⮬ á«ãç ¥ ­  ¢­¥è­¥© áâ®à®­¥ £à ­¨æë á¯à ¢¥¤«¨¢® ᮮ⭮襭¨¥

Et = 
Ht ¤«ï â ­£¥­æ¨ «ì­ëå ª®¬¯®­¥­â ¯®«ï. ˆ­ ç¥ í⮠ᮮ⭮襭¨¥ ¬®¦¥â ¡ëâì § ¯¨á ­® á«¥¤ãî騬 ®¡à §®¬: 22

E~ t = 
(H~  ~n); £¤¥ ~n { ­®à¬ «ì ¢­ãâàì ¬¥â «« . „ ­­®¥ ᮮ⭮襭¨¥ ­ §ë¢ î⠣࠭¨ç­ë¬ ãá«®¢¨¥¬ ‹¥®­â®¢¨ç . Ž­® á¯à ¢¥¤«¨¢® ¯à¨ ᨫ쭮¬ ᪨­-íä䥪â¥, ª®£¤  ⮫騭  ᪨­-á«®ï ¬­®£® ¬¥­ìè¥ ¢á¥å å à ªâ¥à­ëå à §¬¥à®¢, ¢ ⮬ ç¨á«¥ ¤«¨­ë ¢®«­ë ¢ ᢮¡®¤­®¬ ¯à®áâà ­á⢥ ¨ ­ ¨¬¥­ì襣® à ¤¨ãá  ªà¨¢¨§­ë ¯®¢¥àå­®áâ¨. ‚ í⮬ á«ãç ¥ «î¡ãî ¢®«­ã ¢ ¬ «®© ®¡« á⨠¬®¦­® áç¨â âì ¯«®áª®©. ’®«é¨­  ᪨­-á«®ï ¤®«¦­  ¡ëâì â ª¦¥ ¬­®£® ¬¥­ìè¥ â®«é¨­ë ¯« áâ¨­ë ¬¥â «« . “á«®¢¨¥ ¯à¨¬¥­¨¬®á⨠(â.¥. ᨫì­ë© ᪨­-íä䥪â) ¨¬¥¥â ¢¨¤

 !  "0 :

‚ëç¨á«¨¬, ­ ¯à¨¬¥à, ! ¤«ï ¬¥¤¨ ¨ ¯à¨à ¢­ï¥¬ "0 . Žâá ­ ©¤¥¬ ¯à¥¤¥«ì­ãî ç áâ®âã:  = 5:7
107
36 = 6:4
1018 á;1 :  = 5:7
107 ‘¨¬ ; ! = ¬ "0 10;9 f¯à  1018 ƒæ, ç⮠ᮮ⢥âáâ¢ã¥â  = 3
10;8 á¬. Ž¤­ ª® ¯à®¢®¤¨¬®áâì 㦥 ¢ ¨­äà ªà á­®© ®¡« á⨠¤«¨­ ¢®«­ ᨫ쭮 á­¨¦ ¥âáï, â ª çâ® ãá«®¢¨¥ ‹¥®­â®¢¨ç  á¯à ¢¥¤«¨¢® ¤«ï ¤«¨­­®¢®«­®¢®© ¨­äà ªà á­®© ®¡« áâ¨. …᫨ ãáâ६¨âì  ! 1, â®  ! 0,   ¯®â®¬ã Et = 0. à¨ í⮬ ¯®«¥ ¢­ãâਠ⠪®© ¨¤¥ «ì­® ¯à®¢®¤ï饩 á।ë à ¢­® ­ã«î. ’ ª ª ª Bn ­¥¯à¥à뢭 , â® Bn = 0 (¨ Hn = 0) ­  ¢­¥è­¥© áâ®à®­¥ £à ­¨æë. ’ ­£¥­æ¨ «ì­ ï ª®¬¯®­¥­â  Ht ¨¬¥¥â à §àë¢ ¯à¨ ¯¥à¥å®¤¥ ç¥à¥§ £à ­¨æã, â ª ª ª ¯®«¥ ¢­ãâਠ¯à®¢®¤­¨ª  à ¢­® ­ã«î. ®í⮬㠯® ¯®¢¥àå­®á⨠¨¤¥ «ì­®£® ¯à®¢®¤­¨ª  ⥪ãâ ¯®¢¥àå­®áâ­ë¥ ⮪¨, ¯«®â­®áâì ª®â®àëå à ¢­  K = Ht ¯®  ¡á®«îâ­®© ¢¥«¨ç¨­¥.  ¯à ¢«¥­¨¥ ®¯à¥¤¥«ï¥âáï ᮮ⭮襭¨¥¬

K~ = H~  ~n: …᫨ ¯à®¢®¤­¨ª ­¥ ï¥âáï ¨¤¥ «ì­ë¬, â® ¯®¢¥àå­®áâ­ë¥ ⮪¨ ­¥ ¬®£ãâ áãé¥á⢮¢ âì. ® ¬®¦­® ¢¢¥á⨠⠪ ­ §ë¢ ¥¬ë¥ ª¢ §¨¯®¢¥àå­®áâ­ë¥ ⮪¨, ®¯à¥¤¥«ï¥¬ë¥ á«¥¤ãî騬 ®¡à §®¬: 23

K~ =

Z1

J~
dz;

0 ¯à¨ç¥¬ ®áì z ­ ¯à ¢«¥­  ¯¥à¯¥­¤¨ªã«ïà­® ¯®¢¥àå­®á⨠¬¥â «« . ‚ í⮬ á«ãç ¥ ¬®¦­® ¯®ª § âì, çâ® Ht = K ­  ¯®¢¥àå­®áâ¨. „¥©á⢨⥫쭮, ¢­ãâਠ¬¥â «« 

~ rot H~ = j!"E~ + J~  J: ãáâì H~ ¨¬¥¥â ⮫쪮 y-î á®áâ ¢«ïîéãî. ’®£¤  y Jx = rot x H~ = ; @H @z :

®¤áâ ¢«ïï Jx ¢ ¯®¤ë­â¥£à «ì­®¥ ¢ëà ¦¥­¨¥, ­ ©¤¥¬

Kx =

Z1

Jx
dz =

Z1 @Hy ; @z dz

0 0 ’ ª¨¬ ®¡à §®¬, ¨ ¢ í⮬ á«ãç ¥

1

= ; Hy

0



= Hy

z=0

= Ht :

K~ = H~  ~n: ‘®£« á­® ãá«®¢¨î ‹¥®­â®¢¨ç 

~ E~ = 
H~  ~n = 
K: “ç¨â뢠ï í⮠ᮮ⭮襭¨¥, ¢¥«¨ç¨­ã

r !

+j  = (1 + j ) 22 = 1  ­ §ë¢ îâ ¯®¢¥àå­®áâ­ë¬ ᮯà®â¨¢«¥­¨¥¬. “á«®¢¨¥ ‹¥®­â®¢¨ç , â ª¨¬ ®¡à §®¬, á¢ï§ë¢ ¥â ¯®¢¥àå­®áâ­ë© ⮪ á â ­£¥­æ¨ «ì­®© á®áâ ¢«ïî饩 í«¥ªâà¨ç¥áª®£® ¯®«ï ­  ¯®¢¥àå­®áâ¨. €ªâ¨¢­ ï (¢¥é¥á⢥­­ ï) á®áâ ¢«ïîé ï ¯®¢¥àå­®áâ­®£® ᮯà®â¨¢«¥­¨ï ®¯à¥¤¥«ï¥â ¬®é­®áâì ¯®â¥àì ¢ ¬¥â ««¥. Œ®é­®áâì ¯®â¥àì ¢ ¥¤¨­¨æ¥ ¯®¢¥àå­®á⨠®¯à¥¤¥«ï¥âáï ᮮ⭮襭¨¥¬ 24

P1 = 21 Re (E~  H~ )
~n:

’ ª ª ª

E~ = 
(H~  ~n);

â®

h

i

P1 = 12 Re  (H~  ~n)  H~ 
~n = 12 Re  jH j2

¨«¨

1 jH j2 = 1 jK j2 ; P1 = 2 t 2 â.¥. ¬®é­®áâì ¬®¦¥â ¡ëâì ¢ëà ¦¥­  ç¥à¥§ ¯«®â­®áâì ¯®¢¥àå­®áâ1 , ¨¬¥î­®£® ⮪ . ‚¥«¨ç¨­   é ï à §¬¥à­®áâì ᮯà®â¨¢«¥­¨ï, à ¢­  ᮯà®â¨¢«¥­¨î ¯« áâ¨­ë ¢ ä®à¬¥ ª¢ ¤à â  á® áâ®à®­®©, à ¢­®© ¥¤¨­¨æ¥, ⮫騭®©  (á¬. à¨á). ˆ­ ç¥ £®¢®àï, ¯®â¥à¨ â ª®¢ë, ª ª ¥á«¨ ¡ë ⮪ K ¯à®â¥ª « ¢ á«®¥ ⮫騭®© .

K 1 1 111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111

1.8.

δ

«¥ªâத¨­ ¬¨ç¥áª®¥ ¯®¤®¡¨¥

‚ à拉 á«ãç ¥¢ ¤«ï ¨áá«¥¤®¢ ­¨ï í«¥ªâத¨­ ¬¨ç¥áª¨å á¨á⥬ ¯à¨¬¥­ï¥âáï ¬ áèâ ¡­®¥ ¬®¤¥«¨à®¢ ­¨¥. „«ï í⮣® ­¥®¡å®¤¨¬ë ªà¨â¥à¨¨ ¯®¤®¡¨ï. ƒ¥®¬¥âà¨ç¥áª¨ ¯®¤®¡­ë¥ á¨áâ¥¬ë ¬®¦­® áç¨â âì ¯®¤®¡­ë¬¨ í«¥ªâத¨­ ¬¨ç¥áª¨, ¥á«¨ ¨å ¬®¦­® ®¯¨á뢠âì ®¤¨­ ª®¢ë¬¨ ãà ¢­¥­¨ï¬¨ ¯à¨ ®¤¨­ ª®¢ëå £à ­¨ç­ëå ãá«®¢¨ïå. ’ ª ª ª £¥®¬¥âà¨ï £à ­¨æ ¯®¤®¡­ , ­® á¨áâ¥¬ë ®â«¨ç îâáï à §¬¥à ¬¨, â® ãà ¢­¥­¨ï Œ ªá¢¥««  rot E~ = ; j!H~ ; rot H~ = j!"E~ á«¥¤ã¥â ¯à¨¢¥á⨠ª ¡¥§à §¬¥à­®© ä®à¬¥. 25

„«ï í⮣® ¢á¥ «¨­¥©­ë¥ ¢¥«¨ç¨­ë § ¯¨è¥¬ ¢ ¢¨¤¥

l = l0
L; £¤¥ L { ¡¥§à §¬¥à­®¥ ç¨á«®,   l0 { ­¥ª®â®àë© å à ªâ¥à­ë© à §¬¥à á¨á⥬ë. Šà®¬¥ ⮣®, § ¯¨è¥¬ ¯®«¥ ¢ ¢¨¤¥

E~ = e0
E~; H~ = h0
H~ ; £¤¥ e0 ¨ h0 { à §¬¥à­ë¥ ¢¥«¨ç¨­ë,   E~ ¨ H~ { ¡¥§à §¬¥à­ë¥ ¢¥ªâ®àë.

®¤áâ ¢«ïï í⨠¢ëà ¦¥­¨ï ¢ ãà ¢­¥­¨ï Œ ªá¢¥«« , ¯®«ã稬 Rot E~ = ; j! he 0 l0 H~ ; Rot H~ = j!" he0 l0 E~; 0 0 £¤¥ Rot ®§­ ç ¥â ¤¨ää¥à¥­æ¨à®¢ ­¨¥ ¯® ¡¥§à §¬¥à­ë¬ ª®®à¤¨­ â ¬. ®«ã祭­ë¥ ãà ¢­¥­¨ï ¯®«­®áâìî å à ªâ¥à¨§ãîâáï ¡¥§à §¬¥à­ë¬¨ ç¨á« ¬¨ { ªà¨â¥à¨ï¬¨ ¯®¤®¡¨ï a1 = !l0 he 0 ; a2 = !"l0 he0 ; 0 0 ª®â®àë¥ ¤«ï ¯®¤®¡­ëå á¨á⥬ ¤®«¦­ë ¡ëâì ®¤¨­ ª®¢ë¬¨. ’ ª ª ª ¢¥«¨ç¨­ë e0 ¨ h0 ¯à®¨§¢®«ì­ë, â® ¨å ®â­®è¥­¨¥ ¬®¦­® ¢ë¡à âì â ª, ç⮡ë a2 = 1. ’®£¤  h0 = !"l : 0 e0 ®¤áâ ¢«ïï í⮠ᮮ⭮襭¨¥ ¢ a1 , ¯®«ã稬 a1 = !2 " l02 = idem: ’ ª ª ª

!2 " = k2 =

 2  2 

â® ªà¨â¥à¨© ¯®¤®¡¨ï ¡ã¤¥â

l0 = idem: 

ƒà ­¨ç­®¥ ãá«®¢¨¥ ‹¥®­â®¢¨ç  ¨¬¥¥â ¢¨¤ 26

;

r

1 + jH: Et = (1 + j ) ! H ; ¨«¨ E = t t 2  t à¨¢®¤ï í⮠ᮮ⭮襭¨¥ ª ¡¥§à §¬¥à­®© ä®à¬¥, ¯®«ã稬 + j h0 H~ : E~t = 1  e0 t ¨ ¯®¤áâ ¢«ïï á he 0 = !" l0, ¯®«ã稬 0 E~t = (1 + j ) !"l0 H~ t :



Šà¨â¥à¨© ¯®¤®¡¨ï ®âá

!"l0 = idem: 

„®¬­®¦ ï ç¨á«¨â¥«ì ¨ §­ ¬¥­ â¥«ì ­  !¬ =, £¤¥ ¬ { ¬ £­¨â­ ï ¯à®­¨æ ¥¬®áâì ¬ â¥à¨ «  á⥭®ª, ¯®«ã稬 ¬ !2 "l0 = !2 " l2  ¬ : 0 l0   !¬ ’ ª ª ª a1 = !2 "l02 = idem, â® ¢â®à®© ªà¨â¥à¨© ¯à¨¬¥â ¢¨¤

 ¬ l0  = idem:

…᫨ ¬ = 1, â® ¢â®à®© ªà¨â¥à¨© ¯®¤®¡¨ï

 = idem: l0 2.

2.1.

’…ŽˆŸ „‹ˆ›• ‹ˆˆ‰

®¯¥à¥ç­ë¥ ¢®«­ë ¢ «¨­¨¨ ¯¥à¥¤ ç¨

‹¨­¨¥© ¯¥à¥¤ ç¨ í«¥ªâ஬ £­¨â­ëå ¢®«­ ®¡ëç­® ­ §ë¢ îâ í«¥ªâத¨­ ¬¨ç¥áªãî á¨á⥬ã, ®¤­®à®¤­ãî ¢¤®«ì ­¥ª®â®à®© ®á¨ z .  ©¤¥¬ ¯®«ï, 27

ª®â®àë¥ ¬®£ãâ áãé¥á⢮¢ âì ¢ â ª®© á¨á⥬¥. „«ï í⮣® ¨á¯®«ì§ã¥¬ ãà ¢­¥­¨ï Œ ªá¢¥«« . à¨ í⮬ ¯à¥¤¯®«®¦¨¬, çâ® § ¢¨á¨¬®áâì ®â z ®¯¨á뢠¥âáï ¬­®¦¨â¥«¥¬ e; z , â.¥. ¬ë ¨é¥¬ à¥è¥­¨¥ ¢ ¢¨¤¥ ¢®«­, à á¯à®áâà ­ïîé¨åáï ¢ ­ ¯à ¢«¥­¨¨ ®á¨ z . “à ¢­¥­¨ï Œ ªá¢¥««  ¨¬¥îâ ¢¨¤

~ rot E~ = ;j!H; ~ rot H~ = j!"E: ã¤¥¬ ¨áª âì ¢®«­ë,  ­ «®£¨ç­ë¥ ¯® ᢮©á⢠¬ ¯«®áª¨¬ ¢®«­ ¬ ¢ ᢮¡®¤­®¬ ¯à®áâà ­á⢥. ’ ª¨¥ ¢®«­ë å à ªâ¥à¨§ãîâáï ᢮¥© ¯®¯¥à¥ç­®áâìî, â.¥. ¤«ï ­¨å Hz = 0; Ez = 0. “ç¨â뢠ï íâ®,   â ª¦¥ â®, çâ® @=@z = ; , § ¯¨è¥¬ ãà ¢­¥­¨ï Œ ªá¢¥««  ¤«ï ®â¤¥«ì­ëå ª®¬¯®­¥­â:

;j!Hx = Ey ; ;j!Hy = ; Ex ; y ; @Ex ; 0 = @E @x @y

j!"Ex = Hy ; j!"Ey = ; Hx; y ; @Hx : 0 = @H @x @y

ˆ§ ¤ ­­ëå ãà ¢­¥­¨© á«¥¤ãîâ â ª¨¥ ¯ àë ãà ¢­¥­¨©:

Ex ; j!Hy = 0; j!"Ex ; Hy = 0:

Ey + j!Hx = 0; j!"Ey + Hx = 0;

â® | ®¤­®à®¤­ë¥  «£¥¡à ¨ç¥áª¨¥ ãà ¢­¥­¨ï, ®¯à¥¤¥«ïî騥 ¯®¯¥à¥ç­ë¥ ª®¬¯®­¥­âë í«¥ªâ஬ £­¨â­®£® ¯®«ï. ¥âਢ¨ «ì­ë¥ à¥è¥­¨ï áãé¥áâ¢ãîâ, ¥á«¨ ®¯à¥¤¥«¨â¥«ì á¨á⥬ë ãà ¢­¥­¨© à ¢¥­ ­ã«î. à¨à ¢­¨¢ ï ®¯à¥¤¥«¨â¥«¨ ®¡¥¨å á¨á⥬ ­ã«î, ¯®«ã稬 ®¤­® ¨ â® ¦¥ ãà ¢­¥­¨¥, ®¯à¥¤¥«ïî饥 ­¥¨§¢¥áâ­ãî ¯®ª  ¢¥«¨ç¨­ã :

2 + !2 " = 0 : ˆ§ í⮣® ãà ¢­¥­¨ï ­ å®¤¨¬ :

=  j!p" =  jk : …᫨  ¨ " { ¢¥é¥á⢥­­ë, â® { ç¨áâ® ¬­¨¬ ï ¢¥«¨ç¨­ ; à á¯à®áâà -

­¥­¨¥ ¢®«­ë ¢ í⮬ á«ãç ¥ ¯à®¨á室¨â ¡¥§ § âãå ­¨ï. 28

‡ ¬¥â¨¬, çâ® ¢ ¤ ­­®¬ á«ãç ¥ ¬®£ãâ ¡ëâì ®¯à¥¤¥«¥­ë «¨èì ®â­®è¥­¨ï ¯®¯¥à¥ç­ëå ª®¬¯®­¥­â ¯®«ï:

r

r

Ex  Ey  Hy = " ; Hx = ; " : ‡ ¬¥­ïï ¢ ãà ¢­¥­¨¨ @Hy =@x ; @Hx =@y = 0 Hx ¨ Hy ç¥à¥§ Ex ¨ Ey

¨§ íâ¨å ᮮ⭮襭¨©, ¯®«ã稬

@Ex + @Ey = 0; @Ey ; @Ex = 0: @x @y @x @y

‚â®à®¥ ãà ¢­¥­¨¥ ¬®¦¥â ¡ëâì 㤮¢«¥â¢®à¥­®, ¥á«¨ ¯®«®¦¨âì

Ex = ; @@x
e; z ; Ey = ; @@y
e; z ; { ­¥ª®â®à ï ᪠«ïà­ ï äã­ªæ¨ï ¯¥à¥¬¥­­ëå x ¨ y. ®¤áâ ¢«ïï

£¤¥ í⨠¢ëà ¦¥­¨ï ¢ ¯¥à¢®¥ ãà ¢­¥­¨¥, ¯®«ã稬 ãà ¢­¥­¨¥ ¤«ï @ 2 + @ 2 = 0: @x2 @y2 ˆâ ª, ¤«ï E~ ¬ë ¯®«ãç ¥¬ ᮮ⭮襭¨¥

E~ = ; e; z
grad ; ¯à¨ç¥¬ 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î
= 0; £¤¥
{ « ¯« á¨ ­. ƒà ­¨ç­ë¥ ãá«®¢¨ï ¤«ï E~ ¨¬¥îâ ®¡ëç­ë© ¢¨¤ (Et = 0 ­  ¯®¢¥àå­®áâ¨, = const). ’ ª¨¬ ®¡à §®¬, ¬®¦­® ᤥ« âì ¢ë¢®¤, ç⮠㤮¢«¥â¢®àï¥â ¢ ¯®¯¥à¥ç­®¬ á¥ç¥­¨¨ ãà ¢­¥­¨î ‹ ¯« á , ¨, á«¥¤®¢ â¥«ì­®, í«¥ªâà¨ç¥áª®¥ ¯®«¥ ᮢ¯ ¤ ¥â ¯® ª®­ä¨£ãà æ¨¨ á í«¥ªâà®áâ â¨ç¥áª¨¬ ¯®«¥¬.  áᬮâਬ ⥯¥àì á¥ç¥­¨¥ «¨­¨¨. Ž­® ¬®¦¥â ¡ëâì ®¤­®-, ¤¢ã- ¨«¨ ¬­®£®á¢ï§­ë¬. „«ï ®¤­®á¢ï§­®£® á¥ç¥­¨ï í«¥ªâà®áâ â¨ç¥áª®¥ ¯®«¥ ¡¥§ ¯à®¤®«ì­®© á®áâ ¢«ïî饩 áãé¥á⢮¢ âì ­¥ ¬®¦¥â. Œ â¥¬ â¨ç¥áª¨ íâ® á¢ï§ ­® á ⥬, çâ® äã­ªæ¨ï, 㤮¢«¥â¢®àïîé ï ãà ¢­¥­¨î ‹ ¯« á  (£ à¬®­¨ç¥áª ï äã­ªæ¨ï), ¯®áâ®ï­­ ï ­  £à ­¨æ¥, ¯®áâ®ï­­  ¢­ãâਠ¢á¥© ¤¢ã¬¥à­®© ®¡« áâ¨. ®í⮬㠢 «¨­¨ïå á ®¤­®á¢ï§­ë¬ á¥ç¥­¨¥¬ ¯®¯¥à¥ç­ë¥ 29

¢®«­ë áãé¥á⢮¢ âì ­¥ ¬®£ãâ, ¢ â® ¢à¥¬ï ª ª ¢ ¬­®£®á¢ï§­ëå | ¬®£ãâ, â ª ª ª ¯®â¥­æ¨ « à §«¨ç­ëå ¯à®¢®¤­¨ª®¢ ¬®¦¥â à §«¨ç âìáï. ®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï ¯®¯¥à¥ç­ëå ¢®«­ (¢®«­ TEM) ᮢ¯ ¤ ¥â á ¯®áâ®ï­­®© à á¯à®áâà ­¥­¨ï ¯«®áª¨å ¢®«­ ¢ ᢮¡®¤­®¬ ¯à®áâà ­á⢥. ” §®¢ ï ᪮à®áâì ¢®«­ TEM ­¥ § ¢¨á¨â ®â ç áâ®âë: vä = 1=p".  á¯à®áâà ­¥­¨¥ ¢®§¬®¦­® ¤«ï «î¡ëå ç áâ®â ®â 0 ¤® 1. ®á«¥ ⮣® ª ª ­ ©¤¥­® í«¥ªâà¨ç¥áª®¥ ¯®«¥, ¬ £­¨â­®¥ ¯®«¥ ¬®¦¥â ¡ëâì ¯®«ã祭® ¨§ ¯¥à¢®£® ãà ¢­¥­¨ï Œ ªá¢¥«« : ~ H~ = ; 1 rot E:

j!

®¤áâ ¢«ïï á E~ , ¢ëà ¦¥­­®¥ ç¥à¥§ ¯®â¥­æ¨ « ¨ ãç¨âë¢ ï ¨§¢¥áâ­®¥ ᮮ⭮襭¨¥ ¢¥ªâ®à­®£®  ­ «¨§  rot (
F~ ) = 
rot F~ + grad   F~ , ¯®«ã稬

r E~ = ; grad
e; z ; H~ = ; "
e; z
~z0  grad : ‡¤¥áì ~z0 { ¥¤¨­¨ç­ë© ¢¥ªâ®à ¢ ­ ¯à ¢«¥­¨¨ ®á¨ z . ˆ§ ¯®«ã祭­®£® ᮮ⭮襭¨ï á«¥¤ã¥â, ¢ ç áâ­®áâ¨, çâ® ¢¥ªâ®àë E~ ¨ H~ ¢ «î¡®© â®çª¥ ¯¥à¯¥­¤¨ªã«ïà­ë ¤à㣠¤àã£ã.

2.2.

 ¯à殮­¨¥, ⮪ ¨ ¬®é­®áâì ¢ TEM-«¨­¨¨

’ ª ª ª ¢ ¯®¯¥à¥ç­®¬ á¥ç¥­¨¨ ¤¢ãá¢ï§­®© «¨­¨¨ TEM ¯®«¥ | áâ â¨ç¥áª®¥, â® ¢¥«¨ç¨­  ¨­â¥£à «  Z2 U = ; E~
d~l; 1 £¤¥ â®çª¨ 1 ¨ 2 à á¯®«®¦¥­ë ­  ®¤­®¬ ¨ ¤à㣮¬ ¯à®¢®¤­¨ª å ¢ ®¤­®¬ á¥ç¥­¨¨, ­¥ § ¢¨á¨â ®â ¯ã⨠¨­â¥£à¨à®¢ ­¨ï ¯à¨ ãá«®¢¨¨, çâ® íâ®â ¯ãâì «¥¦¨â 楫¨ª®¬ ¢ ¯«®áª®á⨠¯®¯¥à¥ç­®£® á¥ç¥­¨ï. …áâ¥á⢥­­® ­ §¢ âì íâã ¢¥«¨ç¨­ã ­ ¯à殮­¨¥¬ ¢ «¨­¨¨. „ «¥¥, â ª ª ª ¬ £­¨â­®¥ ¯®«¥ ¯®¯¥à¥ç­®, ⮪¨ ¢ ¯à®¢®¤­¨ª å ç¨áâ® ¯à®¤®«ì­ë. ¥âà㤭® ¯®ª § âì, çâ® ¨­â¥£à «

I =

I

30

H~
d~l

¯® ª®­âãàã, ®å¢ â뢠î饬㠢­ãâ७­¨© ¯à®¢®¤­¨ª ¨ «¥¦ é¥¬ã ¢ ¯«®áª®á⨠¯®¯¥à¥ç­®£® á¥ç¥­¨ï, à ¢¥­ ¯®«­®¬ã ⮪㠢® ¢­ãâ७­¥¬ ¯à®¢®¤­¨ª¥ ¨ ­¥ § ¢¨á¨â ®â ¢ë¡®à  í⮣® ª®­âãà . ¥âà㤭® â ª¦¥ ¯®ª § âì, ç⮠⮪ ¢® ¢­¥è­¥¬ ¯à®¢®¤­¨ª¥ à ¢¥­ ⮪㠢® ¢­ãâ७­¥¬ ¯à®¢®¤­¨ª¥ á ®¡à â­ë¬ §­ ª®¬; ¯®í⮬㠬®¦­® ®¤­®§­ ç­® ¢¢¥á⨠⮪ I ¢ «¨­¨¨. ˆâ ª, ¢ TEM-«¨­¨¨ ¬®¦­® ®¤­®§­ ç­® ¢¢¥á⨠⠪¨¥ ¯®­ïâ¨ï, ª ª ­ ¯à殮­¨¥ ¨ ⮪, ª®â®àë¥ ¨¬¥îâ, ª®­¥ç­®, ¡®«¥¥ 㧪¨© á¬ëá«, 祬 ¢ ⥮ਨ 楯¥©, â ª ª ª ¨­â¥£à « ¡¥à¥âáï ®¡ï§ â¥«ì­® ¯® ª®­âãàã, «¥¦ é¥¬ã ¢ ¯®¯¥à¥ç­®¬ á¥ç¥­¨¨.  ¯à殮­¨ï ¨ ⮪¨ ¢ à §«¨ç­ëå á¥ç¥­¨ïå ¤«ï ¡¥£ã饩 ¢®«­ë ¯à¨ ®âáãâá⢨¨ ¯®â¥àì ®â«¨ç îâáï ⮫쪮 ä §®©:

U = U0
e;jkz ; ‚¥«¨ç¨­  ®â­®è¥­¨ï

I = I0
e;jkz :

Z0 = UI = UI 0 0

¯®áâ®ï­­  ¢® ¢á¥å á¥ç¥­¨ïå. ®ª ¦¥¬, çâ® ¯à¨ ®âáãâá⢨¨ ¯®â¥àì ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ TEM«¨­¨¨ Z0 { ¢¥é¥á⢥­­ ï ¢¥«¨ç¨­ , â.¥. U ¨ I ¨¬¥îâ ®¤¨­ ª®¢ãî ä §ã. ãáâì ¯®â¥­æ¨ « = 1 +j 2 | ª®¬¯«¥ªá­ ï äã­ªæ¨ï. ’®£¤  1 ¨ 2 ¤®«¦­ë ¯® ®â¤¥«ì­®á⨠㤮¢«¥â¢®àïâì ãà ¢­¥­¨î
= 0 ¨ £à ­¨ç­ë¬ ãá«®¢¨ï¬ = const ­  ¯¥à¨¬¥âॠ¯®¯¥à¥ç­®£® á¥ç¥­¨ï. ®í⮬㠮⠮­¨ ®â«¨ç îâáï «¨èì ¬­®¦¨â¥«¥¬, ¨ ¢ ª ç¥á⢥ à¥è¥­¨ï ¬®¦­® ¯à¨­ïâì, ­ ¯à¨¬¥à, ¢¥é¥á⢥­­ãî äã­ªæ¨î 1 ¨«¨ 2 . Žâá á«¥¤ã¥â, çâ® ¢® ¢á¥å â®çª å ¤ ­­®£® á¥ç¥­¨ï z ¢¥ªâ®à E~ = ;grad
e;jkz ¨¬¥¥â ®¤­ã ¨ âã ¦¥ ä §ã. ’ ª ª ª 31

Ex Hy =

r

r

 Ey " ¨ Hx = ; " ;

â® ¢¥ªâ®à H~ â ª¦¥ ¢® ¢á¥å â®çª å á¥ç¥­¨ï z ¨¬¥¥â âã ¦¥ ä §ã, çâ® ¨ ¢¥ªâ®à E~ . ’ ª ª ª ­ ¯à殮­¨¥ U ¨ ⮪ I ¢ëà ¦ îâáï ç¥à¥§ ¨­â¥£à «ë ®â 㪠§ ­­ëå ¢¥ªâ®à®¢, â® U ¨ I ­ å®¤ïâáï ¢ á¥ç¥­¨¨ ¢ ®¤­®© ä §¥. ®í⮬ã U=I = Z0 { ¢¥é¥á⢥­­ ï ¢¥«¨ç¨­ .  ©¤¥¬ ­¥ª®â®àë¥ í­¥à£¥â¨ç¥áª¨¥ ᮮ⭮襭¨ï ¤«ï ¢®«­ ¢ «¨­¨¨ TEM. „«ï í⮣® ¢á¯®¬­¨¬, çâ®

r"

Hx = ;  Ey ; Hy = Žâá

r"

 Ex :

Hx2 + Hy2 = " (Ex2 + Ey2 ) = "
E 2 :

‘«¥¤®¢ â¥«ì­®,

H 2 = "E 2 ; 2 2

â® ¥áâì ¯«®â­®áâ¨ í­¥à£¨¨ í«¥ªâà¨ç¥áª®£® ¨ ¬ £­¨â­®£® ¯®«¥© ¢ ¡¥£ã饩 TEM-¢®«­¥ à ¢­ë ¢ «î¡®© â®çª¥. ‡ ¯ áë í«¥ªâà¨ç¥áª®© ¨ ¬ £­¨â­®© í­¥à£¨¨ ¢ ¡¥£ã饩 ¢®«­¥ ­  ¥¤¨­¨æã ¤«¨­ë «¨­¨¨ â ª¦¥ à ¢­ë. „«ï TEM-«¨­¨¨ ¬®¦­® ¢¢¥á⨠¯®­ïâ¨ï ¯®£®­­ëå ¥¬ª®á⨠C1 ¨ ¨­¤ãªâ¨¢­®á⨠L1 á ¯®¬®éìî ᮮ⭮襭¨© 2We = 12 C1 jU j2 ; 2Wh = 12 L1 jI j2 ; £¤¥ We ¨ Wh { á।­¨¥ § ¯ áë í«¥ªâà¨ç¥áª®© ¨ ¬ £­¨â­®© í­¥à£¨© ­  ¥¤¨­¨æã ¤«¨­ë «¨­¨¨. “ç¨â뢠ï, çâ® Wh = We , ¯®«ã稬 C1 jU j2 = L1 jI j2 ; ®âªã¤  32

U = Z = 0 I

rL

1

C1 :

 ©¤¥¬ ⥯¥àì ¬®é­®áâì ¢®«­ë, à á¯à®áâà ­ïî饩áï ¢ TEM-«¨­¨¨: Z 1 P = 2 Re (E~  H~ )
~z0 dS; S

â.¥. ¨­â¥£à « ¢¥ªâ®à  ®©­â¨­£  ¯® á¥ç¥­¨î «¨­¨¨ S . ˆ¬¥¥¬ (E~  H~  )
~z0 =

r"

~ ~  Ex Ex +

r"

~ ~  Ey Ey =

r"

~2  jE j :

®¤áâ ¢«ïï íâ® ¢ ¯®¤ë­â¥£à «ì­®¥ ¢ëà ¦¥­¨¥, ¯®«ã稬 r Z Z 2 P = 21 " jE j2 dS = p1" "E2 dS = S

S

= 2vWe = 2vWh = v (We + Wh ) = vW; £¤¥ W { á।­¨© § ¯ á í­¥à£¨¨ ­  ¥¤¨­¨æã ¤«¨­ë «¨­¨¨, v { ᪮à®áâì à á¯à®áâà ­¥­¨ï ¢®«­ ¢ «¨­¨¨. „ «¥¥ ¡ã¤¥â ¯®ª § ­®, ç⮠᪮à®áâì à á¯à®áâà ­¥­¨ï ¬®¦¥â ¡ëâì ¢ëà ¦¥­  ç¥à¥§ ¯®£®­­ë¥ ¯ à ¬¥âàë: v = p1 : L1 C1 ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï ¬®é­®áâ¨, ¯®«ã稬 rC 1 1 1 1 jU j2 = 1 jU j2 ; 2 P = p C j U j =
1 2 L1 2 Z0 L1 C1 2 ¨«¨ rL 1 1 1 2 p P =
L jI j = 2 C1 jI j2 = 21 jI j2 Z0: L1 C1 2 1 1 ’à¥âì¥ ¢ëà ¦¥­¨¥ ¤«ï ¬®é­®á⨠¡¥£ã饩 ¢®«­ë ¨¬¥¥â ¢¨¤ P = 12 Re U
I  : 33

2.3.

Š® ªá¨ «ì­ ï «¨­¨ï

Ž¤­®© ¨§ ­ ¨¡®«¥¥ à á¯à®áâà ­¥­­ëå «¨­¨© TEM ï¥âáï ª® ªá¨ «ì­ ï «¨­¨ï, ¯à¥¤áâ ¢«ïîé ï ᮡ®© ¤¢  ª® ªá¨ «ì­® à á¯®«®¦¥­­ëå ¯à®¢®¤ïé¨å 樫¨­¤à  (á¬. à¨á.).

1111111111111 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 r 0000000000000 1111111111111 0000 1111 2 0000000000000 1111111111111 0000 1111 0000000000000 1111111111111 r 0000 1111 0000000000000 1111111111111 0000 1111 0000000000000 1111111111111 1 0000 1111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111

¥è¥­¨¥ í«¥ªâà®áâ â¨ç¥áª®© § ¤ ç¨ ¤«ï â ª®© á¨áâ¥¬ë ¨§¢¥áâ­®:

2

=

1

U ln r ; ln rr1 r2 2

£¤¥ U { à §­®áâì ¯®â¥­æ¨ «®¢ ¬¥¦¤ã 樫¨­¤à ¬¨. Žâá

Er = ; @@r = ; Ur 1r = U=r ; ln r1 ln rr2 2 1 r" r " U=r H' =  Er =  r : ln r2 1

’®ª ¢ ¯à®¢®¤­¨ª¥ à ¢¥­

r I = 2r1
Hr=r1 = 2 " Ur : ln r2 1

Žâá ¬®¦­® ­ ©â¨ ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥:

r

Z0 = UI = 21 " ln rr2 Ž¬: 1 …᫨ «¨­¨ï ­¨ç¥¬ ­¥ § ¯®«­¥­ , â®

r Z0 = 21 " 0
ln rr2 = 60 ln rr2 = 138 lg rr2 Ž¬: 0 1 1 1 34

„«ï ®¯à¥¤¥«¥­¨ï ¯®£®­­®© ¥¬ª®á⨠¨ ¨­¤ãªâ¨¢­®á⨠¬®¦­® ¨á¯®«ì§®¢ âì á«¥¤ãî騩 ¯à¨¥¬. ˆ¬¥¥¬:

rL

1 Ž¬; v = p 1 : C1 L1C1 ˆ§ íâ¨å ᮮ⭮襭¨© ¯®«ãç ¥¬ ; L1 = Zv0 ƒ­ C1 = vZ1 ” ¬: 0 ¬ …᫨ á ¯®¤áâ ¢¨âì ¢ëà ¦¥­¨ï ¤«ï v ¨ Z0 , â® ¯®«ã稬

Z0 =

” ; L =  ln r2 ƒ­ : C1 = 2" 1 r 2 r1 ¬ ln r2 ¬ 1

‘«¥¤ã¥â ®â¬¥â¨âì, çâ® ¯®¯¥à¥ç­ ï ¢®«­ , áâண® £®¢®àï, áãé¥áâ¢ã¥â ¢ ª® ªá¨ «ì­®© «¨­¨¨ «¨èì ¯à¨ ãá«®¢¨¨, çâ® «¨­¨ï ¢ á¥ç¥­¨¨ ®¤­®à®¤­® § ¯®«­¥­ . à¨ ç áâ¨ç­®¬ § ¯®«­¥­¨¨ ¯®ï¢«ïîâáï ¯à®¤®«ì­ë¥ ª®¬¯®­¥­âë ¯®«¥©, ¢ á¢ï§¨ á 祬 ¯®ï¢«ï¥âáï ¤¨á¯¥àá¨ï ¨ ¤à㣨¥ ¥­¨ï. 2.4.

„¢ã寫 á⨭ç â ï «¨­¨ï

„¢ã寫 á⨭ç â ï «¨­¨ï ¨¬¥¥â á¥ç¥­¨¥, ¯à¥¤áâ ¢«¥­­®¥ ­  à¨á. …᫨ à §¬¥àë «¨­¨¨ ¢ á¥ç¥­¨¨ ¬ 111111111111 000000000000 «ë ¯® áà ¢­¥­¨î á ¤«¨­®© ¢®«x ­ë, â® ¢ «¨­¨¨ ¬®¦¥â à á¯à®áâà E H b ­ïâìáï ⮫쪮 TEM-¢®«­ . ãáâì y  b. ’®£¤  ¯®«¥ ¬¥¦¤ã ¯« áâ¨111111111111 a­ ¬¨ 000000000000 ¬®¦­® ¯à¨¡«¨¦¥­­® áç¨â âì a ®¤­®à®¤­ë¬.  ¯à殮­¨¥ ¬¥¦¤ã ¯« á⨭ ¬¨ à ¢­®

U = b
Ex ; ⮪ à ¢¥­

I = a
Hy ; â ª ª ª ¯®«¥ ¢­¥ ¯« á⨭ ¢¥á쬠 ¬ «®. 35

Žâá ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ «¨­¨¨

r

Ex = b  Ž¬: Z0 = UI = ab H a " y …¬ª®áâì ¨ ¨­¤ãªâ¨¢­®áâì ­  ¥¤¨­¨æã ¤«¨­ë à ¢­ë Z0 b ƒ­ C1 = vZ1 = "0 ab ” ¬ ; L1 = v = 0 a ¬ : 0 2.5.

„¢ãå¯à®¢®¤­ ï «¨­¨ï

‘¥ç¥­¨¥ ¤¢ãå¯à®¢®¤­®© «¨­¨¨ ¨¬¥¥â ¢¨¤, ¯à¥¤áâ ¢«¥­­ë© ­  à¨á. „«ï í⮩ «¨­¨¨ í«¥ªâà®áâ â¨ª  ¤ ¥â ¢¥«¨ç¨­ã ¯®£®­­®© ¥¬ª®á⨠¢ ¢¨¤¥

11 00 00 11 a 11 00

11 00 00 11 00 11

C1 =

2 ln 4 2ba

+

s"  b 2 2a

3: ; 15

„ ­­ ï ä®à¬ã«  ¯à¨ b  a ã¯à®é ¥âáï:

b

C1  ln"b=a :

‡­ ï ¥¬ª®áâì, ¬®¦­® ­ ©â¨ â ª¦¥ ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥: r Z0 = vC1 = 1 " ln ab Ž¬: 1 2.6.

’¥«¥£à ä­ë¥ ãà ¢­¥­¨ï

“à ¢­¥­¨ï à á¯à®áâà ­¥­¨ï ¢®«­ ¢ «¨­¨ïå TEM ¬®¦­® ¯®«ãç¨âì ª¢ §¨áâ â¨ç¥áª¨¬ ¬¥â®¤®¬ á ãç¥â®¬ ⮣®, çâ® ¢ â ª¨å «¨­¨ïå ¬®£ãâ ¡ëâì ¢¢¥¤¥­ë ¯®­ïâ¨ï ⮪  ¨ ­ ¯à殮­¨ï. ‹¨­¨ï ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­  ¢ ¢¨¤¥ 楯®çª¨, à §¬¥àë ï祥ª ª®â®à®© § â¥¬ ãáâ६«ïîâáï ª ­ã«î (á¬. à¨á.). 36

dz Z 1 dz

U

I

U+dU I+dI

Y1 dz

„«ï â ª®© 楯®çª¨ ¬®£ãâ ¡ëâì á®áâ ¢«¥­ë á«¥¤ãî騥 ãà ¢­¥­¨ï:

dU = ; I Z1 dz; dI = ; U Y1 dz; ®âªã¤ 

dU = ; I Z ; dI = ; U Y : 1 dz 1 dz ˆáª«î稬 ¨§ íâ¨å ãà ¢­¥­¨©, ­ ¯à¨¬¥à, ⮪ I : d2 U ; Z Y U = 0: 1 1 dz 2 Ž¡®§­ ç ï 2 = Z1 Y1 , ¯®«ã稬 d2 U ; 2 U = 0: dz 2 ¥è¥­¨¥ í⮣® ãà ¢­¥­¨ï ¨¬¥¥â ¢¨¤

U = Ae; z + B e z ; â.¥. ¯à¥¤áâ ¢«ï¥â ᮡ®© á㬬㠢®«­, à á¯à®áâà ­ïîé¨åáï ¢ ¯à®â¨¢®¯®«®¦­ëå ­ ¯à ¢«¥­¨ïå. ’®ª ¬®¦¥â ¡ëâì ¯®«ã祭 ¨§ í⮣® ¢ëà ¦¥­¨ï á ¯®¬®éìî ⥫¥£à ä­®£® ãà ¢­¥­¨ï:

e; z ; B e z : I = ; Z1 dU = A Z1 Z1 1 dz ‚¥«¨ç¨­ 

Z0 = Z 1 = p Z1 = Z1 Y1 37

rZ

1

Y1

ï¥âáï ¢®«­®¢ë¬ ᮯà®â¨¢«¥­¨¥¬, â ª ª ª à ¢­  ®â­®è¥­¨î ­ ¯à殮­¨ï ª ⮪㠢 ¡¥£ã饩 ¢®«­¥. …᫨ «¨­¨ï ­¥ ¨¬¥¥â ¯®â¥àì, â® Z1 = j!L1 ; Y1 = j!C1 ; ®âªã¤  á«¥¤ã¥â, çâ® p p

= ;!2 L1C1 = j! L1C1 = jk: ‘ª®à®áâì à á¯à®áâà ­¥­¨ï v = !k = p 1 ; L1 C1   ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ ¢¥é¥á⢥­­®: rL Z0 = C1 : 1  ¯à殮­¨¥ ¨ ⮪ ¬®£ãâ ¡ëâì ¯à¥¤áâ ¢«¥­ë ᮮ⢥âá⢥­­® ¢ ¢¨¤¥ áã¬¬ë ¨ à §­®á⨠¯àאַ© ¨ ®¡à â­®© ¢®«­ U = U¯à + U®¡à; I = I¯à ; I®¡à = Z1 (U¯à ; U®¡à ); 0 ¯à¨ç¥¬ ¢ «¨­¨¨ ¡¥§ ¯®â¥àì Z0 { ¢¥é¥á⢥­­ ï ¢¥«¨ç¨­ . ®«ì§ãïáì í⨬, ¬®¦­® ­ ©â¨ ¬®é­®áâì, à á¯à®áâà ­ïîéãîáï ¢ ¯®«®¦¨â¥«ì­®¬ ­ ¯à ¢«¥­¨¨:  ; U ) = P = 12 Re (U I  ) = 12 Re Z1 (U¯à + U®¡à ) (U¯à ®¡à 0  ; U®¡à U  + U¯àU  ; U  U®¡à ) = = 21 Z1 Re (U¯àU¯à ®¡à ®¡à ¯à 0 2 2 = 12 jUZ¯àj ; 21 jUZ®¡àj = P¯à ; P®¡à ; 0 0 â.¥. ¬®é­®áâì à ¢­  à §­®á⨠¬®é­®á⥩ ¯àאַ© ¨ ®¡à â­®© ¢®«­. â® ¤ ¥â ®á­®¢ ­¨¥ ¯à¨¯¨á âì ª ¦¤®© ¨§ ¢®«­ ¬®é­®áâì 2 P = 12 jUZ j 0 ¨ à áᬠâਢ âì ¤ ­­ë¥ ¢®«­ë ª ª áãé¥áâ¢ãî騥 ­¥§ ¢¨á¨¬®. â® á¯à ¢¥¤«¨¢®, «¨èì ¥á«¨ Z0 { ¢¥é¥á⢥­­ ï ¢¥«¨ç¨­  (­ ¯à¨¬¥à, ¢ «¨­¨¨ ¡¥§ ¯®â¥àì ¨«¨ á ¬ «ë¬¨ ¯®â¥àﬨ). 38

2.7.

Žâà ¦¥­¨¥ ®â ­ £à㧪¨

‘®®â­®è¥­¨¥ ¬¥¦¤ã ¯àאַ© ¨ ®¡à â­®© ¢®«­ ¬¨ ®¯à¥¤¥«ï¥âáï £à ­¨ç­ë¬¨ ãá«®¢¨ï¬¨. ãáâì ¢ ­ ç «¥ «¨­¨¨ ¯à¨á®¥¤¨­¥­ £¥­¥à â®à, ¢®§¡ã¦¤ î騩 ¯àï¬ãî ¢®«­ã (á¬. à¨á.). …᫨ «¨­¨î ®£à ­¨ç¨âì ¨ ­ £à㧨âì ¢®«­®¢ë¬ ᮯà®â¨¢«¥­¨¥¬, U â® ®ç¥¢¨¤­®, çâ® ¢ «¨­¨¨ ¡ã¤¥â áãé¥á⢮¢ âì ⮫쪮 ¡¥£ãé ï ¢®«I ­ . …᫨ ¦¥ ­ £à㧪  ®â«¨ç­  ®â Z Z0 , â® ¢®§­¨ª ¥â ®âà ¦¥­­ ï ¢®«­ , â ª ª ª ¨­ ç¥ ­¥¢®§¬®¦­® 㤮¢«¥â¢®à¨âì £à ­¨ç­ë¬ ãá«®¢¨ï¬. „¥©á⢨⥫쭮, U=I = Z ­  £à ­¨æ¥. ®¤áâ ¢«ïï U = U¯à + U®¡à ¨ I = Z10 (U¯à ; U®¡à ), ¯®«ã稬: + U®¡à = Z Z0
UU¯à ; ¯à U®¡à ¨«¨ U¯à + U®¡à = Z : U¯à ; U®¡à Z0 Žâá, ®¡®§­ ç ï UU®¡à = ; (; { ª®íää¨æ¨¥­â ®âà ¦¥­¨ï), ¯®«ã稬 ¯à 1+; = Z 1;; Z0 ¨«¨, à¥è ï ®â­®á¨â¥«ì­® ª®íää¨æ¨¥­â  ®âà ¦¥­¨ï ;, ; Z0 : ; = ZZ + Z0  áᬮâਬ à á¯à¥¤¥«¥­¨¥ ­ ¯à殮­¨ï ¨ ⮪  ¢¤®«ì «¨­¨¨ ¯à¨ ­ «¨ç¨¨ ®âà ¦¥­­®© ¢®«­ë:

U = U¯à e;jkz + U®¡à e jkz ;

£¤¥ U¯à ¨ U®¡à { ­ ¯à殮­¨ï ¢ ­ ç «¥ «¨­¨¨ (ã £¥­¥à â®à ) (á¬. à¨á.). „ ­­®¥ ᮮ⭮襭¨¥ ¬®¦­® ¯à¥®¡à §®¢ âì: 39

U = e;jkz (U¯à + U®¡à e 2jkz ) : U U

à¨ ¯¥à¥¬¥é¥­¨¨ ¢¤®«ì «¨­¨¨ ®â £¥­¥à â®à  ¢¥ªâ®à U®¡à e 2jkz ¬¥­ï¥â ᢮î ä §ã. à¨ í⮬ ¢ ­¥ª®â®àëå â®çª å ä §ë U¯à ¨ U®¡à ᮢ¯ ¤ îâ, ¨ ®­¨ ᪫ ¤ë¢ îâáï. ‚ íâ¨å â®çª å ­ ¯à殮­¨¥ ¬ ªá¨¬ «ì­®:

U¬ ªá = jU¯àj + jU®¡à j : —¥à¥§ ç¥â¢¥àâì ¢®«­ë, â.¥. ç¥à¥§ z = =4, ­ ¯à殮­¨ï ¢ëç¨â îâáï:

U¬¨­ = jU¯àj ; jU®¡à j: U

z λ 2

’ ª¨¥ ¢®«­ë ­ §ë¢ îâ áâ®ï稬¨ (á¬. à¨á.). ‚¥«¨ç¨­ 

jU®¡àj = 1 + j;j  = jjUU¯à jj + 1 ; j;j ¯à ; jU®¡àj ­ §ë¢ ¥âáï ª®íää¨æ¨¥­â®¬ áâ®ï祩 ¢®«­ë ­ ¯à殮­¨ï (Š‘‚), å à ªâ¥à¨§ã騬 áâ®ïç¥áâì ¢®«­ë. à¨ ç¨áâ® ¡¥£ã饩 ¢®«­¥  = 1,   ¯à¨ áâ®ï祩 ¢®«­¥  = 1. ¥âà㤭® ¢¨¤¥âì, ç⮠⮪ ¢ «¨­¨¨ á® áâ®ï祩 ¢®«­®© ­ ¯à殮­¨ï â ª¦¥ ®¡à §ã¥â áâ®ïçãî ¢®«­ã, ­® ¬ ªá¨¬ã¬ ⮪  ᮮ⢥âáâ¢ã¥â ¬¨­¨¬ã¬ã ­ ¯à殮­¨ï, ¨ ­ ®¡®à®â. 40

2.8.

’à ­áä®à¬ æ¨ï ᮯà®â¨¢«¥­¨© ¨ ¯à®¢®¤¨¬®á⥩

à¥¤¯®«®¦¨¬, çâ® ¬ë ¨¬¥¥¬ ®â१®ª «¨­¨¨ ¤«¨­®© l, ­ £à㦥­­ë© ­  ª®­æ¥ «¨­¨¨ ­¥ª®â®àë¬ á®¯à®â¨¢«¥­¨¥¬ Z . ‚ ¯à®¨§¢®«ì­®¬ á¥ç¥­¨¨ z ⮪ ¨ ­ ¯à殮­¨¥ ¬®£ãâ ¡ëâì § ¯¨á ­ë ¢ ¢¨¤¥

U (z ) = Ae;jkz + B e jkz ; I (z ) = Z1
(Ae;jkz ; B e jkz ): 0 „«ï ®¯à¥¤¥«¥­¨ï A ¨ B ¢®á¯®«ì§ã¥¬áï ãá«®¢¨ï¬¨ ­  ª®­æ¥ «¨­¨¨ (¯à¨ z = l) U (l) = Ae;jkl + B e jkl ; I (l) = Z1
(Ae;jkl ; B e jkl ): 0 ¥è ï í⨠ãà ¢­¥­¨ï ®â­®á¨â¥«ì­® A ¨ B, ¯®«ã稬:

A = U (l) + 2Z0
I (l)
e jkl ; B = U (l) ; Z0
I (l)
e;jkl :

2 ®«ã祭­ë¬¨ १ã«ìâ â ¬¨ ¬®¦­® ¢®á¯®«ì§®¢ âìáï ¤«ï ®¯à¥¤¥«¥­¨ï ­ ¯à殮­¨ï ¨ ⮪  ¢ ­ ç «¥ «¨­¨¨ (z = 0):

U (0) = A + B = U (l) cos kl + jZ0
I (l) sin kl; I (0) = Z1
(A ; B ) = Zj U (l) sin kl + I (l) cos kl: 0 0 „ ­­ë¥ ᮮ⭮襭¨ï ¯®§¢®«ïîâ à¥è¨âì § ¤ çã ® ¢å®¤­®¬ ᮯà®â¨¢«¥­¨¨ ®â१ª  «¨­¨¨, ­ £à㦥­­®£® ᮯà®â¨¢«¥­¨¥¬ Z (l): (0) = U (l) cos kl + jZ0
I (l) sin kl = Z (0) = UI (0) j Z0 U (l) sin kl + I (l) cos kl kl : = Z0
ZZ (l)++jZjZ(l0) tg tg kl 0 41

ˆâ ª,

kl : Z (0) = Z0 ZZ (l)++jZjZ(l0) tg tg kl 0 2.8.1. Š®à®âª®§ ¬ª­ãâ ï «¨­¨ï

à¨ ª®à®âª®¬ § ¬ëª ­¨¨ «¨­¨¨ Z (l) = Z­ = 0. ‘®£« á­® ä®à¬ã«¥ âà ­áä®à¬ æ¨¨ ¢å®¤­®¥ ᮯà®â¨¢«¥­¨¥ ®â१ª  à ¢­®

Z¢å = jZ0
tg kl = jX¢å ; ;0 = ; 1: ˆ§¬¥­¥­¨¥ ¢å®¤­®£® ᮯà®â¨¢«¥­¨ï ¯à¥¤áâ ¢«¥­® ­  à¨á. X

3λ 4

λ 2

ZH= 0

l λ 4

|U|

|I|

l

‚ § ¢¨á¨¬®á⨠®â ¤«¨­ë ®â१ª  ¥£® ᮯà®â¨¢«¥­¨¥ ¬¥­ï¥âáï ®â 0 ¤® 1. ®í⮬㠢¡«¨§¨ ¤«¨­ë (2n + 1)=4 ®­ ¬®¦¥â ¡ëâì 㯮¤®¡«¥­ ¯ à ««¥«ì­®¬ã १®­ ­á­®¬ã ª®­âãàã. Œ ªá¨¬ã¬ë ­ ¯à殮­¨ï ¨ ⮪  ᤢ¨­ãâë ­  =4.   ®â१ª¥ ¤«¨­®© n=4 § ¯ áë í«¥ªâà¨ç¥áª®© ¨ ¬ £­¨â­®© í­¥à£¨© à ¢­ë ¤à㣠¤àã£ã. Žâ१ª¨ «¨­¨¨ ¤«¨­®© =4, § ¬ª­ãâë¥ ­  ª®­æ¥, ¯à¨¬¥­ïîâáï ¢ ª ç¥á⢥ ç¥â¢¥àâ좮«­®¢ëå ¨§®«ïâ®à®¢. 42

2.8.2.  §®¬ª­ãâ ï «¨­¨ï

„«ï à §®¬ª­ã⮩ «¨­¨¨ Z­ = 1 (á¬. à¨á.). ‚室­®¥ ᮯà®â¨¢«¥­¨¥ ®â१ª  à ¢­®

Z¢å = ; jZ0 ctg kl; ;1 = + 1: X

l λ 4

|I|

ZH=

λ 2

8

3λ 4

|U|

l

2.8.3. ‹¨­¨ï, § ¬ª­ãâ ï ­   ªâ¨¢­®¥ ᮯà®â¨¢«¥­¨¥

‚ í⮬ á«ãç ¥ Z­ = R­ (á¬. à¨á. ­  á«¥¤. áâà.): ­ + jZ0 tg kl Z¢å = Z0
R Z0 + jR­ tg kl : ‡¤¥áì ¯à¨ l = 2n=4 Z¢å = R­ ; ¯à¨ l = (2n + 1)=4 Z¢å = Z02 =R­ . Š à⨭  áâ®ïç¨å ¢®«­ § ¢¨á¨â ®â ⮣®, ¡®«ìè¥ R­ 祬 Z0 ¨«¨ ¬¥­ìè¥. ‚ â®çª å ¬ ªá¨¬ã¬®¢ ¨ ¬¨­¨¬ã¬®¢ ¢å®¤­®¥ ᮯà®â¨¢«¥­¨¥ ï¥âáï ç¨áâ®  ªâ¨¢­ë¬ ¨ ¢ § ¢¨á¨¬®á⨠®â à ááâ®ï­¨ï ®â ­ £à㧪¨ à ¢­® R­ ¨«¨ Z02=R­ . 43

Z0

R

R

Z0

R

R |U| R > Z0 Γ> 0 |I|

Z02

R

R

Z02

R

R

|I| R
0

Γ< 0 |U|

2.8.4. ‹¨­¨ï, § ¬ª­ãâ ï ­  ॠªâ¨¢­®¥ ᮯà®â¨¢«¥­¨¥

‚ í⮬ á«ãç ¥ Z­ = jX­ (á¬. à¨á. ­  á«¥¤. áâà.). ‘®£« á­® âà ­áä®à¬ æ¨®­­®© ä®à¬ã«¥ ¢å®¤­®¥ ᮯà®â¨¢«¥­¨¥ à ¢­® ­ + Z0 tg kl Z¢å = jZ0
X Z0 ; X­ tg kl : â® ¢ëà ¦¥­¨¥ ¬®¦¥â ¡ëâì ¯à¥®¡à §®¢ ­® á«¥¤ãî騬 ®¡à §®¬:

X­ + tg kl Z¢å = jZ0
Z0 X = jZ0 tg(kl + ') ; 1 ; Z ­
tg kl 0





  ­ £¤¥ ' = arctg X Z0 , ; 2  '  2 .

’ ª¨¬ ®¡à §®¬, १ã«ìâ â ¯®«ãç ¥âáï ¯ã⥬ ᬥ饭¨ï ¢á¥© ª àâ¨­ë ¯à¨ ª®à®âª®¬ § ¬ëª ­¨¨ ­  㣮« '. ¥âà㤭® ¢¨¤¥âì, çâ® ¢ í⮬ á«ãç ¥ ¢å®¤­®¥ ᮯà®â¨¢«¥­¨¥ ®¡à é ¥âáï ¢ 1 ¨ 0 ¢ â®çª å, ®âáâ®ïé¨å ¤à㣠®â ¤à㣠 ­  =4. ®«®¦¥­¨¥ ¡«¨¦ ©è¥£® ª ­ £à㧪¥ ¬¨­¨¬ã¬  § ¢¨á¨â ®â §­ ª  X­ . …᫨ X­ { ¯®«®¦¨â¥«ì­ ï ¢¥«¨ç¨­ , â® ¡«¨¦ ©è¨© ¬¨­¨¬ã¬ ®âá⮨⠮⠭ £à㧪¨ ­  à ááâ®ï­¨¥, ¡®«ì襥 =4, ­® ¬¥­ì襥 =2; ¥á«¨ 44

¦¥ X­ ®âà¨æ â¥«ì­®, â® ¬¨­¨¬ã¬ ®âá⮨⠮⠭ £à㧪¨ ­  à ááâ®ï­¨¥, ¬¥­ì襥 =4.

|U|

l

|I|

|I|

l

2.9.

X>0 |Γ| = 1

X<0 |Γ| = 1

|U|

’à ­áä®à¬ æ¨ï ᮯà®â¨¢«¥­¨© ®â१ª ¬¨ «¨­¨¨

‘ â®çª¨ §à¥­¨ï âà ­áä®à¬ æ¨¨ ᮯà®â¨¢«¥­¨ï ­ ¨¡®«ì訩 ¨­â¥à¥á ¯à¥¤áâ ¢«ïîâ ¯®«ã¢®«­®¢ë© ¨ ç¥â¢¥àâ좮«­®¢ë© ®â१ª¨ «¨­¨¨. 2.9.1. ®«ã¢®«­®¢ë© ®â१®ª

„«ï ¯®«ã¢®«­®¢®£® ®â१ª  kl = , tg kl = 0. ®í⮬ã

Z¢å = Z­: ‘«¥¤®¢ â¥«ì­®, ¯®«ã¢®«­®¢ë© ®â१®ª ¯à¥®¡à §ã¥â ᮯà®â¨¢«¥­¨¥ á ª®íää¨æ¨¥­â®¬ âà ­áä®à¬ æ¨¨ 1. â® ᢮©á⢮ ç áâ® ¨á¯®«ì§ã¥âáï ¢ â¥å­¨ª¥. 2.9.2. —¥â¢¥àâ좮«­®¢ë© ®â१®ª

‚ í⮬ á«ãç ¥ kl = 2 , tg kl = 1. ®í⮬ã 2 Z¢å = ZZ0 : ­

45

—¥â¢¥àâ좮«­®¢ë© ®â¥§®ª ¯à¥®¡à §ã¥â ¨­¤ãªâ¨¢­®áâì ¢ ¥¬ª®áâì ¨ ­ ®¡®à®â, १®­ ­á { ¢  ­â¨à¥§®­ ­á ¨ ®¡à â­®. —¥â¢¥àâ좮«­®¢ë© ®â१®ª ¨á¯®«ì§ã¥âáï ¤«ï ᮣ« á®¢ ­¨ï «¨­¨© á à §«¨ç­ë¬ ¢®«­®¢ë¬ ᮯà®â¨¢«¥­¨¥¬. ’ ª®¥ ¯à¨¬¥­¥­¨¥ ®á­®¢ ­® ­  ⮬, çâ® ¥á«¨ ¢ª«îç¨âì ­  ª®­æ¥ ç¥â¢¥àâ좮«­®¢®£® ®â१ª   ªâ¨¢­®¥ ᮯà®â¨¢«¥­¨¥, â® ¢å®¤­®¥ ᮯà®â¨¢«¥­¨¥ â ª¦¥ ¡ã¤¥â  ªâ¨¢­ë¬.  áᬮâਬ â ª®© ¯à¨¬¥à. ¥®¡å®¤¨¬® ᮣ« á®¢ âì «¨­¨¨ á ¢®«­®¢ë¬¨ ᮯà®â¨¢«¥­¨ï¬¨ Z01 ¨ Z02 (á¬. à¨á.).  ©¤¥¬ ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ ç¥â¢¥àâ좮«­®¢®£® ®â१ª  Z03 , ᮣ« áãî饣® í⨠«¨­¨¨ ¨§ ãá«®¢¨ï Z032 = Z : 01 Z02

λ/4 Z

01

Z

03

Z

02

Žâá

Z03 =

p

Z01
Z02 :

”¨§¨ç¥áª¨ ᮣ« á®¢ ­¨¥ ¬®¦­® ®¡êïá­¨âì ⥬, çâ® ¢®«­ë, ®âà ¦¥­­ë¥ ®â ¤¢ãå áâ몮¢ (á¬. à¨á.), ¢® ¢å®¤­®© «¨­¨¨ ®ª §ë¢ îâáï ¢ ¯à®â¨¢®ä §¥, ¨ ¥á«¨  ¬¯«¨âã¤ë à ¢­ë, â® ®­¨ ã­¨ç⮦ îâ 11111111111111 ¨å 00000000000000 ¤à㣠¤à㣠. Žâ¬¥â¨¬, çâ® â λ/4 ª®¥ ᮣ« á®¢ ­¨¥ ¡ã¤¥â ¨¬¥âì ¬¥áâ® «¨èì ¢ áà ¢­¨â¥«ì­® 㧪®© ¯®«®á¥ ç áâ®â. „«ï è¨à®ª®¯®«®á­®£® ᮣ« á®¢ ­¨ï ¯à¨¬¥­ïîâ ¬­®£®áâ㯥­ç âë¥ ¯¥à¥å®¤ë, á®áâ ¢«¥­­ë¥ ¨§ ç¥â¢¥àâ좮«­®¢ëå ®â१ª®¢.

11111111111111 00000000000000

46

2.10.

à¥¤áâ ¢«¥­¨¥ ®â१ª  «¨­¨¨ T-®¡à §­ë¬ ç¥âëà¥å¯®«îá­¨ª®¬

Z1

Žâ१®ª «¨­¨¨ ¯à¥¤áâ ¢«ï¥â ᮡ®© ç¥âëà¥å¯®«îá­¨ª. „«ï à áç¥â®¢ ç áâ® ¡ë¢ ¥â 㤮¡­® ¥£® § ¬¥­¨âì T-®¡à §­ë¬ ç¥âëà¥å¯®«îá­¨ª®¬ á á®á।®â®ç¥­­ë¬¨ ¯ à ¬¥âà ¬¨ (á¬. à¨á.). „«ï í⮣® ¢ëç¨á«¨¬ ¢å®¤­®¥ ᮯà®â¨¢«¥­¨¥ â ª®£® ç¥âëà¥å¯®«îá­¨ª :

Z1

Z2

ZH

Z­ + Z1 ZZ11++2ZZ22 Z ( Z + Z ) 2 1 ­ Z¢å = Z1 + Z + Z + Z = : 1 2 ­ 1 + Z Z+­ Z 1 2 ‘à ¢­¨¬ íâ® ¢ëà ¦¥­¨¥ á ä®à¬ã«®© ¤«ï ¢å®¤­®£® ᮯà®â¨¢«¥­¨ï ®â१ª  «¨­¨¨, ­ £à㦥­­®£® ­  ᮯà®â¨¢«¥­¨¥ Z­ : Z¢å = Z­ +j jZ0 tg kl : 1 + Z
Z­ tg kl 0 „ ­­ë¥ ¢ëà ¦¥­¨ï ᮢ¯ ¤ îâ, ¥á«¨

jZ0 tg kl = Z1 ZZ1 ++ 2ZZ2 ; 1 2 1 : j tg kl = Z0 Z1 + Z2 ®«ã祭­ë¥ ãà ¢­¥­¨ï ¬®£ãâ ¡ëâì à¥è¥­ë ®â­®á¨â¥«ì­® Z1 ¨ Z2 (à¥è¥­¨¥ ¤¢ã§­ ç­®):

Z1 = jZ0 tg kl

¯à¨ kl 6= (2n + 1);

Z2 = ; jZ0 csc kl 47

Z1 = ; jZ0 ctg kl

¯à¨ kl 6= 2n:

Z2 = jZ0 csc kl

‘«¥¤ã¥â ¨¬¥âì ¢ ¢¨¤ã, çâ® í⨠ᮮ⭮襭¨ï ®¡¥á¯¥ç¨¢ îâ íª¢¨¢ «¥­â­®áâì «¨èì ­  ®¤­®© ç áâ®â¥, â ª ª ª ॠ«ì­ë¥ á®á।®â®ç¥­­ë¥ ᮯà®â¨¢«¥­¨ï ­¥ ¬®£ãâ ¨§¬¥­ïâìáï ¢ § ¢¨á¨¬®á⨠®â ç áâ®âë ¯® 㪠§ ­­®¬ã § ª®­ã. 2.11.

®¬®£à ¬¬  ¯®«­ëå ᮯà®â¨¢«¥­¨© (¤¨ £à ¬¬  ‘¬¨â )

2.11.1. à¥®¡à §®¢ ­¨¥ ª®íää¨æ¨¥­â  ®âà ¦¥­¨ï ®â१ª®¬ «¨­¨¨

Š®íää¨æ¨¥­â ®âà ¦¥­¨ï à ¢¥­ ; = UU®âà : ¯ ¤ …᫨ ¯¥à¥©â¨ ­  à ááâ®ï­¨¥ l ¡«¨¦¥ ª £¥­¥à â®àã, â® ¢ í⮬ á¥ç¥­¨¨ ¡ã¤¥¬ ¨¬¥âì

U®âà (l) = U®âà e;jkl ; ¯®í⮬ã

U¯ ¤(l) = U¯ ¤ e jkl ;

;(l) = UU®âà e;2jkl = ; e;2jkl : ¯ ¤ Žâá ¢¨¤­®, çâ® j;j = const, â.¥. ­¥ § ¢¨á¨â ®â ¤«¨­ë ®â१ª ,   ä §  ®âá⠥⠭  2kl. 2.11.2. ®¬®£à ¬¬  ¯®«­ëå ᮯà®â¨¢«¥­¨©

„«ï ¯ áᨢ­ëå 楯¥© «î¡®¥ ª®¬¯«¥ªá­®¥ ᮯà®â¨¢«¥­¨¥ ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­® â®çª®©, «¥¦ é¥© ¢ ¯à ¢®© ¯®«ã¯«®áª®á⨠(á¬. à¨á. ­  á«¥¤. áâà.). 48

‚ª«î祭¨¥ ®â१ª  «¨­¨¨ ¯à¥®¡à §ã¥â ᮯà®â¨¢«¥­¨¥ ­ £à㧪¨ ¢ ᮯà®â¨¢«¥­¨¥ ¢å®¤  ¯® ¯à¨¢¥¤¥­­®© à ­¥¥ ä®à¬ã«¥, ª®â®à ï ¤®áâ â®ç­® á«®¦­ . ‘ ¤à㣮© áâ®à®­ë, ¯à¨ § ¤ ­­®¬ ¢®«­®¢®¬ ᮯà®â¨¢«¥­¨¨ ®â१ª  «¨­¨¨ ª ¦¤®¬ã ᮯà®â¨¢«¥­¨î Z­ ᮮ⢥âáâ¢ã¥â ®¯à¥¤¥«¥­­ë© ª®íää¨æ¨¥­â ®âà ¦¥­¨ï ;:

Z­ ; 1 1 Z ; Z ­ 0 = zz ; ; = Z + Z = ZZ0 + 1: ­ + 1 ­ 0 Z0 à¥®¡à §®¢ ­¨î ᮯà®â¨¢«¥­¨ï ᮮ⢥âáâ¢ã¥â ¯à¥®¡à §®¢ ­¨¥ ª®íää¨æ¨¥­â  ®âà ¦¥­¨ï. Ž¤­ ª® ¯à¥®¡à §®¢ ­¨¥ ª®íää¨æ¨¥­â  ®âà ¦¥­¨ï ¯à®¨§¢®¤¨âáï ¯® ¡®«¥¥ ¯à®áâë¬ ä®à¬ã« ¬. ®í⮬㠤«ï £à ä¨ç¥áª¨å ¯®áâ஥­¨© ¢ë£®¤­® ¯¥à¥©â¨ ¢ ¯«®áª®áâì ª®íää¨æ¨¥­â®¢ ®âà ¦¥­¨ï ¯® ¯à¨¢¥¤¥­­®© ¢ëè¥ ä®à¬ã«¥. ’ ª ª ª j;j  1, â® ª®íää¨æ¨¥­â ®âà ¦¥­¨ï «î¡®© ­ £à㧪¨ ¬®¦¥â ¡ëâì ¨§®¡à ¦¥­ â®çª®© ¢­ãâਠ®ªà㦭®á⨠¥¤¨­¨ç­®£® à ¤¨ãá . ‘®®â­®è¥­¨¥ ¬¥¦¤ã ; ¨ Z­ ¯à¥¤áâ ¢«ï¥â ᮡ®© ¤à®¡­®-«¨­¥©­®¥ ¯à¥®¡à §®¢ ­¨¥ ¯®«ã¯«®áª®á⨠R  0 ­  ªà㣠¥¤¨­¨ç­®£® à ¤¨ãá . â® ¯à¥®¡à §®¢ ­¨¥ ï¥âáï ª®­ä®à¬­ë¬ ¨ ¯à¥®¡à §ã¥â ¯àï¬ë¥ «¨­¨¨ ¢ ®ªà㦭®áâ¨. â¨ á®®¡à ¦¥­¨ï ¯®§¢®«ïîâ ¯®áâநâì «¨­¨¨ X = const ¨ R = const ¢ ¯«®áª®á⨠ª®íää¨æ¨¥­â®¢ ®âà ¦¥­¨ï.

49

X = const

to s ou

rce

X>0

-1

X=0

+1 Z=

8

Z=0

to loa d

R = const

0

R=

X<0

  ¯à¨¢¥¤¥­­®© ­®¬®£à ¬¬¥ ­ ­¥á¥­  á¥âª  «¨­¨© X = const ¨ R = const. ¥à¥á¥ç¥­¨¥ ¤¢ãå â ª¨å «¨­¨© ®¯à¥¤¥«ï¥â ᮯà®â¨¢«¥­¨¥ ­ £à㧪¨, ®â­¥á¥­­®¥ ª Z0 . ‚ â® ¦¥ ¢à¥¬ï ª®íää¨æ¨¥­â ®âà ¦¥­¨ï ¯® ¬®¤ã«î ¨ ä §¥ ®¯à¥¤¥«ï¥âáï à ¤¨ãá-¢¥ªâ®à®¬ â®çª¨ ¯¥à¥á¥ç¥­¨ï. à¨ ¯¥à¥å®¤¥ ¢ ¤à㣮¥ á¥ç¥­¨¥ à ¤¨ãá-¢¥ªâ®à ¯®¢®à ç¨¢ ¥âáï ­  㣮«

' = 2kl = 4 l :

…£® ª®­¥æ ®¯à¥¤¥«ï¥â ¯à¥®¡à §®¢ ­­®¥ ᮯà®â¨¢«¥­¨¥. ’ ª ï ­®¬®£à ¬¬  ¨«¨, ª ª ¥¥ ­ §ë¢ îâ, ¤¨ £à ¬¬  ‘¬¨â  (¨«¨ ªà㣮¢ ï ¤¨ £à ¬¬ ) è¨à®ª® ¯à¨¬¥­ï¥âáï ¯à¨ à áç¥â å á ¤«¨­­ë¬¨ «¨­¨ï¬¨. Žâ¬¥â¨¬, çâ® íâ  ¦¥ ¤¨ £à ¬¬  ¬®¦¥â ¡ëâì ¨á¯®«ì§®¢ ­  ¤«ï à áç¥â  ¯à®¢®¤¨¬®á⥩, ­® ¤«ï í⮣® ¤¨ £à ¬¬ã ­ã¦­® ¯®¢¥à­ãâì ­  180o, § ¬¥­¨¢ X ­  B ¨ R ­  G. ¥§ã«ìâ â ¯®«ãç ¥âáï ¢ ¢¨¤¥ ¯à®¢®¤¨¬®áâ¨, ®â­¥á¥­­®© ª ¢®«­®¢®© ¯®¢®¤¨¬®á⨠Y0 . 2.12.

Œ­®£®¯à®¢®¤­ë¥ «¨­¨¨ TEM

Šà®¬¥ «¨­¨©, á®áâ®ïé¨å ¨§ ¤¢ãå ¯à®¢®¤­¨ª®¢, ¯à¨¬¥­ïîâ â ª¦¥ «¨­¨¨, á®áâ®ï騥 ¨§ ¡®«ì襣® ç¨á«  ¯à®¢®¤­¨ª®¢ { ¬­®£®¯à®¢®¤­ë¥ «¨­¨¨. ‚ ç áâ­®áâ¨, â ª¨¥ «¨­¨¨ ¯à¨¬¥­ïîâ ¢ ­ ¯à ¢«¥­­ëå ®â¢¥â¢¨â¥«ïå. ’¥®à¨ï ¬­®£®¯à®¢®¤­ëå «¨­¨© ¨á¯®«ì§ã¥âáï â ª¦¥ ¯à¨ à áç¥â¥ ­¥ª®â®àëå 50

⨯®¢ § ¬¥¤«ïîé¨å á¨á⥬. à¨¬¥à á¥ç¥­¨ï ¬­®£®¯à®¢®¤­®© «¨­¨¨ ¯à¨¢¥¤¥­ ­  à¨á.  á¯à®áâà ­¥­¨¥ ¢®«­ ¢ ¬­®£®¯à®¢®¤­®© «¨­¨¨ ¬®¦¥â ¡ëâì ®¯¨á ­® 00 11 00 11 00 11 000 111 á ¯®¬®éìî ⥫¥£à ä­ëå ãà ¢­¥00 11 00 11 00 11 000 111 ­¨©. „«ï í⮣® ¢¢®¤ïâáï ¯®£®­­ë¥ 00 11 11 00 11 00 111 000 ¯ à ¬¥âàë á ¯®¬®éìî ᮮ⭮襭¨© (¢ áâ â¨ç¥áª®¬ ¯à¨¡«¨¦¥­¨¨)

Qi =

X k

Cik
Uk ; Us =

X m

Psm
Qm ;

£¤¥ Qi { «¨­¥©­ ï ¯«®â­®áâì § à鸞 ­  i-¬ ¯à®¢®¤­¨ª¥, Us { ¯®â¥­æ¨ « s-£® ¯à®¢®¤­¨ª , Cik { ¯®£®­­ë¥ ç áâ¨ç­ë¥ ¥¬ª®á⨠(®âà¨æ â¥«ì­ë¥ ¯à¨ i 6= k), Psm { ¯®£®­­ë¥ ¯®â¥­æ¨ «ì­ë¥ ª®íää¨æ¨¥­âë. à¨¢¥¤¥­­ë¥

ᮮ⭮襭¨ï ¬®£ãâ ¡ëâì § ¯¨á ­ë ¢ ¬ âà¨ç­®© ä®à¬¥:

Q

C
U ; U = P
Q ; £¤¥ Q = fQig; U = fUs g; C = fCik g; P = fPsm g { ᮮ⢥âáâ¢ãî騥 ¬ âà¨æë. Œ âà¨æë C ¨ P á¢ï§ ­ë ᮮ⭮襭¨¥¬ C
P = 1 ; £¤¥ 1 { ¥¤¨­¨ç­ ï ¬ âà¨æ , â.¥. C ¨ P { ¢§ ¨¬­® ®¡à â­ë¥ ¬ âà¨æë: P = C ;1 . =

‘ ¤à㣮© áâ®à®­ë, ¬®¦­® ¢¢¥á⨠⠪¦¥ ¬ âà¨æã ¯®£®­­ëå ¨­¤ãªâ¨¢­®á⥩ á ¯®¬®éìî ᮮ⭮襭¨ï i =

X k

Lik
Ik ;

£¤¥ i { ¬ £­¨â­ë© ¯®â®ª ­  ¥¤¨­¨æã ¤«¨­ë «¨­¨¨, á楯«¥­­ë© á i-¬ ¯à®¢®¤­¨ª®¬, Lik { ¯®£®­­ë¥ ¢§ ¨¬­ë¥ ¨­¤ãªâ¨¢­®áâ¨. Œ âà¨ç­ ï § ¯¨áì í⮣® ᮮ⭮襭¨ï ¨¬¥¥â ¢¨¤

 = L
I: ’¥«¥£à ä­ë¥ ãà ¢­¥­¨ï á®áâ ¢«ïîâáï â ª ¦¥, ª ª ¨ ¢ á«ãç ¥ ¤¢ãå¯à®¢®¤­®© «¨­¨¨: 51

dU = ; j!L
I ; dz dI = ; j!C
U dz (§ ¯¨áì ¢ ¬ âà¨ç­®© ä®à¬¥). „«ï à¥è¥­¨ï ¨áª«îç ¥¬ ⮪ ¨§ ¯¥à¢®£® ãà ¢­¥­¨ï: d2 U = ; j!L
dI = ; !2 LC
U ; dz 2 dz

£¤¥ LC { ¬ âà¨æ  (¯à®¨§¢¥¤¥­¨¥ ¬ âà¨æ L ¨ C ). ‚ëç¨á«¨¬ ¬ âà¨æã LC , ­ ©¤ï x i á¢ï§ì ¬¥¦¤ã ¬ âà¨æ ¬¨ L ¨ P . k „«ï í⮣® à áᬮâਬ ¤¢  ¯à®¢®¤000 111 000 111 000 111 111 000 ­¨ª  á ­®¬¥à ¬¨ i ¨ k ¯à¨ i 6= k 0a 1 (á¬. à¨á.). 0 1  ©¤¥¬ ¯®â¥­æ¨ « k-£® ¯à®¢®¤­¨0 1 111111111111 000000000000 000000000000 111111111111 ¥á«¨ ¯®â¥­æ¨ « i-£® ¯à®¢®¤­¨000000000000 111111111111 y ª , ª  à ¢¥­ Ui :

000000000000 111111111111

Uki =

Za

; Eix
dx;

0 £¤¥ Eix { x-ï ª®¬¯®­¥­â  í«¥ªâà¨ç¥áª®£® ¯®«ï, ᮧ¤ ¢ ¥¬®£® i-¬ ¯à®¢®¤­¨ª®¬. ‚¥«¨ç¨­ã Eix ¬®¦­® ®¯à¥¤¥«¨âì ¨§ ãà ¢­¥­¨ï Œ ªá¢¥««  Eix = 1 rot x H~ i ;

j!"

£¤¥ H~ i { ¬ £­¨â­®¥ ¯®«¥, ᮧ¤ ¢ ¥¬®¥ i-¬ ¯à®¢®¤­¨ª®¬. ’ ª ª ª Hz = 0, â® rot H~ = ; @Hiy ¨ E = ; 1 @Hiy : x i

ix

@z

’ ª¨¬ ®¡à §®¬, 52

j!" @z

1 @Hiy dx = 1 @ Z H dx: j!" @z j!" @z iy 0 0 R ® ; Hiy dx = ik { ¯®â®ª, ᮧ¤ ¢ ¥¬ë© i-¬ ¯à®¢®¤­¨ª®¬, ®å¢ â뢠î騩 k-© ¯à®¢®¤­¨ª. â®â ¯®â®ª ik = Lki Ii , ¯®í⮬ã U i = ; 1 L @Ii :

Uki =

Za

a

k

j!" ki @z

i à®¨§¢®¤­ãî @I @z ¬®¦­® ¢ëç¨á«¨âì á ¯®¬®éìî ãà ¢­¥­¨ï ­¥¯à¥à뢭®áâ¨, ª®â®à®¥ ¢ í⮬ á«ãç ¥ ¨¬¥¥â ¢¨¤

@Q + @I = 0: @t @z

Žâá

@Ii = ; j!Q : i @z ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï Uki , ¯®«ã稬 1 L
(;j!Q ) = Lki
Q : Uki = ; j!" ki i " i ‚¥«¨ç¨­  " = v12 , £¤¥ v { ᪮à®áâì à á¯à®áâà ­¥­¨ï ¯®¯¥à¥ç­ëå ¢®«­ ¢ «¨­¨¨. ˆâ ª, ¨¬¥¥¬ Uki = v2
Lki
Qi : ‘à ¢­¨¢ ï á ®¯à¥¤¥«¥­¨¥¬ í«¥¬¥­â®¢ ¬ âà¨æë P , ­ å®¤¨¬ Pki = v2
Lki ¯à¨ i 6= k:

€­ «®£¨ç­ë¥ à áá㦤¥­¨ï ¯à¨¢®¤ïâ ª â ª®¬ã ¦¥ ᮮ⭮襭¨î ¯à¨

i = k. ®í⮬ã

P

= v2 L ; 53

L

= vP2 :

®¤áâ ¢«ïï íâ® ¢ëà ¦¥­¨¥ ¢ ¯®«ã祭­®¥ ¢ëè¥ ¤¨ää¥à¥­æ¨ «ì­®¥ ãà ¢­¥­¨¥, ­ å®¤¨¬ d2 U = ;  ! 2 PC
U = ; k2 U : dz 2 v “à ¢­¥­¨¥ ¯à¨®¡à¥â ¥â ¢¨¤ d2 U + k 2 U = 0 : dz 2 â® ãà ¢­¥­¨¥ ¯à¥¤áâ ¢«ï¥â ᮡ®© ªà âªãî § ¯¨áì ¯®á«¥¤®¢ â¥«ì­®á⨠ãà ¢­¥­¨© ¤«ï Ui , ¯à¨ç¥¬ ¢ ª ¦¤®¥ ãà ¢­¥­¨¥ ¢å®¤¨â «¨èì ®¤¨­ ¯®â¥­æ¨ « Ui , â.¥. ¯¥à¥¬¥­­ë¥ à §¤¥«ïîâáï. ¥è¥­¨¥ íâ¨å ãà ¢­¥­¨© ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¬ âà¨ç­®© ä®à¬¥

U

=

Ae;jkz

+

Be jkz ;

£¤¥ U , A, B { ᮮ⢥âáâ¢ãî騥 ¬ âà¨æë. „«ï ⮪  ¨§ ¯à¥¤ë¤ãé¨å ãà ¢­¥­¨© ¨¬¥¥¬ I = ; 1 L;1 dU = Z ;1 ;Ae;jkz ; Be jkz
¨«¨

j!

0

dz

Z0 I

Ae;jkz ; Be jkz : ‡¤¥áì ¢¢¥¤¥­® ®¡®§­ ç¥­¨¥ !k
L = Z0 (Z0 { ¢¥é¥á⢥­­ ï ¬ âà¨æ , =

¨¬¥îé ï á¬ëá« ¬ âà¨æë ¢®«­®¢®£® ᮯà®â¨¢«¥­¨ï). ‚®á¯®«ì§ã¥¬áï ¯®«ã祭­ë¬¨ ᮮ⭮襭¨ï¬¨ ¤«ï ¢ëïá­¥­¨ï, ª ª ¯à¥®¡à §ãîâáï ­ ¯à殮­¨¥ ¨ ⮪ ¢ ¬­®£®¯à®¢®¤­®© «¨­¨¨. ®« £ ï z = 0, ­ ©¤¥¬ ­ ¯à殮­¨ï ¨ ⮪ ¢ ­ ç «¥ «¨­¨¨:

U (0) Z0I (0) „«ï ®¯à¥¤¥«¥­¨ï A ¨

z = l:

= =

A + B; A ; B:

B § ¯¨è¥¬ ãá«®¢¨ï ¢ ª®­æ¥ «¨­¨¨, â.¥. 54

¯à¨

U (l) Z0I (l)

= =

Ae;jkl + Be jkl ; Ae;jkl ; Be jkl ;

A B

U (l) + Z0I (l) ; 2 U ( l ) ; Z0 I ( l ) : ; jkl e

®âªã¤  = e jkl =

2

®¤áâ ¢«ïï A ¨ B ¢ ¢ëà ¦¥­¨ï ¤«ï U (0) ¨ I (0), ­ å®¤¨¬:

U (0) I (0)

= U (l) cos kl + j Z0 I (l) sin kl; = j Z0 ;1 U (l) sin kl + I (l) cos kl:

’ ª¨¬ ®¡à §®¬, ¬ë ¯®«ã稫¨ ãà ¢­¥­¨ï,  ­ «®£¨ç­ë¥ ãà ¢­¥­¨ï¬ ¤«ï ¤¢ãå¯à®¢®¤­®© «¨­¨¨, § ¯¨á ­­ë¥ ¢ ¬ âà¨ç­®© ä®à¬¥. Žâ¬¥â¨¬, çâ® ¢ íâ¨å ãà ¢­¥­¨ïå à §¤¥«¥­¨¥ ¯¥à¥¬¥­­ëå 㦥 ­¥ ¨¬¥¥â ¬¥áâ  ¢ ®â«¨ç¨¥ ®â ¨á室­®£® ¤¨ää¥à¥­æ¨ «ì­®£® ãà ¢­¥­¨ï.  £à㧪  ¬­®£®¯à®¢®¤­®© «¨­¨¨ ®¯à¥¤¥«ï¥âáï ¬ âà¨æ¥© ­ £à㧮筮£® ᮯà®â¨¢«¥­¨ï, á¢ï§ë¢ î饣® ¬¥¦¤ã ᮡ®© ¯®â¥­æ¨ «ë ¯à®¢®¤®¢ ­  ¢ë室¥ ¨ ⮪¨, ¯à®â¥ª î騥 ¢ íâ¨å ¯à®¢®¤ å. 2.13.

‹¨­¨ï á ¬ «ë¬¨ ¯®â¥àﬨ

 ¨¡®«¥¥ ¯à®áâ® ¢ëïá­¨âì ¢«¨ï­¨¥ ¬ «ëå ¯®â¥àì ­  à á¯à®áâà ­¥­¨¥ ¢®«­ á ¯®¬®éìî ⥫¥£à ä­ëå ãà ¢­¥­¨©. ®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï ¢ í⮬ á«ãç ¥ ¡ã¤¥â à ¢­ 

= =

p p Z1
Y1 = (j!L1 + R1 )(j!C1 + G1 ) s    R1 G1 ;!2L1 C1
1 + j!L 1

p

= j! L1 C1

s

R1 1 + j!L 1 55

1 + j!C 1



=



G1 : 1 + j!C 1

=

…᫨ R1  !L1 ¨ G1  !C1 , â® ¯à¨¡«¨¦¥­­®

p

(





1 R1 + G1 + 1

 j! L1 C1 1 + 2j! L1 C1 8!2 (  R1 G1 2) 1  R1 1 = jk 1 + 8!2 L ; C + 2 Z + 1 1 0

r

r

 R1

G1 2 ; L1 C1

G1 Y0



)

=

= jk0 + ;

L1 , Y0 = C1 { ¢®«­®¢ë¥ ᮯà®â¨¢«¥­¨¥ ¨ ¯à®¢®¤¨¬®áâì £¤¥ Z0 = C L1 1 «¨­¨¨ ¡¥§ ¯®â¥àì. ‡¤¥áì ᪮à®áâì à á¯à®áâà ­¥­¨ï (v = !=k0) § ¢¨á¨â ®â ç áâ®âë, â.¥. ¨¬¥¥â ¬¥áâ® ¤¨á¯¥àá¨ï. Šà®¬¥ ⮣®, ¯®ï¢«ï¥âáï § âãå ­¨¥, ᮮ⢥âáâ¢ãî饥 ¢¥é¥á⢥­­®© ç á⨠¯®áâ®ï­­®© à á¯à®áâà ­¥­¨ï :   U = U0 e;z
e;jk0 z ;  = 12 RZ1 + GY 1 : 0 0  «¨ç¨¥ ¤¨á¯¥àᨨ ¯à¨¢®¤¨â ª ¯®ï¢«¥­¨î ¨áª ¦¥­¨© ¨¬¯ã«ìᮢ ¢ «¨­¨¨. â¨ ¨áª ¦¥­¨ï ¬®£ãâ ¡ëâì ®á®¡¥­­® ᨫì­ë¬¨ ¢ ¤«¨­­ëå «¨­¨ïå. ˆáª ¦¥­¨© ­¥â ¢ ç áâ­®¬ á«ãç ¥, ª®£¤  R1 = G1 : L1 C1 ‚ í⮬ á«ãç ¥



p R1

= j! L1 C1 1 + j!L 1



1 = jk + R Z0 : ’ ª ª ª ¢ ⥫¥£à ä­ëå ª ¡¥«ïå ¯à¥®¡« ¤ îâ ¯®â¥à¨ §  áç¥â ᮯà®â¨¢«¥­¨ï ¯à®¢®¤®¢ (R1 ), â® ¤«ï 㬥­ì襭¨ï ¨áª ¦¥­¨© ­  ­¨§ª¨å ç áâ®â å 㢥«¨ç¨¢ îâ ¨­¤ãªâ¨¢­®áâì, ¢ª«îç ï ç¥à¥§ ­¥ª®â®àë¥ ¨­â¥à¢ «ë ª âã誨 (ª âã誨 ã¯¯¨­ ). …᫨ ¯®â¥à¨ ®¡ãá«®¢«¥­ë ⮫쪮 ᮯà®â¨¢«¥­¨¥¬ ¯à®¢®¤®¢, â® ¯à¨ R1  !L1 § âãå ­¨¥ à ¢­®

 = 2RZ1 : 0

‚¥«¨ç¨­  R1 ®¯à¥¤¥«ï¥âáï ª ª á㬬  ᮯà®â¨¢«¥­¨ï ¯àאַ£® ¨ ®¡à â­®£® ¯à®¢®¤®¢ á ãç¥â®¬ ⮫騭ë ᪨­-á«®ï, ¯® ª®â®à®¬ã ¯à®â¥ª ¥â ⮪. 56

 «¨ç¨¥ ¯®â¥àì ¯à¨¢®¤¨â ª ⮬ã, çâ® ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ áâ ­®¢¨âáï ª®¬¯«¥ªá­ë¬. ˆáª«î祭¨¥ á®áâ ¢«ï¥â «¨èì «¨­¨ï ¡¥§ ¨áª ¦¥­¨©, 1 G1 . ¥âà㤭® ¢¨¤¥âì, çâ® ¤«ï ª®â®à®© ¢ë¯®«­ï¥âáï ᮮ⭮襭¨¥ R L1 = Cp 1 ¢ í⮬ á«ãç ¥ ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ à ¢­® L1 =C1 , â.¥. ¢®«­®¢®¬ã ᮯà®â¨¢«¥­¨î «¨­¨¨ ¡¥§ ¯®â¥àì. à¨¬¥à.  áç¥â § âãå ­¨ï ª® ªá¨ «ì­®£® ª ¡¥«ï. „¨ ¬¥âàë ¯à®¢®¤­¨ª®¢: d1 =0.6 ¬¬, d2 =4.0 ¬¬; ‚®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ Z0 =75 Ž¬; — áâ®â  f =1000 Œƒæ. Žâá ᮯà®â¨¢«¥­¨¥ ¢­ãâ७­¥£® ¯à®¢®¤­¨ª  R10 = 0:0175
Sl , £¤¥ l { ¤«¨­  (1 ¬); S { ¯«®é ¤ì á¥ç¥­¨ï ¢­ãâ७­¥£® ¯à®¢®¤­¨ª , ¬¬2 (S = d1
,  { ⮫騭  ᪨­-á«®ï). ’®«é¨­  ᪨­-á«®ï ¢ ¬¥¤¨ ¤«ï ç áâ®âë f =1 000 Œƒæ à ¢­  2:1
10;3 ¬¬. ‚ १ã«ìâ â¥ ¯®«ãç ¥¬: R10  4:46 Ž¬/¬, ¨  ­ «®£¨ç­® ᮯà®â¨¢«¥­¨¥ ­ à㦭®£® ¯à®¢®¤­¨ª  R100  0:65 Ž¬/¬, ¨â®£® R1  5:1 Ž¬/¬. 1 = 0:031 ­¥¯¥à/¬ = 0:031
8:69 = 0:29 ¤/¬. Žâá  = 25
:75 3.

3.1.

‚Ž‹Ž‚Ž„›

‚ëà ¦¥­¨¥ ¢¥ªâ®à®¢ ¯®«ï ç¥à¥§ ¯®â¥­æ¨ «ì­ë¥ ä㭪樨. E- ¨ H-¬®¤ë

Šà®¬¥ à áᬮâ७­ëå à ­¥¥ ç¨áâ® ¯®¯¥à¥ç­ëå ¢®«­ ¢ «¨­¨ïå ¯¥à¥¤ ç¨ ¬®£ã⠯ਠ­¥ª®â®àëå ãá«®¢¨ïå à á¯à®áâà ­ïâìáï ¢®«­ë, ¨¬¥î騥 ¯à®¤®«ì­ãî á®áâ ¢«ïîéãî í«¥ªâà¨ç¥áª®£® ¨«¨ y ¬ £­¨â­®£® ¯®«¥©. ‚ ç áâ­®áâ¨, ¢ «¨­¨z ïå á ®¤­®á¢ï§­ë¬ á¥ç¥­¨¥¬ (á¬. à¨á.) ¬®£ãâ áãé¥á⢮¢ âì ⮫쪮 â ª¨¥ ¢®«­ë. ‚ «¨­¨ïå á ¤¢ã- ¨«¨ ¬­®£®á¢ï§­ë¬ á¥ç¥­¨¥¬ ªà®¬¥ ¯®¯¥à¥ç­ëå ¢®«­ ¬®£ãâ à á¯à®áâà ­ïâìáï â ª¦¥ ¢®«­ë á ¯à®¤®«ì­ë¬¨ á®áâ ¢«ïî騬¨ ¯®«ï, ¥á«¨ ç áâ®â  ¯à¥¢®á室¨â ­¥ª®â®à®¥ ªà¨â¨ç¥áª®¥ §­ ç¥­¨¥. x

57

«¥ªâ஬ £­¨â­®¥ ¯®«¥ ¢ ¢®«­®¢®¤¥ 㤮¢«¥â¢®àï¥â ®¤­®à®¤­ë¬ ãà ¢­¥­¨ï¬ Œ ªá¢¥«« 

~ rot H~ = j!"E: ~ rot E~ = ; j!H; ˆáª«îç ï ¯®®ç¥à¥¤­® í«¥ªâà¨ç¥áª®¥ ¨ ¬ £­¨â­®¥ ¯®«ï, ¬®¦­® ¯¥à¥©â¨ ª ãà ¢­¥­¨ï¬ ƒ¥«ì¬£®«ìæ  ¤«ï ª ¦¤®£® ¨§ ¯®«¥©:
E~ + k2 E~ = 0;
H~ + k2 H~ = 0: ’ ª ª ª Ez ¨ Hz { ¤¥ª à⮢ë á®áâ ¢«ïî騥, â® ¤«ï ­¨å ãà ¢­¥­¨ï ƒ¥«ì¬£®«ìæ  ¤ îâ 2
? Ez + @@zE2z + k2 Ez = 0; 2
? Hz + @@zH2z + k2 Hz = 0;

£¤¥
? { « ¯« á¨ ­, ¤¥©áâ¢ãî騩 «¨èì ­  ¯®¯¥à¥ç­ë¥ ª®®à¤¨­ âë. “à ¢­¥­¨ï ¬®¦­® à¥è âì à §¤¥«¥­¨¥¬ ¯¥à¥¬¥­­ëå:

Ez = E? (x; y)
Ek (z ); Hz = H? (x; y)
Hk (z ): ®¤áâ ¢«ïï ¨ à §¤¥«ïï ¯¥à¥¬¥­­ë¥, ¯®«ã稬

d2 E k 2 2 2 dz 2 ; Ek = 0;
? E? + (k + )E? = 0; d2 Hk 2 2 2 dz 2 ; Hk = 0;
? H? + (k + )H? = 0; £¤¥ { ¯®áâ®ï­­ ï à §¤¥«¥­¨ï. ˆ§ íâ¨å ãà ¢­¥­¨© á«¥¤ã¥â, çâ® § ¢¨á¨¬®áâì ¯®«ï ®â z íªá¯®­¥­æ¨ «ì­ : e z . à¨ í⮬ 6= jk, â ª ª ª ¢ á«ãç ¥ à ¢¥­á⢠ ¯®¯¥à¥ç­®¥ ¯®«¥ ¤®«¦­® ¡ëâì áâ â¨ç¥áª¨¬, ª ª íâ® á«¥¤ã¥â ¨§ ¯®«ã祭­ëå ãà ¢­¥­¨©. Šà®¬¥ ãà ¢­¥­¨© Œ ªá¢¥«« , ¯®«ï 㤮¢«¥â¢®àïî⠣࠭¨ç­ë¬ ãá«®¢¨ï¬, ª®â®àë¥ ¤«ï ¨¤¥ «ì­® ¯à®¢®¤ïé¨å á⥭®ª ¨¬¥îâ ¢¨¤

E~ t = 0; Hn = 0; 58

£¤¥ Et { â ­£¥­æ¨ «ì­ ï á®áâ ¢«ïîé ï í«¥ªâà¨ç¥áª®£® ¯®«ï ­  ¯®¢¥àå­®á⨠¬¥â «« , Hn { ­®à¬ «ì­ ï á®áâ ¢«ïîé ï ¬ £­¨â­®£® ¯®«ï ­  í⮩ ¯®¢¥àå­®áâ¨.  ©¤¥¬ à¥è¥­¨ï ãà ¢­¥­¨© Œ ªá¢¥«« . „«ï í⮣® § ¯¨è¥¬ ¨å ¤«ï ¤¥ª à⮢ëå á®áâ ¢«ïîé¨å ¢¥ªâ®à®¢ ¯®«ï, ãç¨â뢠ï íªá¯®­¥­æ¨ «ì­ãî § ¢¨á¨¬®áâì ¨å ®â z (¤¨ää¥à¥­æ¨à®¢ ­¨î ¯® z ᮮ⢥âáâ¢ã¥â 㬭®¦¥­¨¥ ­  ; ): z ;j!Hx = @E @y + Ey ; z ;j!Hy = ; Ex ; @E @x ; y ; @Ex ; ;j!Hz = @E @x @y

z j!"Ex = @H @y + Hy ; z j!"Ey = ; Hx ; @H @x ; @Hx y j!"Ez = @H @x ; @y :

‡ ¯¨á ­­ë¥ ãà ¢­¥­¨ï ¯®§¢®«ïîâ ¢ëà §¨âì ¯®¯¥à¥ç­ë¥ ª®¬¯®­¥­âë

Ex , Ey , Hx , Hy ç¥à¥§ ¯à®¤®«ì­ë¥ Ez , Hz . „«ï í⮣® ¯®¤áâ ¢¨¬ ¢ ¯¥à¢®¥ ãà ¢­¥­¨¥ Ey , ¢§ï⮥ ¨§ ¯ï⮣®. ’®£¤  ¯®«ã稬 @Hz z (!2 " + 2 )Hx = j!" @E @y ; @x ¨«¨ (®¡®§­ ç ï !2 " = k2 ) @Ez ;

@Hz : Hx = k2 j!" 2 2 + @y k + 2 @x €­ «®£¨ç­®

@Ez ;

@Hz ; Hy = ; k2 j!" 2 2 + @x k + 2 @y @Hz

@Ez Ex = ; k2 j! + 2 @y ; k2 + 2 @x ; 59

@Hz

@Ez Ey = k2 j! + 2 @x ; k2 + 2 @y : ‘®áâ ¢«ïî騥 Ez ¨ Hz ¤®«¦­ë 㤮¢«¥â¢®àïâì ãà ¢­¥­¨ï¬ ƒ¥«ì¬-

£®«ìæ :


Ez + k2 Ez = 0;
Hz + k2 Hz = 0;

@ = ; , ¯®«ãç ¥¬ ¨«¨, ãç¨â뢠ï, çâ® @z

d2 Ez + d2 Ez + (k2 + 2 )E = 0; z dx2 dy2 d2 Hz + d2 Hz + (k2 + 2 )H = 0: z dx2 dy2

„ ­­ë¥ ãà ¢­¥­¨ï ¬®£ãâ ¡ëâì ¯®«ãç¥­ë ¨ ­¥¯®á।á⢥­­® ¨§ âà¥â쥣® ¨ è¥á⮣® ¨§ ¢ë襭 ¯¨á ­­ëå ãà ¢­¥­¨©, ¥á«¨ ¯®¤áâ ¢¨âì ¢ ­¨å Ex , Ey , Hx ¨ Hy . ‚¢¥¤¥¬ á«¥¤ãî騥 ®¡®§­ ç¥­¨ï:

Ez = (k2 + 2 )
(x; y)
e; z ; Hz = (k2 + 2 )
(x; y)
e; z ; £¤¥ (x; y), (x; y) { ­¥ª®â®àë¥ áª «ïà­ë¥ ä㭪樨 ¯®¯¥à¥ç­ëå ¯¥à¥¬¥­­ëå. â¨ ä㭪樨 ¤®«¦­ë, ®ç¥¢¨¤­®, 㤮¢«¥â¢®àïâì ãà ¢­¥­¨ï¬
x;y  + (k2 + 2 ) = 0;
x;y + (k2 + 2) = 0; ¨«¨, ®¡®§­ ç ï k2 + 2 = g2 ,
x;y  + g2  = 0;
x;y + g2 = 0: ‘®áâ ¢«ïî騥 ¯®«¥© ¢ëà ¦ îâáï ç¥à¥§ ä㭪樨  ¨ : 60

Hx =

 @  j!" @y ; @@x
e; z ;   @ @

;j!" @x ; @y
e; z ;   @ @ Ex = ;j! @y ; @x
e; z ;

Hy = Ey =





; z j! @@x ; @ @y
e ;

¥âà㤭® ¯à®¢¥à¨âì, çâ® í⨠ᮮ⭮襭¨ï ¬®£ãâ ¡ëâì § ¯¨á ­ë ¢ ¢¨¤¥ H~ = j!"
rot (
~z0
e; z ) + grad div (
~z0
e; z ) + k2
e; z
~z0 ; E~ = ; j!
rot (
~z0
e; z ) + grad div (
~z0
e; z ) + k2 
e; z
~z0 ; £¤¥ ~z0 { ¥¤¨­¨ç­ë© ¢¥ªâ®à ¢ ­ ¯à ¢«¥­¨¨ ®á¨ z . „¥©á⢨⥫쭮, ¥á«¨ ¨á室¨âì ¨§ ¨§¢¥áâ­ëå ¢¥ªâ®à­ëå ⮦¤¥á⢠grad (
) = 
grad +
grad ; rot (
F~ ) = 
rot F~ + grad   F~ ; div (
F~ ) = 
div F~ + grad 
F~ ; ¬®¦­® ¯®«ãç¨âì rot (
~z0
e; z ) = e; z
grad   ~z0 ; grad div (
~z0
e; z ) = ; e; z grad

+ 2
~z0 e; z :

®¤áâ ¢«ïï í⨠¢ëà ¦¥­¨ï, ¯®«ã稬

H~ = j!"
e; z grad   ~z0 + (k2 + 2 )e; z
~z0 ; e; z grad ; E~ = ; j!
e; z grad  ~z0 + (k2 + 2 )e; z 
~z0 ; e; z grad : „ ­­ë¥ ᮮ⭮襭¨ï, ¥á«¨ ¨å § ¯¨á âì ¢ ¤¥ª à⮢ëå á®áâ ¢«ïîé¨å, ¯à¨¢®¤ïâáï ª ¯®«ã祭­ë¬ ¢ëè¥. 61

‚¥ªâ®àë

~ = 
~z0
e; z ¨ ~ 0 = 


~z0
e; z |

íâ® ¢¥ªâ®àë ƒ¥àæ . ’ ª¨¬ ®¡à §®¬, ¢¥ªâ®àë ¯®«ï ¢ëà ¦ îâáï ç¥à¥§ í«¥ªâà¨ç¥áª¨© ¨ ¬ £­¨â­ë© ¢¥ªâ®àë ƒ¥àæ , ¨¬¥î騥 ¢ ¤ ­­®¬ á«ãç ¥ ⮫쪮 ¯à®¤®«ì­ë¥ (z ) á®áâ ¢«ïî騥.  ©¤¥¬ £à ­¨ç­ë¥ ãá«®¢¨ï ­  ¯®¢¥àå­®á⨠¤«ï ¯®â¥­æ¨ «ì­ëå ä㭪権  ¨ . „«ï í⮣® ­¥®¡å®¤¨¬® ¢ëà §¨âì â ­£¥­æ¨ «ì­ãî ª®¬¯®­¥­âã í«¥ªâà¨ç¥áª®£® ¯®«ï E~ t ¨ ­®à¬ «ì­ãî ¬ £­¨â­®£® ¯®«ï Hn ­  ¯®¢¥àå­®á⨠ç¥à¥§  ¨ . ’ ­£¥­æ¨ «ì­ãî ª®¬¯®­¥­âã E~ t ¬®¦­® à §«®¦¨âì ­  á®áâ ¢«ïîéãî ¢ ­ ¯à ¢«¥­¨¨ ®á¨ z ¨ á®áâ ¢«ïîéãî, «¥¦ éãî ¢ ¯«®áª®á⨠á¥ç¥­¨ï, â.¥. ª á â¥«ì­ãî ª ª®­âãàã ¯®¯¥à¥ç­®£® á¥ç¥­¨ï. Ž¡¥ á®áâ ¢«ïî騥 ¤®«¦­ë ¡ëâì à ¢­ë ­ã«î. ˆ§ à ¢¥­á⢠ Ez = 0 ­  ¯®¢¥àå­®á⨠¯®«ãç ¥¬  = 0 ­  ª®­âãॠá¥ç¥­¨ï C . —â®¡ë ­ ©â¨ ª á â¥«ì­ãî ª ª®­âãàã á¥ç¥­¨ï á®áâ ¢«ïîéãî E~ s , ¢¢¥¤¥¬ ¢ ¯à®¨§¢®«ì­®© â®çª¥ ª®­âãà  ¤¢  ®àâ  ~s ¨ ~n, ¯à¨ç¥¬ ~n ­ ¯à ¢«¥­ ¢­ãâàì ¢®«­®¢®¤  (á¬. à¨á.). ’®£¤  ¯®  ­ «®£¨¨ á ¢ëà ¦¥­¨¥¬ ¤«ï Ey (®â®¦¤¥á⢫ïï ­ n(x) ¯à ¢«¥­¨¥ y á s,   ­ ¯à ¢«¥­¨¥ x { á n) ¯®«ã稬

1111111111111 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 s (y 0000000000000 1111111111111 )

Es = j! @@n ; @ @s :

’ ª ª ª  = 0 ­  C , â® ¤«ï ⮣®, ç⮡ë Es = 0, ­¥®¡å®¤¨¬® ¨ ¤®áâ â®ç­®, ç⮡ë @ =@n = 0 ­  ª®­âãॠC . ®ª ¦¥¬, çâ® í⨠ãá«®¢¨ï ®¡¥á¯¥ç¨¢ îâ â ª¦¥ à ¢¥­á⢮ Hn = 0 ­  C . „«ï í⮣® § ¯¨è¥¬ ¢ëà ¦¥­¨¥ ¤«ï Hn ¯®  ­ «®£¨¨ á ¢ëà ¦¥­¨¥¬ ¤«ï Hx :

@ Hn = ; j!" @ @s ; @n : ˆâ ª, £à ­¨ç­ë¥ ãá«®¢¨ï ¤«ï ä㭪権  ¨ ¨¬¥îâ ¢¨¤ 62

 = 0 ­  C; @ = 0 ­  C; @n

  á ¬¨ ä㭪樨 㤮¢«¥â¢®àïîâ ãà ¢­¥­¨ï¬
x;y  + g2  = 0;
x;y + g2 = 0; £¤¥ g2 = k2 + 2 . ‡ ¤ ç  à á¯ ¤ ¥âáï ­  ¤¢¥: ®â¤¥«ì­® ¤«ï  ¨ . ‚ ¨â®£¥ ¨¬¥¥¬ ¤¢¥ á¨á⥬ë à¥è¥­¨©. „«ï ®¤­®© ¯®«¥ ¢ëà ¦ ¥âáï ⮫쪮 ç¥à¥§ äã­ªæ¨î , ¯à¨ í⮬ ®â«¨ç­  ®â ­ã«ï z -ª®¬¯®­¥­â  í«¥ªâà¨ç¥áª®£® ¯®«ï, ¤«ï ¤à㣮© ¯®«¥ ¢ëà ¦ ¥âáï ç¥à¥§ -äã­ªæ¨î, ®â«¨ç­  ®â ­ã«ï z -ª®¬¯®­¥­â  ¬ £­¨â­®£® ¯®«ï. ˆ§¢¥áâ­®, çâ® â ª¨¥ § ¤ ç¨ ¨¬¥îâ ­¥âਢ¨ «ì­ë¥ à¥è¥­¨ï ¯à¨ ®¯à¥¤¥«¥­­ëå §­ ç¥­¨ïå g2 { ᮡá⢥­­ëå §­ ç¥­¨ïå § ¤ ç¨. ‚ ®¡é¥¬ á«ãç ¥ í⨠§­ ç¥­¨ï à §«¨ç­ë ¤«ï  ¨ ä㭪権. ”㭪樨  ¨ , ¯à¥¤áâ ¢«ïî騥 ᮡ®© ­¥âਢ¨ «ì­ë¥ à¥è¥­¨ï 㪠§ ­­®© ¢ëè¥ § ¤ ç¨, ­ §ë¢ îâ ᮡá⢥­­ë¬¨ äã­ªæ¨ï¬¨. Œ®¦­® ¯®ª § âì, ç⮠ᮡá⢥­­ë¥ §­ ç¥­¨ï ¢¥é¥á⢥­­ë ¨ ¯®«®¦¨â¥«ì­ë. „«ï í⮣® ¬®¦­® ¨á室¨âì ¨§ ä®à¬ã«ë ƒà¨­  ¤«ï ¯à®¨§¢®«ì­ëå ä㭪権  ¨

Z S

(

+ grad 
grad ) dS =

I

C


@@n dC;

£¤¥ S { ®¡« áâì ­  ¯«®áª®áâ¨, C { ª®­âãà, ®£à ­¨ç¨¢ î騩 íâã ®¡« áâì. ‡ ¬¥­¨¬  ­   . ’®£¤ 

Z S

( 

+ jgrad j2 ) dS =

I

C


@ dC = 0

@n

¢á«¥¤á⢨¥ £à ­¨ç­ëå ãá«®¢¨©. Šà®¬¥ ⮣®,
= ;g2 ¨§ ãà ¢­¥­¨ï. ®¤áâ ¢«ïï íâ® ¢ à ¢¥­á⢮, ¯®«ã祭­®¥ ¨§ ä®à¬ã«ë ƒà¨­ , ­ å®¤¨¬ 63

g2 =

R jgrad j2 dS S R j j2 dS ; S

®âªã¤  á«¥¤ã¥â, çâ® g2 > 0. ®«ã祭­ ï ä®à¬ã«  ¯®§¢®«ï¥â ­ ©â¨ ᮡá⢥­­®¥ §­ ç¥­¨¥, ¥á«¨ ¨§¢¥áâ­  ᮮ⢥âáâ¢ãîé ï ᮡá⢥­­ ï äã­ªæ¨ï. ‘®¡á⢥­­ë¥ §­ ç¥­¨ï ®¡à §ãîâ ¢®§à áâ îéãî áç¥â­ãî ¯®á«¥¤®¢ â¥«ì­®áâì ¯®«®¦¨â¥«ì­ëå ç¨á¥«, á।¨ ª®â®àëå ¨¬¥¥âáï ®â«¨ç­®¥ ®â ­ã«ï ­ ¨¬¥­ì襥 ç¨á«®: g12; g22 ; : : : ; gn2 ; : : :. â  ¯®á«¥¤®¢ â¥«ì­®áâì ­¥ ¨¬¥¥â â®ç¥ª á£ã饭¨ï, §  ¨áª«î祭¨¥¬ 1. Š ¦¤®¬ã ᮡá⢥­­®¬ã §­ ç¥­¨î ᮮ⢥âáâ¢ã¥â ®¤­  ¨«¨ ¡®«ìè¥ á®¡á⢥­­ëå ä㭪権. Œ®¦­® ¯®ª § âì, ç⮠ᮡá⢥­­ë¥ ä㭪樨, ᮮ⢥âáâ¢ãî騥 à §«¨ç­ë¬ ᮡá⢥­­ë¬ §­ ç¥­¨ï¬, ®à⮣®­ «ì­ë ­  á¥ç¥­¨¨ ¢®«­®¢®¤  S: Z Z n
m dS = 0; ¥á«¨ gn2 6= gm2 : n
m dS = 0; S

S

„«ï ¤®ª § â¥«ìá⢠ ¢®á¯®«ì§ã¥¬áï ¢â®à®© ä®à¬ã«®© ƒà¨­ :

Z S

(n

m ; m

n ) dS =

I

C

m ; 
@n ) dC: (n
@ m @n @n

à ¢ ï ç áâì í⮣® à ¢¥­á⢠ ®¡à é ¥âáï ¢ ­ã«ì ¢ ᨫ㠣࠭¨ç­ëå ãá«®¢¨© ­  ª®­âãॠá¥ç¥­¨ï ¢®«­®¢®¤ 

Z S

(n

m ; m

n ) dS = 0:

®¤áâ ¢«ïï á ¨§ ãà ¢­¥­¨©
n = ;gn2
n ,
m = ;gm2
m , ¯®«ã稬 (gn2 ; gm2 )

Z S

n
m dS = 0:

’ ª ª ª ¯® ¯à¥¤¯®«®¦¥­¨î gn2 6= gm2 , â®

Z S

n
m dS = 0: 64

€­ «®£¨ç­® ¤®ª §ë¢ ¥âáï, çâ®

Z S

n
m dS = 0:

‚ á«ãç ¥ ¢ë஦¤¥­¨ï, â.¥. ª®£¤  n ¨ m ᮮ⢥âáâ¢ãîâ ®¤­®¬ã ¨ ⮬㠦¥ ᮡá⢥­­®¬ã §­ ç¥­¨î, ®­¨ ¬®£ãâ ¨ ­¥ ¡ëâì ®à⮣®­ «ì­ë. ˆ§¢¥áâ­®, çâ® ¢ í⮬ á«ãç ¥ ¢®§¬®¦¥­ ¯à®æ¥áá ®à⮣®­ «¨§ æ¨¨, â.¥. ¬®¦­® ¯®¤®¡à âì â ª¨¥ «¨­¥©­ë¥ ª®¬¡¨­ æ¨¨ ¨§ íâ¨å ä㭪権, ª®â®àë¥ ¡ã¤ãâ ®à⮣®­ «ì­ë ¤à㣠ª ¤àã£ã, ®áâ ¢ ïáì ¯à¨ í⮬ ᮡá⢥­­ë¬¨ äã­ªæ¨ï¬¨ (ᮮ⢥âáâ¢ãî騬¨ ®¤­®¬ã ᮡá⢥­­®¬ã §­ ç¥­¨î). ®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï ¤«ï ¤ ­­®© ¬®¤ë n ¬®¦¥â ¡ëâì ®¯à¥¤¥«¥­ , ¥á«¨ ¨§¢¥áâ­® ᮡá⢥­­®¥ §­ ç¥­¨¥ gn2 :

p

n2 = gn2 ; k2 ; n =  gn2 ; k2 :  á¯à®áâà ­¥­¨¥ ¡¥§ § âãå ­¨ï ¨¬¥¥â ¬¥áâ®, ¥á«¨ n { ¬­¨¬ ï ¢¥«¨-

稭 , â.¥. ¥á«¨ ¢ë¯®«­ï¥âáï ­¥à ¢¥­á⢮ k2 > gn2 :

â® ãá«®¢¨¥ ¬®¦­® § ¯¨á âì ¨­ ç¥: !2 " > gn2 ¨«¨

n = ! | ! > pg" n ªà¨â¨ç¥áª ï ç áâ®â  ¤«ï n-© ¬®¤ë. …᫨ íâ® ãá«®¢¨¥ ¢ë¯®«­ï¥âáï, â®

¯®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï à ¢­ 

p

n = j n = j k2 ; gn2 : Žâá ¬®¦­® ­ ©â¨ ¤«¨­ã ¢®«­ë ¢ ¢®«­®¢®¤¥ ¨ ä §®¢ãî ᪮à®áâì. „¥©á⢨⥫쭮, n = 2 ; k = 2 ; n £¤¥ n { ¤«¨­  ¢®«­ë ¢ ¢®«­®¢®¤¥,  { ¤«¨­  ¢®«­ë ¢ ­¥®£à ­¨ç¥­­®¬ ¯à®áâà ­á⢥ á ¯ à ¬¥âà ¬¨ , ". 65

®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï n , ¯®«ã稬 (gn = 2 ): 1 n = ®âªã¤  n =

s

n

1 1 2 ; 2n ;

r 

2: 1 ; 2 n

” §®¢ ï ᪮à®áâì ¬®¦¥â ¡ëâì ­ ©¤¥­  ¯® ¨§¢¥áâ­®© ¤«¨­¥ ¢®«­ë: vä = ! = !2n = r c 2 > c: n 1 ; 2 n ˆâ ª, ¢®«­ë, à á¯à®áâà ­ïî騥áï ¢ ¢®«­®¢®¤¥, ¬®£ãâ ¡ëâì ¯à¥¤áâ ¢«¥­ë ¢ ¢¨¤¥ á㬬ë í«¥ªâà¨ç¥áª¨å (E ¨«¨ TM,  0) ¨ ¬ £­¨â­ëå (H ¨«¨ TE,   0) ¢®«­. ˆ¬ ᮮ⢥âáâ¢ãîâ ¤¢¥ á¨áâ¥¬ë ¢¥ªâ®à­ëå ä㭪権, í«¥ªâà¨ç¥áª¨¥ { E~ n(e) ; H~ n(e) ¨ ¬ £­¨â­ë¥ { E~ n(h) ; H~ n(h). Žâ¬¥â¨¬, çâ® ¢¥ªâ®à­ë¥ ä㭪樨 ¢§ ¨¬­® ®à⮣®­ «ì­ë ¢ ⮬ á¬ë«¥, çâ®

Z S

E~ n
E~ m dS = 0;

Z S

H~ n
H~ m dS = 0;

Z S

(E~ n  H~ m )
~z0 dS = 0

¯à¨ gn2 6= gm2 . ˆ­â¥£à «ë ¡¥àãâáï ¯® á¥ç¥­¨î ¢®«­®¢®¤ . „ ­­®¥ ᢮©á⢮ ¯®§¢®«ï¥â à áᬠâਢ âì ¢®«­ë à §«¨ç­ëå ¬®¤ ª ª ­¥§ ¢¨á¨¬ë¥, â ª ª ª ¡« £®¤ àï ®à⮣®­ «ì­®áâ¨ í­¥à£¨ï ¨ ¬®é­®áâì ᪫ ¤ë¢ îâáï ¨§ í­¥à£¨¨ ¨ ¬®é­®á⨠®â¤¥«ì­ëå ¬®¤. ‡ ¯¨è¥¬ ¢ëà ¦¥­¨¥ ¤«ï H-¢®«­ ç¥à¥§ ¯®â¥­æ¨ «ì­ãî äã­ªæ¨î :

E~ = ; j!
e; z grad  ~z0 ; H~ = ;
e; z grad + (k2 + 2 ) e; z
~z0 : ®¯¥à¥ç­ë¥ á®áâ ¢«ïî騥 í«¥ªâà¨ç¥áª®£® ¨ ¬ £­¨â­®£® ¢¥ªâ®à®¢ ¯¥à¯¥­¤¨ªã«ïà­ë, çâ® á«¥¤ã¥â ¨§ à ¢¥­á⢠ ­ã«î ¨å ᪠«ïà­®£® ¯à®¨§¢¥¤¥­¨ï 66

E~ ¯®¯
H~ ¯®¯ = grad
(grad  ~z0 ) = 0: ˆ§ ¯à¨¢¥¤¥­­ëå ¢ëè¥ ä®à¬ã« á«¥¤ã¥â, çâ® ¯®¯¥à¥ç­ë¥ á®áâ ¢«ïî騥 ¯®«ï á¢ï§ ­ë ¬¥¦¤ã ᮡ®© ᮮ⭮襭¨¥¬ (¤«ï ª ¦¤®© ¬®¤ë)

E~ ¯®¯ = Zh
H~ ¯®¯  ~z0 ; £¤¥ Zh = j!= . „«ï à á¯à®áâà ­ïîé¨åáï ¬®¤ Zh ¢¥é¥á⢥­­®:

Zh = ! = vä =

p="

p1 ; 2 =2 : n

„«ï E-¢®«­ ¯®«¥ ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¢¨¤¥

E~ = ; e; z grad  + (k2 + 2) e; z 
~z0; H~ = j!"
e; z grad   ~z0 : ‡¤¥áì E~ ¯®¯ ¨ H~ ¯®¯ â ª¦¥ ¯¥à¯¥­¤¨ªã«ïà­ë. Šà®¬¥ ⮣®, E~ ¯®¯ = Ze
H~ ¯®¯  ~z0 ; ¯à¨ç¥¬ Ze = =j!". „«ï à á¯à®áâà ­ïîé¨åáï ¢®«­ Ze ¢¥é¥á⢥­­®:

= 1 = Ze = !" vä
"

rs

2 : 1 ; " 2n

 ­¥¥ ¡ë«® ¯®ª § ­®, çâ® ¢®«­  ¢ ¢®«­®¢®¤¥ ¬®¦¥â à á¯à®áâà ­ïâìáï, ¥á«¨ ¢ë¯®«­ï¥âáï ãá«®¢¨¥ n ; ! > !n = pg"

¯à¨ç¥¬ á।¨ gn ¨¬¥¥âáï ­ ¨¬¥­ì襥, ­ ¯à¨¬¥à g1 . Žâá á«¥¤ã¥â, çâ® ¯® ¤ ­­®¬ã ¢®«­®¢®¤ã ¯à¨ § ¤ ­­®© ç áâ®â¥ ¬®¦¥â à á¯à®áâà ­ïâìáï «¨èì ª®­¥ç­®¥ ç¨á«® ¬®¤; íâ® ç¨á«® à áâ¥â á à®á⮬ ç áâ®âë. ‚ ç áâ­®áâ¨, áãé¥áâ¢ã¥â ¨­â¥à¢ « ç áâ®â, ¢ ª®â®à®¬ ¢ ¢®«­®¢®¤¥ ¬®¦¥â à á¯à®áâà ­ïâìáï «¨èì ®¤­  ¬®¤ .

67

3.2.

àאַ㣮«ì­ë¥ ¢®«­®¢®¤ë

3.2.1. ‚®«­ë H-⨯ 

„«ï H-¢®«­ ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥ (á¬. à¨á.) ¨¬¥¥â ¬¥áâ® ãà ¢­¥­¨¥ @ 2 + @ 2 + g2 = 0 @x2 @y2

y a

b

¯à¨ ãá«®¢¨¨ ­  £à ­¨æ å @@n = 0. ¥è¥­¨¥ ­ ©¤¥¬ ¯® ¬¥â®¤ã ”ãàì¥ ¢ ¢¨¤¥ ¯à®¨§¢¥¤¥­¨ï

b x a

(x; y) = X (x)
Y (y) : ’®£¤  ãà ¢­¥­¨¥ ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­® ¢ ¢¨¤¥ 1 @ 2 X + 1 @ 2Y = ; g2 X @x2 Y @y2 ¨ ¯¥à¥¬¥­­ë¥ à §¤¥«ïîâáï, â.¥. ãà ¢­¥­¨¥ à á¯ ¤ ¥âáï ­  ¤¢ : @ 2 X + g2 X = 0 ¨ @ 2 Y + g2 Y = 0; x y @x2 @y2 ¯à¨ç¥¬ gx2 + gy2 = g2 . ®«ã祭­ë¥ ãà ¢­¥­¨ï ¨¬¥îâ à¥è¥­¨ï

X = A cos gx x + B sin gxx; Y = C cos gy y + D sin gy y:   £à ­¨æ å ¨¬¥¥¬: ¯à¨ x = 0 dX dx = 0, ®âªã¤  á«¥¤ã¥â, çâ® B = 0; X = cos gx x; ¯à¨ y = 0 dY dy = 0, ®âªã¤  á«¥¤ã¥â, çâ®

D = 0; Y = cos gy y; 68

¯à¨ x = a dX dx = 0, â.¥. sin gxa = 0, ®âªã¤ 

gx a = n ¨ gx = n a ; n = 0; 1; 2; : : : ; ¯à¨ y = b dY dy = 0, â.¥. sin gy b = 0, ®âªã¤  gy b = m ¨ gy = m b ; m = 0; 1; 2; : : :: ’¥¯¥àì ­ å®¤¨¬ (x; y): m (x; y) = cos n a x
cos b y

á â®ç­®áâìî ¤® ¯à®¨§¢®«ì­®£® ¬­®¦¨â¥«ï. 2 à ¢­ë ‘®¡á⢥­­ë¥ §­ ç¥­¨ï gnm     2 = n 2 + m 2 n = 0; 1; 2; : : :; m = 0; 1; 2; : : :: gnm a b

‡ ¬¥â¨¬, çâ® ¨­¤¥ªáë n ¨ m ­¥ ¬®£ãâ ®¤­®¢à¥¬¥­­® ®¡à é âìáï ¢ ­ã«ì, â ª ª ª ¢ í⮬ á«ãç ¥ = const ¨ ¢á¥ ª®¬¯®­¥­âë ¯®«ï ®¡à é îâáï ¢ ­ã«ì. Žâá ¬®£ãâ ¡ëâì ­ ©¤¥­ë ªà¨â¨ç¥áª¨¥ ¤«¨­ë ¢®«­ (¤«ï ¢ ªã㬠): nm = g2 = s 2 2  2 = s 2 2  2 ; nm n + m n + m a b a b â.¥. ªà¨â¨ç¥áª ï ¤«¨­  ¢®«­ë ®¯à¥¤¥«ï¥âáï à §¬¥à ¬¨ ¢®«­®¢®¤  ¨ ­®¬¥à®¬ ¬®¤ë. ®á«¥ ⮣® ª ª ­ ©¤¥­  äã­ªæ¨ï (x; y), ¬®¦­® ­ ©â¨ ¢á¥ á®áâ ¢«ïî騥 ¯®«ï (¡¥§ íªá¯®­¥­âë e; z ):

Hz = (k2 + g2 ) (x; y) =

 n 2  m 2 n m + cos x cos y; a

a b n m n n m Hx = ; @@x = n a sin a x cos b y = j a sin a x cos b y; n m m n m Hy = ; @@y = m b cos a x sin b y = j b cos a x sin b y; 69

b

n m Ex = ; j! @@y = j! m b cos a x sin b y; n m Ey = j! @@x = ; j! n a sin a x cos b y: ‚ ¤ «ì­¥©è¥¬ ¤«ï ®¯à¥¤¥«¥­­®á⨠¡ã¤¥¬ ¯à¥¤¯®« £ âì, çâ® a > b.  áᬮâਬ áâàãªâãàã ¯®«¥© à §«¨ç­ëå ¬®¤. à¥¦¤¥ ¢á¥£®, ¥é¥ à § ­ ¯®¬­¨¬, çâ® n ¨ m ­¥ ¬®£ãâ ®¤­®¢à¥¬¥­­® ¯à¨­¨¬ âì ­ã«¥¢ë¥ §­ ç¥­¨ï. ® ®¤­® ¨§ íâ¨å ç¨á¥« ¬®¦¥â ¡ëâì à ¢­® ­ã«î. ¥âà㤭® ¢¨¤¥âì, çâ® ¥á«¨ a > b, â® ­ ¨¡®«ìèãî ªà¨â¨ç¥áªãî ¢®«­ã ¨¬¥¥â ¬®¤  H10 :

10 = 2a: Œ®¤ã á ­ ¨¡®«ì襩 ªà¨â¨ç¥áª®© ¢®«­®© ­ §ë¢ î⠮᭮¢­®© ¤«ï ¤ ­­®£® ¢®«­®¢®¤ . ‘ãé¥áâ¢ã¥â ¤¨ ¯ §®­ ç áâ®â, ¢ ª®â®à®¬ ¢ ¤ ­­®¬ ¢®«­®¢®¤¥ ¬®¦¥â à á¯à®áâà ­ïâìáï ⮫쪮 ®á­®¢­ ï ¬®¤ , ¢ ®â«¨ç¨¥ ®â ¢ëáè¨å ¬®¤. Œ®¤¥ H10 ᮮ⢥âáâ¢ãîâ á«¥¤ãî騥 §­ ç¥­¨ï á®áâ ¢«ïîé¨å ¯®«¥©:

Hz Hx Hy Ex Ey

= = = = =

  2

 a cos a x; j a sin a x; 0; 0; ;j! a sin a x:

®«ï §¤¥áì ®¯à¥¤¥«¥­ë á â®ç­®áâìî ¤® ¬­®¦¨â¥«ï, ®¤¨­ ª®¢®£® ¤«ï ¢á¥å ª®¬¯®­¥­â. â®â ¬­®¦¨â¥«ì ®¯à¥¤¥«ï¥âáï, ¥á«¨ § ¤ ­  ¬®é­®áâì ¢®«­ë, à á¯à®áâà ­ïî饩áï ¯® ¢®«­®¢®¤ã. ‘âàãªâãà  ¢®«­ë H10 ¬®¦¥â ¡ëâì ¢ëïá­¥­  á ¯®¬®éìî £à ä¨ª®¢

70

(á¬. à¨á.): y

y

111111111111111111111111111 000000000000000000000000000 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111

Ey 111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111

Hx

111111111111111111111111111 000000000000000000000000000 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111

Ey

Hz 1111111111111 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 000000000000011111111111111 1111111111111 00000000000000 00000000000000 11111111111111 00000000000000 11111111111111

x

x

„«ï ãïá­¥­¨ï áâàãªâãàë ¯®«ï á«¥¤ã¥â â ª¦¥ ãç¨â뢠âì, çâ® Hz ¨ Hx ᤢ¨­ãâë ¢® ¢à¥¬¥­¨ ¯® ä §¥ ­  90o. Š à⨭  ᨫ®¢ëå «¨­¨© ¬ £-

­¨â­®£® ¯®«ï ¯à¥¤áâ ¢«¥­  ­  à¨á.

‘âàãªâãà  ¯®«ï ¬®¤ë Hn0 ¬®¦¥â ¡ëâì ¯®«ã祭  ¯ã⥬ n-ªà â­®£® ¯®¢â®à¥­¨ï áâàãªâãàë ¬®¤ë H10 ¢ ­ ¯à ¢«¥­¨¨ ®á¨ x. Œ®¤ë ⨯  Hn0 å à ªâ¥à­ë ⥬, çâ® ¯®«¥ ­¥ § ¢¨á¨â ®â ª®®à¤¨­ âë y. Šà®¬¥ ⮣®, ªà¨â¨ç¥áª ï ¤«¨­  ¢®«­ë ¨, á«¥¤®¢ â¥«ì­®, ¤«¨­  ¢®«­ë ¢ ¢®«­®¢®¤¥ ­¥ § ¢¨áï⠮⠢ëá®âë ¢®«­®¢®¤  b:

n0 = 2 a=n; n = 1; 2; : : : : 71

‚ â¥å­¨ª¥ ­ ¨¡®«ì襥 ¯à¨¬¥­¥­¨¥ ­ è«¨ ¯àאַ㣮«ì­ë¥ ¢®«­®¢®¤ë á ¢®«­®© H10 . à¨ í⮬ à §¬¥àë ¨å ®¡ëç­® ¢ë¡¨à îâáï â ª, çâ®¡ë ¢ à ¡®ç¥¬ ¤¨ ¯ §®­¥ ç áâ®â ¢ëá訥 ¬®¤ë ­¥ ¬®£«¨ à á¯à®áâà ­ïâìáï.  áᬮâਬ ⥯¥àì áâàãªâãàã ¢®«­ë H11 . ‚ í⮬ á«ãç ¥ á®áâ ¢«ïî騥 ¯®«ï § ¢¨áïâ â ª¦¥ ®â y (á¬. à¨á.).

‘âàãªâãà  ¯®«ï §¤¥áì ¡®«¥¥ á«®¦­ , â ª ª ª ®â«¨ç­ë ®â ­ã«ï ¢á¥ á®áâ ¢«ïî騥, ªà®¬¥ Ez . ‘âàãªâãà  ¯®«¥© ⨯  Hnm ¬®¦¥â ¡ëâì ¯®«ã祭  ¯ã⥬ n-ªà â­®£® ¯®¢â®à¥­¨ï áâàãªâãàë H11 ¢ ­ ¯à ¢«¥­¨¨ x ¨ m-ªà â­®£® ¯®¢â®à¥­¨ï ¢ ­ ¯à ¢«¥­¨¨ y. ˆ¬¥¥â á¬ëá« à áᬮâà¥âì â ª¦¥ ⮪¨ ¢ á⥭ª å. «®â­®áâì ⮪  ¢ á⥭ª å ¬®¦¥â ¡ëâì ­ ©¤¥­  ¨§ £à ­¨ç­®£® ãá«®¢¨ï

K~ = H~  ~n; £¤¥ ~n { ¥¤¨­¨ç­ë© ¢¥ªâ®à ­®à¬ «¨, ­ ¯à ¢«¥­­ë© ¢ ¬¥â ««. â® §­ ç¨â, çâ® ¯®¢¥àå­®áâ­ ï ¯«®â­®áâì ⮪  ç¨á«¥­­® à ¢­  â ­£¥­æ¨ «ì­®© á®áâ ¢«ïî饩 ­ ¯à殮­­®á⨠¬ £­¨â­®£® ¯®«ï ¨ ­ ¯à ¢«¥­  ¯¥à¯¥­¤¨ªã«ïà­® ¬ £­¨â­®¬ã ¯®«î, ¯®í⮬㠫¨­¨¨ ⮪  ¢ á⥭ª å ¬®£ãâ ¡ëâì ¯®áâ஥­ë ª ª «¨­¨¨, ®à⮣®­ «ì­ë¥ ᥬ¥©áâ¢ã «¨­¨© ¬ £­¨â­®£® ¯®«ï ­  ¯®¢¥àå­®á⨠¢®«­®¢®¤ . ‚­ ç «¥ ¯®áâந¬ «¨­¨¨ ⮪  ¤«ï ¢®«­ë H10 (á¬. à¨á. ­  á«¥¤. áâà.).

72

„«ï ¢®«­ë H11 «¨­¨¨ ⮪  ­  è¨à®ª®© ¨ 㧪®© á⥭ª å  ­ «®£¨ç­ë «¨­¨ï¬ ⮪  ­  è¨à®ª®© á⥭ª¥ ¤«ï ¢®«­ë H10 . ‹¨­¨¨ ⮪  ¤«ï ¢®«­ Hnm ¯®«ãç îâáï ¯ã⥬ ¯®¢â®à¥­¨ï ª à⨭ë, ¯®«ã祭®© ¤«ï ¢®«­ë H11 , n ¨ m à § ­  à §«¨ç­ëå á⥭ª å. ‚ëç¨á«¨¬ â ª¦¥ ¬®é­®áâì, à á¯à®áâà ­ïîéãîáï ç¥à¥§ ¢®«­®¢®¤ á ¢®«­®© H10 . â® ¯®§¢®«¨â ­ ©â¨ ­®à¬¨àãî騩 ¬­®¦¨â¥«ì. Œ®é­®áâì ®¯à¥¤¥«ï¥âáï ¯®â®ª®¬ ¢¥ªâ®à  ®©­â¨­£  ç¥à¥§ á¥ç¥­¨¥ ¢®«­®¢®¤ : Z Z 1 1  ~ ~ P = 2 Re (E  H )
~z0 dS = 2 Re (Ex Hy ; Ey Hx ) dS: S

S

„«ï ¬®¤ë H10 ®â«¨ç­ë ®â ­ã«ï «¨èì Ey ¨ Hx:

Ey = ; Aj! a sin a x; Hx = Aj a sin a : ®¤áâ ¢«ïï Ey ¨ Hx ¯®¤ ¨­â¥£à «, ¯®«ã稬 2 P = A2 !4
ab ; ®âªã¤  ¬®¦¥â ¡ëâì ®¯à¥¤¥«¥­ ­®à¬¨àãî騩 ¬­®¦¨â¥«ì:

A = 2

s

P a !
b :

Œ®é­®áâì ¬®¦¥â ¡ëâì ¢ëà ¦¥­  â ª¦¥ ç¥à¥§ ­ ¯à殮­­®áâì í«¥ªâà¨ç¥áª®£® ¯®«ï ¢ 業âॠ¢®«­®¢®¤  E0 : 73

s

£¤¥  =

p=".

2 2 P = E40 ab 1 ; 2 ; ªà

3.2.2. ‚®«­ë E-⨯ 

„«ï ä㭪樨 , ®¯à¥¤¥«ïî饩 í«¥ªâà¨ç¥áª¨¥ ¢®«­ë, ¨¬¥¥â ¬¥áâ® ãà ¢­¥­¨¥ @ 2  + @ 2  + g2  = 0 @x2 @y2 ¯à¨ ãá«®¢¨¨ ­  £à ­¨æ¥  = 0. ¥è¥­¨¥ ãà ¢­¥­¨ï ­ ©¤¥¬ ¢ ¢¨¤¥

(x; y) = X (x)
Y (y): „«ï ä㭪権 X ¨ Y  ­ «®£¨ç­® ¯à¥¤ë¤ã饬㠬®¦­® § ¯¨á âì

X = A cos gx x + B sin gxx; Y = C cos gy y + D sin gy y: ƒà ­¨ç­ë¥ ãá«®¢¨ï ⥯¥àì ¤à㣨¥. € ¨¬¥­­®, ¯à¨ x = 0 X = 0, ®âªã¤  á«¥¤ã¥â, çâ®

A = 0; X = sin gx x; ¯à¨ y = 0 Y = 0, ®âªã¤ 

C = 0; Y = sin gy y; ¯à¨ x = a X = 0, â.¥. sin gx a = 0, ®âªã¤ 

gxa = n ¨ gx = n a ; n = 0; 1; 2; : : : ; ¯à¨ y = b Y = 0, â.¥. sin gy b = 0, ®âªã¤  gy b = m ¨ gy = m b ; m = 0; 1; 2; : : :: 74

”ã­ªæ¨ï (x; y) à ¢­ 

m (x; y) = sin n a x
sin b y: ‡ ¬¥â¨¬, çâ® ­¨ ®¤¨­ ¨§ ¨­¤¥ªá®¢ n ¨ m ¢ í⮬ á«ãç ¥ ­¥ ¬®¦¥â

¯à¨­¨¬ âì ­ã«¥¢®¥ §­ ç¥­¨¥, â ª ª ª íâ® ¯à¨¢¥¤¥â ª ⮦¤¥á⢥­­®¬ã ®¡à é¥­¨î ¢ ­ã«ì ä㭪樨 . Š ª ¨ ¤«ï ¬ £­¨â­ëå ¬®¤, ᮡá⢥­­ë¥ §­ ç¥­¨ï ®¯à¥¤¥«ïîâáï ᮮ⭮襭¨¥¬     2 = n 2 + m 2 n = 1; 2; : : :; m = 1; 2; : : : : gnm

a

b

Žâá ¬®£ãâ ¡ëâì ­ ©¤¥­ë ªà¨â¨ç¥áª¨¥ ¤«¨­ë ¢®«­ (¤«ï ¢ ªã㬠): nm = g2 = s 2 2  2 = s 2 2  2 : nm n + m n + m a b a b ’¥¯¥àì ¬®¦­® § ¯¨á âì â ª¦¥ ¢ëà ¦¥­¨ï ¤«ï á®áâ ¢«ïîé¨å ¯®«ï:

Hz = 0 ; m sin n x
cos m y; Hx = j!" @ = j!" @y b a b @ n n Hy = ;j!" @x = ; j!" a cos a x
sin m y;  n 2  mb 2 n m y; Ez = (k2 + 2 )
(x; y) = +
sin x
sin a b a b @ n n m Ex = ; @x = ; j a cos a x
sin b y; m sin n x
cos m y: = ; j Ey = ; @ @y b a b ’ ª ª ª n ¨ m ­¥ ¬®£ã⠯ਭ¨¬ âì ­ã«¥¢ë¥ §­ ç¥­¨ï, â® á ¬®© ­¨§ª®ç áâ®â­®© ¬®¤®© ¡ã¤¥â E11 , ¤«ï ª®â®à®© ªà¨â¨ç¥áª ï ¤«¨­  ¢®«­ë à ¢­ 

11 =

s 2 2  2 1 1 a

+

b

75

= p 22ab 2 : a +b

„ ­­ ï ªà¨â¨ç¥áª ï ¤«¨­  ¢®«­ë ¬¥­ìè¥ ªà¨â¨ç¥áª®© ¤«¨­ë ¢®«­ë á ¬®© ­¨§ª®ç áâ®â­®© ¬ £­¨â­®© ¬®¤ë H10 , à ¢­®© 2a. 2 ¨ ᮮ⢥⎡à â¨¬ ¢­¨¬ ­¨¥ ­  â®, ç⮠ᮡá⢥­­ë¥ §­ ç¥­¨ï gnm á⢥­­® ªà¨â¨ç¥áª¨¥ ¤«¨­ë ¢®«­ nm ¤«ï ¢ëáè¨å í«¥ªâà¨ç¥áª¨å ¨ ¬ £­¨âëå ¬®¤ ᮢ¯ ¤ îâ. ’ ª¨¬ ®¡à §®¬, ¨¬¥¥â ¬¥áâ® ¤¢ãªà â­®¥ ¢ë஦¤¥­¨¥ ¤«ï ¬®¤ á ­¥­ã«¥¢ë¬¨ ¨­¤¥ªá ¬¨. „«ï ¢®«­ë E11 ¨¬¥¥¬

Hz = 0; Hx = j!" b sin a x
cos b y; Hy = ;j!" a cos a x
sin b y;   2   2   y; Ez = +
sin x
sin a b a b    Ex = ;j a cos a x
sin b y; Ey = ;j b sin a x
cos b y:  áᬮâਬ áâàãªâãàã ¯®«ï E11 (á¬. à¨á.). y

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11

11111111111 00000000000 00000000000 11111111111 00000000000111111111111 11111111111 000000000000 000000000000 111111111111 E x111111111111 000000000000

y

x

y

111111111111111111111 000000000000000000000 000000000000000000000 111111111111111111111 00 11 000000000000000000000 111111111111111111111 00 11 00 11

11 00 00 11 00 11 00 11 00 11 00 11 000 111 000 111 000 111 000 111 000 111 000 111

11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 Ey

76

x

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11

x

Ez

y

b

x a

Λ 2

®«ï ¢ëáè¨å ¬®¤ ¯®«ãç îâáï ¯ã⥬ ¬­®£®ªà â­®£® ¯®¢â®à¥­¨ï ª àâ¨­ë ¯®«ï E11 .  ©¤¥¬ â ª¦¥ «¨­¨¨ ⮪®¢ ¢ á⥭ª å. ’ ª ª ª ¬ £­¨â­®¥ ¯®«¥ ¤«ï E-¬®¤ ¯®¯¥à¥ç­®, â®, ®ç¥¢¨¤­®, ⮪¨ ¬®£ãâ ¡ëâì ⮫쪮 ¯à®¤®«ì­ë¬¨ (á¬. à¨á.).

11111111111111111111111111111111111111 00000000000000000000000000000000000000 00000000000000000000000000000000000000 11111111111111111111111111111111111111 00000000000000000000000000000000000000 11111111111111111111111111111111111111 00000000000000000000000000000000000000 11111111111111111111111111111111111111 00000000000000000000000000000000000000 11111111111111111111111111111111111111 77

3.3.

Šàã£«ë¥ ¢®«­®¢®¤ë

3.3.1. Œ £­¨â­ë¥ ¬®¤ë

à¨ ¨áá«¥¤®¢ ­¨¨ ªà㣫®£® ¢®«­®¢®¤  « ¯« á¨ ­ ¢ ãà ¢­¥­¨ïå ¤«ï ä㭪権 ¨  㤮¡­® § ¯¨á뢠âì ¢ ¯®«ïà­®© á¨á⥬¥ ª®®à¤¨­ â:  @  1 @2 @ 1
xy = r @r r @r + r2 @'2 : ®¤áâ ¢«ïï íâ®â « ¯« á¨ ­ ¢ ãà ¢­¥­¨¥ ¤«ï , ¯®«ã稬 @ 2 + 1 @ + 1 @ 2 + g2 = 0: @r2 r @r r2 @'2 “à ¢­¥­¨¥ à¥è ¥âáï ¬¥â®¤®¬ à §¤¥«¥­¨ï ¯¥à¥¬¥­­ëå. „«ï í⮣® ¯®« £ ¥¬, çâ® = R(r)
('): ®¤áâ ¢«ïï ¢ ¯à¥¤ë¤ã饥 ãà ¢­¥­¨¥ ¨ ¤¥«ï ­  R
, ¯®«ã稬 1 d2 R + 1 dR + 1 d2  + g2 = 0: R dr2 rR dr r2  d'2 „ ­­®¥ ãà ¢­¥­¨¥ à á¯ ¤ ¥âáï ­  ¤¢ : 1 d2  = ; 2 ;  d'2 1 d2 R + 1 dR ; 2 + g2 = 0: R dr2 rR dr r2 ¥à¢®¥ ãà ¢­¥­¨¥ ¯à¨¢®¤¨âáï ª ¢¨¤ã d2  + 2  = 0: d'2 ¥è¥­¨¥ â ª®£® ãà ¢­¥­¨ï ¨¬¥¥â ¢¨¤  '  = sin cos ' : ’ ª ª ª à¥è¥­¨¥ ¯¥à¨®¤¨ç­® ¯® ', â® ¤«ï ' = 2 ¤®«¦­® ¢ë¯®«­ïâìáï ãá«®¢¨¥ 78

2 = 2m; m = 0; 1; 2; : : :; ®âªã¤ 

 = m; £¤¥ m ¯à¨­¨¬ ¥â 㪠§ ­­ë¥ §­ ç¥­¨ï. “à ¢­¥­¨¥ ¤«ï R ¯à¨­¨¬ ¥â ¢¨¤ d2 R + 1 dR + (g2 ; m2 ) = 0: dr2 R dr r2 â® ãà ¢­¥­¨¥ ¥áᥫï. …£® à¥è¥­¨ï¬¨ ïîâáï ä㭪樨 ¥áá¥«ï ¨ ¥©¬ ­  Jm (gr) ¨ Nm (gr). ‚â®à®¥ à¥è¥­¨¥ ¤®«¦­® ¡ëâì ®â¡à®è¥­®, â ª ª ª ®¡à é ¥âáï ¢ ¡¥áª®­¥ç­®áâì ¢ 業âॠ(¯à¨ r = 0). ®í⮬ã

R = Jm (gr): „«ï ¬ë ¨¬¥«¨ ãá«®¢¨¥ @@n = 0 ­  £à ­¨æ¥ á¥ç¥­¨ï ¢®«­®¢®¤ . ‚ ¤ ­­®¬ á«ãç ¥ @@n = @@r . ®í⮬㠣࠭¨ç­®¥ ãá«®¢¨¥ âॡã¥â, ç⮡ë Jm0 (ga) = 0; £¤¥ a { à ¤¨ãá ¢®«­®¢®¤ . Š®à­¨ ¯à®¨§¢®¤­®© ä㭪樨 ¥áá¥«ï ®¡®§­ ç¨¬ ç¥à¥§ t0mn : 0 a = t0 ; gmn mn

®âªã¤ 

0 = t0mn ; n = 1; 2; : : : : gmn a

„ ­­ ï ä®à¬ã«  ¤ ¥â ᮡá⢥­­ë¥ §­ ç¥­¨ï § ¤ ç¨, ª®â®àë¥ § ¢¨áï⠮⠤¢ãå ¨­¤¥ªá®¢. ‘®¡á⢥­­ë¥ ä㭪樨 mn ¢ ¤ ­­®¬ á«ãç ¥ ¨¬¥îâ ¢¨¤ 0 r) cos m' : mn = Jm (gmn sin m' 79

0 ᮮ⢥âáâ¢ãîâ ¤¢¥ ᮡŒë ¢¨¤¨¬ §¤¥áì, çâ® ª ¦¤®¬ã §­ ç¥­¨î gmn á⢥­­ë¥ ä㭪樨, ªà®¬¥ á«ãç ï m = 0, â.¥. ¨¬¥¥â ¬¥áâ® ¤¢ãªà â­®¥ ¢ë஦¤¥­¨¥. ˆ¬¥ï ¢ëà ¦¥­¨¥ ¤«ï ä㭪樨 , ¬ë ¬®¦¥¬ ­ ©â¨ ⥯¥àì ¢á¥ ª®¬¯®­¥­âë ¯®«¥©. „«ï í⮣® ¢®á¯®«ì§ã¥¬áï ¯®«ã祭­ë¬¨ à ­¥¥ ᮮ⭮襭¨ï¬¨ ¤«ï ¬ £­¨â­ëå ¬®¤

H~ = ;
grad + g2
~z0 ; E~ = ; j!
grad  ~z0 : “ç¨â뢠ï, çâ® á®áâ ¢«ïî騥 ¢ ¯®«ïà­®© á¨á⥬¥ ª®®à¤¨­ â äã­ª@ ; 1 @ ) ¨ grad  ~z0 ( 1 @ ; ; @ ), ¯®«ã稬 権 grad ( @r r @' r @' @r 0 2
Jm (g0 r) cos m' ; Hz = gmn mn sin m' 0
J 0 (g0 r) cos m' ; Hr = ; @@r = ; j gmn m mn sin m' sin m' 0 H' = ; r @@' =  j m r Jm (gmn r) cos m' ; @ =  j!m J (g0 r) sin m' ; Er = ; j! r @' r m mn cos m' 0
J 0 (g0 r) cos m' : E' = j! @@r = j!gmn m mn sin m'

Šà¨â¨ç¥áª ï ¤«¨­  ¢®«­ë ¬®¦¥â ¡ëâì ­ ©¤¥­  ¯® ä®à¬ã«¥  = 2 = 2a : mn

0 gmn

t0mn

Žá­®¢­ ï ¬®¤  ᮮ⢥âáâ¢ã¥â ­ ¨¬¥­ì襩 ¢¥«¨ç¨­¥ t0mn . ’ ª®© ¬®¤®© ï¥âáï ¬®¤  H11 , ¤«ï ª®â®à®© t011  1:84. â® ­ ¨¬¥­ì訩 ª®à¥­ì ãà ¢­¥­¨ï J10 (t) = 0 (â ¡«. 1). à¨ m = 1 ¨ n = 1 ¯®«ãç ¥âáï ¬®¤  H11 , ¯® áâàãªâãॠ¯®«ï தá⢥­­ ï ¬®¤¥ H10 ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥. Š®¬¯®­¥­âë ¯®«ï ¤«ï í⮩ ¬®¤ë à ¢­ë 80

n

m 1 2 3

’ ¡«¨æ  1: Š®à­¨ ãà ¢­¥­¨ï

0 Jm (t) = 0.

0 1 2 3 4 5 3.83 1.84 3.05 4.20 5.32 6.42 7.016 5.33 6.71 8.01 9.286 10.52 10.17 8.53 9.972 11.355 12.68 13.99 0 2
J1 (g0 r) cos ' ; Hz = g11 11 sin ' 0
J 0 (g0 r) cos ' ; Hr = ;j g11 1 11 sin ' 0 r) sin ' ; H' =  j J ( g 1 11 cos ' r 0 r) sin ' ; Er =  j! J ( g 1 11 cos ' r 0
J 0 (g0 r) cos ' : E' = j!g11 1 11 sin '

‘âàãªâãà  ¯®«ï H11 ¬®¤ë ¯à¥¤áâ ¢«¥­  ­  à¨á. H 11 H

E

‚ëè¥ ã¦¥ ®â¬¥ç «®áì, çâ® ¢ ªà㣫®¬ ¢®«­®¢®¤¥ ¬®¤  H11 ï¥âáï ¤¢ãªà â­® ¢ë஦¤¥­­®©. …© ᮮ⢥âáâ¢ãîâ ¤¢¥ ¯ àë ¢¥ªâ®à­ëå ä㭪権, ®â«¨ç îé¨åáï ®¤­  ®â ¤à㣮© ¯®«ïਧ æ¨¥© ­  90o. ‚®«­ , ¯à®¨§¢®«ì­® ¯®«ïਧ®¢ ­­ ï, ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­  ¢ ¢¨¤¥ «¨­¥©­®© ª®¬¡¨­ æ¨¨ ¤¢ãå ®á­®¢­ëå ᮡá⢥­­ëå ä㭪権. ‚ë஦¤¥­¨¥ ¯à¨¢®¤¨â ª àï¤ã ­¥ã¤®¡á⢠¯à¨ ¨á¯®«ì§®¢ ­¨¨ ªà㣫ëå ¢®«­®¢®¤®¢ á ¬®¤®©

81

H11 ¤«ï ¯¥à¥¤ ç¨ ‘‚—-ª®«¥¡ ­¨©.

à¨ ­ «¨ç¨¨ ­¥â®ç­®á⥩ ¨ ­¥®¤­®à®¤­®á⥩ ¢ë஦¤¥­¨¥ á­¨¬ ¥âáï, ¨ ¯®áâ®ï­­ë¥ à á¯à®áâà ­¥­¨ï ¤¢ãå ⨯®¢ ¢®«­ áâ ­®¢ïâáï à §«¨ç­ë¬¨. â® ¯à¨¢®¤¨â ª ¨§¬¥­¥­¨î ä §®¢ëå ᮮ⭮襭¨© á®áâ ¢«ïîé¨å ¨ ¢ १ã«ìâ â¥ ª ¨§¬¥­¥­¨î ¢¨¤  ¨ ¯«®áª®á⨠¯®«ïਧ æ¨¨. „«ï ⮣® ç⮡ë í⮣® ¨§¡¥¦ âì, ¨­®£¤  ¯à¥¤­ ¬¥à¥­­® ᨫ쭮 ¨áª ¦ îâ ¯®¯¥à¥ç­®¥ á¥ç¥­¨¥ ¢®«­®¢®¤  (á¬. à¨á.), çâ®¡ë ¬ «ë¥ ­¥â®ç­®á⨠­¥ ¬®£«¨ áãé¥á⢥­­® ¨áª §¨âì ª à⨭ã. ‚®«­ë ⨯  Hm1 ¬®£ãâ ¡ëâì ¯®«ãç¥­ë ¨§ ¢®«­ë H11 ¯ã⥬  §¨¬ãâ «ì­®© ¤¥ä®à¬ æ¨¨ (á¬. à¨á.).

‚®«­ë H1n ¯®«ãç îâáï ­ à é¨¢ ­¨¥¬ ¢ à ¤¨ «ì­®¬ ­ ¯à ¢«¥­¨¨ (á¬. à¨á.). ‚®«­ë ⨯  Hmn ¯à¨ m > 1, n > 1 ¯®«ãç îâáï  ­ «®£¨ç­ë¬ ®¡à §®¬. Žá®¡ë© ¨­â¥à¥á ¯à¥¤áâ ¢«ïîâ ᨬ¬¥âà¨ç­ë¥ ¬®¤ë, ¨§ ª®â®àëå ¯¥à¢®© ï¥âáï ¬®¤  H01 (n = 1). Š®¬¯®­¥­âë ¯®«¥© í⮩ ¬®¤ë à ¢­ë

82

Hz Hr H' Er E'

0 2
J0 (g0 r); g01 01 ;j g010
J00 (g010 r) = j g010 J1 (g010 r); 0; 0; ;j!g010
J00 (g010 r) = j!g010 J1 (g010 r):

= = = = =

‘âàãªâãà  ¯®«ï ¯à¥¤áâ ¢«¥­  ­  à¨á. E H

Žâ¬¥â¨¬, çâ® ­  á⥭ª å ¢®«­®¢®¤  ¬ £­¨â­®¥ ¯®«¥ ¨¬¥¥â ⮫쪮 ¯à®¤®«ì­ãî á®áâ ¢«ïîéãî. â® §­ ç¨â, ç⮠⮪¨ ¢ á⥭ª å ¨¬¥îâ ⮫쪮 ¯®¯¥à¥ç­ãî á®áâ ¢«ïîéãî ¨ ®¡à §ãîâ § ¬ª­ãâë¥ «¨­¨¨ (á¬. à¨á.). Žá®¡¥­­®áâìî H01 ¬®¤ë ïîâáï ®ç¥­ì ¬ «ë¥ ¯®â¥à¨ ¢ á⥭ª å, ª®â®àë¥ ã¬¥­ìè îâáï á à®á⮬ ç áâ®âë. ‚ á¢ï§¨ á í⨬ ¬®¤  H01 ¯à¥¤áâ ¢«ï¥â ¨­â¥à¥á ¤«ï ¯¥à¥¤ ç¨ í­¥à£¨¨ ­  ¡®«ì訥 à ááâ®ï­¨ï (­ ¯à¨¬¥à, ¢ ¢®«­®¢®¤­ëå «¨­¨ïå í«¥ªâய¥à¥¤ ç¨). Ž¤­ ª® á¨âã æ¨ï ®á«®¦­ï¥âáï ⥬, çâ® ¬®¤  H01 ­¥ ï¥âáï Λ 2 ®á­®¢­®© ¢ ªà㣫®¬ ¢®«­®¢®¤¥. 3.3.2. «¥ªâà¨ç¥áª¨¥ ¬®¤ë

“à ¢­¥­¨¥ ¤«ï ä㭪樨  ¨¬¥¥â â®â ¦¥ ¢¨¤, çâ® ¨ ¤«ï ä㭪樨 , ¯®í⮬ã à¥è¥­¨¥  ­ «®£¨ç­®, â.¥. 83

m' = Jm (gr) cos sin m' : ƒà ­¨ç­®¥ ãá«®¢¨¥ ¢ í⮬ á«ãç ¥ ¨¬¥¥â ¢¨¤  = 0 ¯à¨ r = a; â.¥. Jm (ga) = 0; £¤¥ a { à ¤¨ãá ¢®«­®¢®¤ . Š®à­¨ ä㭪樨 ¥áᥫï m-£® ¯®à浪  ®¡®§­ ç¨¬ ç¥à¥§ tmn : gmn a = tmn ; ®âªã¤  Žâá ¨¬¥¥¬

gmn = tmn a ; n = 1; 2; : : : :

m' mn = Jm (gmn r) cos sin m' : Š ª ¢¨¤¨¬, ä㭪樨  ¯à¨ m 6= 0 â ª¦¥ ¤¢ãªà â­® ¢ë஦¤¥­ë. ‘ ¯®¬®éìî ä㭪樨  ¬®¦­® ⥯¥àì ¯®«ãç¨âì á®áâ ¢«ïî騥 ¯®«ï

¤«ï E-¢®«­:

Hz = 0; @ j!"m sin m' Hr = j!" r @' =  r Jm (gmn r) cos m' ; cos m' 0 H' = ;j!" @ @r = ; j!"
gmn Jm (gmn r) sin m' ; 2  = gmn 2 Jm (gmnr) cos m' ; Ez = gmn sin m' cos m' 0 Er = ; @ @r = ; j gmn Jm (gmn r) sin m' ; @ =  j m J (g r) sin m' : E' = ; r @' r m mn cos m' 84

 ¨¡®«¥¥ ¯à®á⮩ áâàãªâãன ®¡« ¤ ¥â ᨬ¬¥âà¨ç­ ï E01 ¬®¤ :

Hz Hr H' Ez Er E'

= = = = = =

0; 0;

;j!"
g01 J00 (g01 r) = j!"g01J1 (g01 r); 2 J0 (g01 r); g01 ;j g01 J00 (g01 r) = j g01 J1 (g01 r); 0:

‘âàãªâãà  ¯®«ï ¨§®¡à ¦¥­  ­  à¨á. E

01

‘âàãªâãà  ¯®«¥© ⨯  E0n ¬®¦¥â ¡ëâì ¯®«ã祭  ᦠ⨥¬ ¯® à ¤¨ãáã.  áᬮâਬ â ª¦¥ áâàãªâãàã ¯®«¥© E11 ¬®¤ë (á¬. à¨á.).

Š®à­¨ ä㭪樨 ¥áᥫï Jm (x) ¯à¨¢¥¤¥­ë ¢ â ¡«¨æ¥ 2.

85

n

m 1 2 3

’ ¡«¨æ  2: Š®à­¨ ãà ¢­¥­¨ï

Jm (t) = 0.

0 1 2 3 4 5 2.40 3.83 5.14 6.38 7.59 8.77 5.52 7.016 8.42 9.76 11.06 12.34 8.65 10.17 11.62 13.015 14.37

3.3.3. ‚®«­®¢®¤­ë¥ ¬®¤ë ¢ ª® ªá¨ «ì­®© «¨­¨¨

‚ ª® ªá¨ «ì­®© «¨­¨¨ ­ àï¤ã á TEM-¢®«­®© ¬®£ãâ à á¯à®áâà ­ïâìáï â ª¦¥ ¢®«­®¢®¤­ë¥ ¬®¤ë, ¨¬¥î騥 ¯à®¤®«ì­ãî ª®¬¯®­¥­âã í«¥ªâà¨ç¥áª®£® ¨«¨ ¬ £­¨â­®£® ¯®«ï.  áᬮâਬ ª®à®âª® H-¢®«­ë ¢ ª® ªá¨ «ì­®© «¨­¨¨. “à ¢­¥­¨¥ ¤«ï ä㭪樨 ¯®-¯à¥¦­¥¬ã ¨¬¥¥â ¢¨¤ @ 2 + 1 @ + 1 @ 2 + g2 = 0: @r2 r @r r2 @'2 „ ­­®¥ ãà ¢­¥­¨¥ à á¯ ¤ ¥âáï ­  ¤¢ . ‚ ç áâ­®áâ¨, ¤«ï R(r) ¬ë ¨¬¥«¨ ãà ¢­¥­¨¥ ¥áᥫï d2 R + 1 dR + (g2 ; m2 ) = 0: dr2 R dr r2

’ ª ª ª á¥ç¥­¨¥ «¨­¨¨ ¤¢ãá¢ï§­® ¨ 業âà (r = 0) ­¥ ¢å®¤¨â ¢ ®¡« áâì ¯®«ï, â® à¥è¥­¨¥ ¤®«¦­® ¡ëâì ¯à¥¤áâ ¢«¥­® ¢ ¢¨¤¥ «¨­¥©­®© ª®¬¡¨­ æ¨¨ ä㭪樨 ¥áᥫï Jm (gr) ¨ ä㭪樨 ¥©¬ ­  Nm (gr):

R = A
Jm (gr) + B
Nm(gr): „ ­­ ï äã­ªæ¨ï ¤®«¦­  㤮¢«¥â¢®àïâì £à ­¨ç­ë¬ ãá«®¢¨ï¬ dR dr = 0 ¯à¨ r = a ¨ r = b (b > a), â.¥.

A
Jm0 (ga) + B
Nm0 (ga) = 0; A
Jm0 (gb) + B
Nm0 (gb) = 0: ®«ã祭­ ï á¨á⥬  ®¤­®à®¤­ëå «¨­¥©­ëå ãà ¢­¥­¨© ®â­®á¨â¥«ì­®

A ¨ B ᮢ¬¥áâ­  «¨èì ¯à¨ à ¢¥­á⢥ ­ã«î ®¯à¥¤¥«¨â¥«ï á¨á⥬ë 86

Jm0 (ga)
Nm0 (gb) ; Jm0 (gb)
Nm0 (ga) = 0: 0 , ®¯à¥¤¥«ïî騥 ˆ§ ¤ ­­®£® ãà ¢­¥­¨ï ¬®¦­® ®¯à¥¤¥«¨âì ª®à­¨ gmn ᮡá⢥­­ë¥ ä㭪樨 mn . â¨ ª®à­¨ § ¢¨áïâ ®â à ¤¨ãá  a ¨ ®â­®è¥­¨ï b=a (á¬. à¨á.). 2a

111 000 000 111 000 111 2b

‘âàãªâãà  ¯®«¥© ¬®¦¥â ¡ëâì ¯®«ã祭  ¨§ áâàãªâãàë ᮮ⢥âáâ¢ãîé¨å ¬®¤ ¢ ªà㣫®¬ ¢®«­®¢®¤¥. €­ «®£¨ç­® í⮬ã, ¤«ï í«¥ªâà¨ç¥áª¨å ¬®¤ ¬®¦­® ¯®«ãç¨âì ¤¨á¯¥àᨮ­­®¥ ãà ¢­¥­¨¥

Jm (ga)
Nm (gb) ; Jm (gb)
Nm(ga) = 0: à¥¤áâ ¢«ï¥â ¨­â¥à¥á ¨¬¥âì ¯à¨¡«¨¦¥­­ãî ®æ¥­ªã ¤«ï ªà¨â¨ç¥áª®© ¤«¨­ë ¢®«­ë ­¨§è¨å ¢®«­®¢®¤­ëå ¬®¤.  ¨¡®«ì襩 ªà¨â¨ç¥áª®© ¤«¨­®© ¢®«­ë ®¡« ¤ ¥â ¬ £­¨â­ ï ¬®¤  H11 , ¤«ï ª®â®à®© 11  2 a +2 b ; â® ¥áâì ¤«¨­¥ á।­¥© ®ªà㦭®áâ¨. „«ï ⮣® çâ®¡ë ¢®«­®¢®¤­ë¥ ¬®¤ë ¢ ª® ªá¨ «ì­®© «¨­¨¨ ­¥ à á¯à®áâ࠭﫨áì, ­¥®¡å®¤¨¬® ­  ª®à®âª¨å ¢®«­ å 㬥­ìè âì ¯®¯¥à¥ç­ë¥ à §¬¥àë íâ¨å «¨­¨©. „ ­­®¥ âॡ®¢ ­¨¥ ®£à ­¨ç¨¢ ¥â ¤®¯ãáâ¨¬ë¥ ¯®¯¥à¥ç­ë¥ à §¬¥àë ª® ªá¨ «ì­®© «¨­¨¨ ¢ § ¢¨á¨¬®á⨠®â à ¡®ç¥£® ¤¨ ¯ §®­  ¤«¨­ ¢®«­. 87

3.4.

‚®«­®¢®¤ë á á¥ç¥­¨¥¬ á«®¦­®© ä®à¬ë ¨ ª¢ §¨áâ â¨ç¥áª¨¥ ¢®«­®¢®¤ë

Šà®¬¥ ¯àאַ㣮«ì­ëå ¨ ªà㣫ëå ¢®«­®¢®¤®¢ â®ç­®¥ à¥è¥­¨¥ ¬®¦¥â ¡ëâì ¯®«ã祭® â ª¦¥ ¤«ï í««¨¯â¨ç¥áª¨å ¢®«­®¢®¤®¢. „«ï ¢á¥å ®áâ «ì­ëå ¢®«­®¢®¤®¢ ¯à¨å®¤¨âáï ¯®«ì§®¢ âìáï ¨áªãáá⢥­­ë¬¨, ¯à¨¡«¨¦¥­­ë¬¨ ¨«¨ ¢ëç¨á«¨â¥«ì­ë¬¨ ¬¥â®¤ ¬¨.  §à ¡®â ­ë ¢ëç¨á«¨â¥«ì­ë¥ ¯à®£à ¬¬ë, ¯®§¢®«ïî騥 à ááç¨â뢠âì ¯®«ï ¨ ªà¨â¨ç¥áª¨¥ ç áâ®âë ¤«ï ­¥ª®â®à®£® ç¨á«  ¬®¤ ­¨§è¨å ¯®à浪®¢. ‚ à拉 á«ãç ¥¢ à¥è¥­¨¥ ¬®¦¥â ¡ëâì ¯®«ã祭® ¨§ à¥è¥­¨ï ¤«ï ªà㣫®£® ¨«¨ ¯àאַ㣮«ì­®£® ¢®«­®¢®¤®¢ ¯ã⥬ ¢¢¥¤¥­¨ï ¢ ¯®á«¥¤­¨¥ ¯à®¢®¤ïé¨å ¯¥à¥£®à®¤®ª.  áᬮâਬ ­¥ª®â®àë¥ á¥ç¥­¨ï, ¯®«ãç ¥¬ë¥ â ª¨¬ ®¡à §®¬. Š¢ ¤à â­ë© ¢®«­®¢®¤ ¥âà㤭® ¯®ª § âì, çâ® ¤«ï â ª®£® ⨯  ¢®«­ë í«¥ªâà¨ç¥áª¨¥ ᨫ®¢ë¥ «¨­¨¨ ¯¥à¯¥­¤¨ªã«ïà­ë ¤¨ £®­ «ì­ë¬ ¯«®áª®áâï¬,   «¨­¨¨ ¬ £­¨â­®£® ¯®«ï ª á â¥«ì­ë ª í⨬ ¯«®áª®áâï¬ (á¬. à¨á.). ®í⮬ã, ¥á«¨ ¢¢¥á⨠¯à®¢®¤ï騥 ¯®¢¥àå­®á⨠¢ ¤¨ £®­ «ì­ëå ¯«®áª®áâïå, â® £à ­¨ç­ë¥ ãá«®¢¨ï ­¥ ­ àãè îâáï,   ¢®«­®¢®¤ à á¯ ¤ ¥âáï ¯à¨ í⮬ ­  ¢®«­®¢®¤ë âà¥ã£®«ì­®£® á¥ç¥­¨ï, ¨¬¥î騥 âã ¦¥ ªà¨â¨ç¥áªãî ¢®«­ã, çâ® ¨ ¨á室­ë© ¢®«­®¢®¤.

H11

Šàã£«ë© ¢®«­®¢®¤ (á¬. à¨á.) H11

H01

88

E 11

Š® ªá¨ «ì­ë© ¢®«­®¢®¤ (á¬. à¨á.) H11

Žâ¤¥«ì­ë© ª« áá ¢®«­®¢®¤®¢ á®áâ ¢«ïîâ ¢®«­®¢®¤ë á á¥ç¥­¨¥¬, ᢥà­ãâë¬ ¨§ á¥ç¥­¨ï ¯àאַ㣮«ì­®£® ¢®«­®¢®¤  (á¬. à¨á.). a

’ ª¨¬ ®¡à §®¬ ¬®£ãâ ¡ëâì ¯®«ãç¥­ë ¢®«­®¢®¤ë, ¨¬¥î騥 ¯®¯¥à¥ç­®¥ á¥ç¥­¨¥, ¬ «®¥ ¯® áà ¢­¥­¨î á ¤«¨­®© ¢®«­ë.  ¨¡®«ì襥 ¯à¨¬¥­¥­¨¥ ¨§ íâ¨å ¢®«­®¢®¤®¢ ¯®«ã稫¨ { ¨ { ®¡à §­ë¥ ¢®«­®¢®¤ë (á¬. à¨á.).

 ª®­¥æ, ¨¬¥îâáï â ª¦¥ ª¢ §¨áâ â¨ç¥áª¨¥ ¢®«­®¢®¤ë, ®¡à §ã¥¬ë¥ ¨§ { ®¡à §­ëå ¯ã⥬ 㬥­ì襭¨ï § §®à  (á¬. à¨á.). ’ ª¨¥ ¢®«­®¢®¤ë ¢¤ «¨ ®â ªà¨â¨ç¥áª®© ¢®«­ë ¡«¨§ª¨ ª ¯«®áª®© «¨­¨¨. Ž­¨ ¨¬¥îâ ªà¨â¨ç¥áªãî ¢®«­ã ¬­®£® ¡®«ìè¥ à §¬¥à®¢ ¯®¯¥à¥ç­®£® á¥ç¥­¨ï. Š¢ §¨áâ â¨ç¥áª¨¬¨ ïîâáï â ª¦¥ 饫¥¢ë¥ ¢®«­®¢®¤ë. ’ ª ª ª â ª¨¥ ¢®«­®¢®¤ë ¨¬¥îâ ¯®¯¥à¥ç­®¥ á¥ç¥­¨¥, ¬ «®¥ ¯® áà ¢­¥­¨î á ¤«¨­®© ¢®«­ë, â® ¯®«¥ ¢ ­¨å ª¢ §¨áâ æ¨®­ à­® ¨ ¤«ï à áç¥â®¢ ¬®¦­® ¨á¯®«ì-

89

§®¢ âì ⥫¥£à ä­ë¥ ãà ¢­¥­¨ï.  áᬮâਬ, ­ ¯à¨¬¥à, 饫¥¢®© ¢®«­®¢®¤ (á¬. à¨á.). ‚ ­¥¬ ¢ 饫¨ ¯®«ï ¯®¯¥à¥ç­ë,   ¢ ®áâ «ì­®© ç á⨠¬ £­¨â­®¥ ¯®«¥ ¯à®¤®«ì­® ¨ ¯®¯¥à¥ç­®,   í«¥ªâà¨ç¥áª¨¬ ¯®«¥¬ ¬®¦­® ¯à¥­¥¡¥çì. ª¢¨¢ «¥­â­ ï á奬  ï祩ª¨ ¡ã¤¥â ¨¬¥âì á«¥¤ãî騩 ¢¨¤ (á¬. à¨á.) ‡¤¥áì

Z1 = j!L¯à®¤ 1 ; Y1 = j!C1 + j!L1¯®¯ : 1 ®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï à ¢­ 

=

p

Z1 Y1 =

s

s



j!L¯à®¤ j!C1 + j!L1¯®¯ 1 1





=



1 ; !C : = !L¯à®¤ 1 1 !L¯®¯ 1 Žç¥¢¨¤­®, çâ® à á¯à®áâà ­¥­¨¥ ¨¬¥¥â ¬¥áâ® ¯à¨ ãá«®¢¨¨ 1 ; !C < 0; â:¥: ! > p 1 = !ªà: 1 !L¯®¯ L¯®¯ 1 1 C1 …᫨ íâ® ãá«®¢¨¥ ¢ë¯®«­¥­®, â®

s



s







1 !C1 ; !L1¯®¯ = j! L¯à®¤

= j !L¯à®¤ 1 1 C1 1 ; !2 L¯®¯C1 : 1 1 2 , L¯à®¤ ‚ â® ¦¥ ¢à¥¬ï L¯®¯1 C = !ªà C = c12 . 1 1 1 1 ®¤áâ ¢«ïï í⨠¢ëà ¦¥­¨ï ¢ ä®à¬ã«ã ¤«ï , ¯®«ã稬 90

r

!2

= j !c 1 ; !ªà2 = j :

„«¨­  ¢®«­ë ¢ ¢®«­®¢®¤¥ à ¢­   = 2 = s 2



2 ! 1 ; !ªà c !2

=

s 

2 1 ; 2 ªà

;

â.¥. ¬ë ¨¬¥¥¬ ᮮ⭮襭¨¥,  ­ «®£¨ç­®¥ ®¡ëç­®¬ã ¤«ï ¢®«­®¢®¤®¢. à¨¬¥­¨¬ ¯®«ã祭­®¥ ᮮ⭮襭¨¥ ª -®¡à §­®¬ã ¢®«­®¢®¤ã á ¬ «ë¬ § §®à®¬ (á¬. à¨á.). b

2a

b

g

h

‡¤¥áì

C1 = "0gS1 = "0ga ; L¯®¯ 1 = 0 Sá¥ç = 0 bh (l = 1) ; 1 g gc2 : 2 = !ªà = = "0 a
0 bh 0 "0
abh abh g Žâá ªà¨â¨ç¥áª ï ¤«¨­  ¢®«­ë à ¢­  91

s

ªà = 2!c = 2 abh g : ªà ˆ§ ¯®«ã祭­®© ä®à¬ã«ë ¢¨¤­®, çâ®, 㬥­ìè ï à §¬¥à g, ¬®¦­® ᪮«ì 㣮¤­® 㢥«¨ç¨âì ªà¨â¨ç¥áªãî ¤«¨­ã ¢®«­ë. Žâ¬¥â¨¬, çâ® ªà¨â¨ç¥áª ï ¢®«­  ᮮ⢥âáâ¢ã¥â १®­ ­áã ¥¬ª®áâ¨ á ¯®¯¥à¥ç­®© ¨­¤ãªâ¨¢­®áâìî. â® { ­¥ á«ãç ©­®áâì, ªà¨â¨ç¥áª ï ¢®«­  ¢® ¢á¥å á«ãç ïå ¬®¦¥â ¡ëâì ®¯à¥¤¥«¥­  ª ª १®­ ­á­ ï ¤«ï ¯®¯¥à¥ç­®£® á¥ç¥­¨ï. à¨¬¥à: ¬®¤  H10 ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥ (á¬. à¨á.). a 2

a = ªà !  = 2a : ªà 2 4

3.5.

­¥à£¥â¨ç¥áª¨¥ ᮮ⭮襭¨ï ¤«ï ¢®«­®¢®¤®¢

‚ëç¨á«¨¬ ¬®é­®áâì ¢®«­ë, à á¯à®áâà ­ïî饩áï ¢ ¢®«­®¢®¤¥: Z P = 12 Re (E~  H~  )
~z0 dS: S

‚ëç¨á«¨¬ ¬®é­®áâì ¤«ï H-¬®¤. „«ï ­¨å

E~ = ;j! grad  ~z0
e; z ; H~ = (; grad + g2 ~z0 )
e; z : ®¤áâ ¢«ïï íâ® ¢ ¯®¤ë­â¥£à «ì­®¥ ¢ëà ¦¥­¨¥, ¢ëç¨á«¨¬  (E~  H~  )
~z0 = j!  [(grad  ~z0 )  grad ] ~z0
e; z
e; z =  = j!  (grad )2
e;( + )z :

92

®¤áâ ¢«ïï íâ® ¯®¤ ¨­â¥£à «, ­ ©¤¥¬ ¬®é­®áâì Z P = 12 Re j! 
(grad )2
e;( + )z dS: S

à¨  < ªà { ¬­¨¬ ï ¢¥«¨ç¨­ : = j ,  = ;j . ’®£¤  Z Z P = 12 ! (grad )2 dS = 21 ! gm2 2 dS: S

S

à¨  > ªà { ¢¥é¥á⢥­­ ï ¢¥«¨ç¨­ . ’®£¤  P = 0. „«ï E -¬®¤ (í«¥ªâà¨ç¥áª¨å)  ­ «®£¨ç­® Z P = 12 !" gm2 2 dS ¯à¨  < ªà : S

Ž¡à â¨¬ ¢­¨¬ ­¨¥ ­  â®, çâ® á ¯à¨¡«¨¦¥­¨¥¬ ª ªà¨â¨ç¥áª®© ¢®«­¥ áâ६¨âáï ª ­ã«î,   ¬®é­®áâì ¯à¨ ¯®áâ®ï­­®©  ¬¯«¨â㤥 E~ ¨«¨ H~ ã¡ë¢ ¥â ¤® ­ã«ï. â® §­ ç¨â, çâ® ¤«ï ¯¥à¥¤ ç¨ § ¤ ­­®© ¬®é­®á⨠­ ¯à殮­­®áâì ¯®«¥© ¯à¨å®¤¨âáï 㢥«¨ç¨¢ âì, çâ® ¯à¨¢®¤¨â ª ¯à®¡®ï¬ ¨ ¡®«ì襬㠧 âãå ­¨î. ®í⮬ã à ¡®â  ¢¡«¨§¨ ªà¨â¨ç¥áª®© ¢®«­ë ­¥¢ë£®¤­ .  áᬮâਬ ᮮ⭮襭¨ï ¤«ï ¯«®â­®áâ¨ í­¥à£¨¨. «®â­®áâì í«¥ªâà¨ç¥áª®© ¨ ¬ £­¨â­®© í­¥à£¨¨ ¤«ï H -¬®¤ à ¢­ 

"jE j2 = " (!)2
(grad )2 ; 2 2 jH j2 =  fg4
2 + 2 (grad )2 g: 2 2 m ˆ§ íâ¨å ᮮ⭮襭¨© ¢¨¤­®, çâ® ¯«®â­®áâì í­¥à£¨¨ ¬ £­¨â­®£® ¨ í«¥ªâà¨ç¥áª®£® ¯®«¥© ­¥ à ¢­ë ¬¥¦¤ã ᮡ®© ¢ «î¡®© â®çª¥ ¢­ãâਠ¢®«­®¢®¤ . ‚ â® ¦¥ ¢à¥¬ï ¬®¦­® ¯®ª § âì, çâ® § ¯ áë í«¥ªâà¨ç¥áª®© ¨ ¬ £­¨â­®© í­¥à£¨¨ ­  ¥¤¨­¨æã ¤«¨­ë ¢®«­®¢®¤  (â.¥. ¢ á।­¥¬ ¯® á¥ç¥­¨î) à ¢­ë ¤à㣠¤àã£ã. „¥©á⢨⥫쭮,

We =

Z "jE j2 dS S

2

Z Z = 2"
(!)2 (grad )2 dS = 2" (!)2
gm2 2 dS; S

S

93

jH j dS =  g4 2 dS + 
2 (grad )2 dS = 2 2 m 2 SZ S S Z Z    4 2 2 2 2 2 dS + 2 gm dS = 2 gm (gm2 + 2 ) 2 dS: = 2 gm

Wh =

S

S

S

‚ â® ¦¥ ¢à¥¬ï gm2 = k2 + 2 = k2 ; 2 = !2 " ; 2 : ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨ï ¤«ï Wh , ¯®«ã稬

Z Z  " 2 2 2 2 2 Wh = 2 gm ! " dS = 2 (!)
gm 2 dS: S

S

‘à ¢­¥­¨¥ ¯®ª §ë¢ ¥â, çâ® ¤¥©á⢨⥫쭮 Wh = We . „«ï ­¥à á¯à®áâà ­ïîé¨åáï ¬®¤ { ¢¥é¥á⢥­­ ï ¢¥«¨ç¨­ , ¯®í⮬㠯८¡« ¤ ¥â ¬ £­¨â­ ï (¤«ï H-¬®¤) ¨«¨ í«¥ªâà¨ç¥áª ï (¤«ï E-¬®¤) í­¥à£¨ï, ¨ ¢å®¤­®¥ ᮯà®â¨¢«¥­¨¥ ®ª §ë¢ ¥âáï ॠªâ¨¢­ë¬. 3.6.

” §®¢ ï ¨ £à㯯®¢ ï ᪮à®áâ¨

ˆ§ ¯®«ã祭­ëå à ­¥¥ ᮮ⭮襭¨© ¬®¦­® ­ ©â¨ ä §®¢ãî ᪮à®áâì (§­ ï ¤«¨­ã ¢®«­ë):

vä = f =

s

1

f

  2 ; ªà

=

c

s

1

  2 > c: ; ªà

’ ª¨¬ ®¡à §®¬, ä §®¢ ï ᪮à®áâì ¡®«ìè¥ áª®à®á⨠ᢥâ , çâ®, ®¤­ ª®, ­¥ ¯à®â¨¢®à¥ç¨â ⥮ਨ ®â­®á¨â¥«ì­®áâ¨.  ©¤¥¬ £à㯯®¢ãî ᪮à®áâì. Ž­  ¬®¦¥â ¡ëâì ®¯à¥¤¥«¥­  ª ª ᪮à®áâì ¯¥à¥¬¥é¥­¨ï ¬ ªá¨¬ã¬  ¡¨¥­¨© ¢®«­ ¤¢ãå ¡«¨§ª¨å ç áâ®â. ãáâì ¬ë ¨¬¥¥¬ á㬬㠤¢ãå ¢®«­ ¡«¨§ª¨å ç áâ®â n o e j(!1 t; 1 z) + e j(!2 t; 2 z) = e j(!1 t; 1 z)
1 + e j[(!2 ;!1 )t ; ( 2 ; 1 )z] : Œ ªá¨¬ã¬ ᮮ⢥âáâ¢ã¥â ãá«®¢¨î (!2 ; !1 ) t ; ( 2 ; 1 ) z = 0: 94

Žâá, á¡«¨¦ ï ! ¨ ¨ ¡¥àï ¬ «ë¥ ¨­â¥à¢ «ë t ¨ z , ¯®«ã稬 ¢ ¯à¥¤¥«¥
!

t =


z = 0; ®âªã¤ 

d! = v£à = dz dt d ¢ ®â«¨ç¨¥ ®â ä §®¢®© ᪮à®á⨠vä = ! .

 ©¤¥¬ £à㯯®¢ãî ᪮à®áâì ¤«ï ¢®«­®¢®¤  1 v£à = d! d = d =d! :

s

= 2 = 2 1 ;

  2 2 s ªà 2 ; ªà = ªà  s 2

1 =

! = 2 !ªà ; 1 : ªà 2 d = 2 s 2!=!ªà 1 =  2! = 2ªà s 2   2 d! ªà ! ªà ! ;1 2 !! !ªà ; 1 ªà 1 1 =  2!
s = 1c
s     2 : 2 ªà ªà ! ªà 1; ! 1;  ªà Žâá s  2 v£à = c 1 ;  < c: ªà

95

„«ï ¢®«­®¢®¤  ¯®«ãç ¥¬ ¯à®á⮥ ᮮ⭮襭¨¥, á¢ï§ë¢ î饥 ä §®¢ãî ¨ £à㯯®¢ãî ᪮à®á⨠(á¬. à¨á.):

v£à
vä = c2 : v

ˆ­®£¤  㯮âॡ«ïîâ ¤à㣮¥ ®¯à¥¤¥«¥­¨¥ £à㯯®¢®© ᪮à®áâ¨:

Φ

c

v

P; v£à = W

Γp

λkp

£¤¥ P { ¬®é­®áâì ¢ ¢®«­®¢®¤¥, W { á।­¨© § ¯ á í­¥à£¨¨ ­  ¥¤¨­¨æã ¤«¨­ë ¢®«­®¢®¤ . ˆá¯®«ì§ãï ¯®«ã祭­ë¥ à ­¥¥ ᮮ⭮襭¨ï, ¨¬¥¥¬ (¤«ï H-¢®«­)

Z

P = 12 ! gm2 2 dS; S Z " 2 W = (!)
g2 2 dS: m

2

S

Žâá

P = : v£à = W !" ” §®¢ ï ᪮à®áâì, ª ª ¡ë«® ®¯à¥¤¥«¥­® ¢ëè¥,

vä = ! : ®¤áâ ¢«ïï != ¢ ¢ëà ¦¥­¨¥ ¤«ï v£à , ­ å®¤¨¬ 1 = c2 : v£à
vä = " ‘à ¢­¨¢ ï á ­ ©¤¥­­ë¬ à ­¥¥ ᮮ⭮襭¨¥¬, ­ å®¤¨¬, çâ® ®¡  ®¯à¥¤¥«¥­¨ï £à㯯®¢®© ᪮à®á⨠¤ îâ ®¤¨­ ª®¢ë© १ã«ìâ â. 96

3.7.

‚®«­ë à¨««îí­ 

‚®«­ë ¢ ¢®«­®¢®¤¥ ¬®¦­® à á¬®âà¥âì ¥é¥ ¢ ®¤­®¬  á¯¥ªâ¥. à¨««îí­®¬ ¡ë«® ®â¬¥ç¥­®, çâ® áãé¥áâ¢ã¥â á¢ï§ì ¬¥¦¤ã ®¤­®à®¤­ë¬¨ ¯«®áª¨¬¨ ¢®«­ ¬¨ ¨ ¢®«­ ¬¨ ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥. â  á¢ï§ì ¬®¦¥â ¡ëâì ®¡­ à㦥­  â ª¦¥ ¢ ¢®«­®¢®¤ å ¤à㣮© ä®à¬ë.  ¯à¨¬¥à, ¤«ï ¬®¤ë H10 ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥

8  x  > < j a ; z ; j! e ; 2a > :

x ;j z = Ey = ; j! a sin a
e



e

;j x + z

a

9 > = ; > ;

â® ¥áâì á㬬  ¤¢ãå ¯«®áª¨å ¢®«­.  áᬮâਬ ¢®«­ã H10 ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥. ‚ í⮬ á«ãç ¥ ¯®«¥ ¬®¦­® ¯à¥¤áâ ¢¨âì ¢ ¢¨¤¥ ¤¢ãå ¯«®áª¨å ¢®«­, à á¯à®áâà ­ïîé¨åáï ¯®¤ 㣫®¬ ¤à㣠ª ¤àã£ã ¬¥¦¤ã ¤¢ã¬ï ¯«®áª®áâﬨ (®¡à §®¢ ­­ë¬¨ è¨à®ª¨¬¨ á⥭ª ¬¨) (á¬. à¨á.).

Λ λ π

2

θ θ

x

θ

z

—⮡ë íâ® ¯®ª § âì, à áᬮâਬ ¤¢¥ ¯«®áª¨¥ ¢®«­ë, à á¯à®áâà ­ïî騥áï ¬¥¦¤ã ¯ à ««¥«ì­ë¬¨ ¯à®¢®¤ï騬¨ ¯«®áª®áâﬨ ¯®¤ 㣫 ¬¨  ª ®á¨ z . ’®£¤  ­¥âà㤭® ¢¨¤¥âì, çâ®   = cos ,  = sin . ¯à®¤ ¯®¯

Λ

‚ â® ¦¥ ¢à¥¬ï ¤¢¥ ¢®«­ë ¢ ¯®¯¥à¥ç­®¬ ª z á¥ç¥­¨¨ ¤ îâ áâ®ïçãî ¢®«­ã, ¯à¨ç¥¬ à ááâ®ï­¨¥ ¬¥¦¤ã ¬¨­¨¬ã¬ ¬¨ à ¢­® ¯®¯ 2 . …᫨ 㣮«  ¢ë¡à ­ â ª, ç⮡ë à áâ®ï­¨¥ ¬¥¦¤ã ¬¨­¨¬ã¬ ¬¨ ¡ë«® à ¢­® a, â® ­  «¨­¨¨ ¬¨­¨¬ã¬  ¬®¦­® ¯®¬¥áâ¨âì ¯à®¢®¤ï騥 á⥭ª¨, ­¥ ­ àãè ï ª à⨭㠯®«ï. ’ ª¨¬ ®¡à §®¬, ¯®«ãç ¥¬ ãá«®¢¨¥ ¯®¯ = a; ¨«¨ sin  =  : 2 2a Žç¥¢¨¤­®, çâ® â ª®¥ ᮮ⭮襭¨¥ ¬®¦­® ¢ë¯®«­¨âì «¨èì ¯à¨ ãá«®¢¨¨, çâ® 97

 < 2a; â:¥: ªà = 2a:

’¥¯¥àì ¬®¦­® ®¯à¥¤¥«¨âì ¤«¨­ã ¢®«­ë ¢ ­ ¯à ¢«¥­¨¨ ®á¨ z : ¯à®¤ = cos  =

p 

= 1 ; sin2 



s

1;

  2 : ªà

Šà®¬¥ ⮣®, ¬®¦­® ®â¬¥â¨âì, çâ® ¯à¨ ¯à¨¡«¨¦¥­¨¨ ª ªà¨â¨ç¥áª®© ¢®«­¥ 㣮«  ¯à¨¡«¨¦ ¥âáï ª 2 , â.¥. ¢¡«¨§¨ ªà¨â¨ç¥áª®© ¢®«­ë à á¯à®áâà ­¥­¨¥ ¢ ®á­®¢­®¬ ¯®¯¥à¥ç­®. ®«ã祭­ë¥ ᮮ⭮襭¨ï ¯®§¢®«ïîâ ­ ©â¨ â ª¦¥ ¨ £à㯯®¢ãî ᪮à®áâì s  2 p 2 v£à = c
cos  = c
1 ; sin  = c
1 ;  : ªà „«ï ¢®«­ Hmn ¨ Emn ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥ ¯®«¥ ¢ ®¡é¥¬ á«ãç ¥ ¬®¦¥â ¡ëâì ¯®«ã祭® ¢ ¢¨¤¥ á㬬ë ç¥âëà¥å ¯«®áª¨å ¢®«­, à á¯à®áâà ­ïîé¨åáï ¯®¤ ®¤¨­ ª®¢ë¬ 㣫®¬ ª ®á¨ z . ‚ á«ãç ¥ ¡®«¥¥ á«®¦­ëå ¢®«­®¢®¤®¢ ¯®«¥ ¬®¦¥â ¡ëâì ¯®«ã祭® á㬬¨à®¢ ­¨¥¬ ¡¥áª®­¥ç­®£® ç¨á«  ¯«®áª¨å ¢®«­, ­ ¯à ¢«¥­¨¥ à á¯à®áâà ­¥­¨ï ª®â®àëå ®¡à §ãîâ ª®­ãá á 㣫®¬  ¯à¨ ¢¥à設¥. …᫨ § à ­¥¥ ¨áª âì à¥è¥­¨¥ ¢ ¢¨¤¥ áã¬¬ë ¯«®áª¨å ¢®«­, â® ¬®¦­® ¯®«ãç¨âì ¨­â¥£à «ì­®¥ ãà ¢­¥­¨¥ ¤«ï ä㭪権 ¨ . 3.8.

‡ âãå ­¨¥ ¢ ¢®«­®¢®¤ å

‚ ¢®«­®¢®¤¥ á ¬¥â ««¨ç¥áª¨¬¨, ­¥¨¤¥ «ì­® ¯à®¢®¤ï騬¨ á⥭ª ¬¨ ç áâì ¬®é­®á⨠â¥àï¥âáï ¢ á⥭ª å. â® ¯à¨¢®¤¨â ª § âãå ­¨î à á¯à®áâà ­ïîé¨åáï ¢®«­. ”®à¬ «ì­® íâ® ¢ëà ¦ ¥âáï ¢ ⮬, çâ® ¤«ï ç áâ®â ¢ëè¥ ªà¨â¨ç¥áª®© ¢¥«¨ç¨­  ¯®áâ®ï­­®© à á¯à®áâà ­¥­¨ï ®ª §ë¢ ¥âáï 㦥 ­¥ ç¨áâ® ¬­¨¬®©,   ¯®ï¢«ï¥âáï ¢¥é¥á⢥­­ ï ç áâì, ®¡ãá«®¢«¨¢ îé ï § âãå ­¨¥. ˆ§¬¥­ï¥âáï â ª¦¥ ¨ ¬­¨¬ ï ç áâì ¢á«¥¤á⢨¥ ¯à®­¨ª­®¢¥­¨ï ¯®«ï ¢ á⥭ªã ­  ⮫騭ã ᪨­-á«®ï. ˆ§¬¥­¥­¨¥ ¯®áâ®ï­­®© à á¯à®áâà ­¥­¨ï ¢á«¥¤á⢨¥ ¯®â¥àì ¬®¦¥â ¡ëâì ¢ëç¨á«¥­® ¯à¨¡«¨¦¥­­® ¢ à ¬ª å ⥮ਨ ¢®§¬ã饭¨©. à¥¤¯®«®¦¨¬, çâ® ­¥ª®â®à ï ¬®¤  ¯à¨ ®âáãâá⢨¨ ¯®â¥àì ®¯¨á뢠¥âáï ¢¥ªâ®à­ë¬¨ äã­ªæ¨ï¬¨ E~m
e; 0 z , H~m
e; 0 z . à¨ ¯®ï¢«¥­¨¨ ¬ «ëå ¯®â¥àì 98

¢¥ªâ®à­ë¥ ä㭪樨 ­¥áª®«ìª® ¨§¬¥­ïîâáï: E
e; z , H
e; z . à¥¤¯®«®¦¨¬, çâ® ¯à¨ ¬ «®¬ § âãå ­¨¨ í⨠ä㭪樨 ¬ «® ®â«¨ç îâáï ®â ­¥¢®§¬ã饭­ëå. ‚® ¢á类¬ á«ãç ¥, íâ® ¨¬¥¥â ¬¥áâ® ¯à¨ ®âáãâá⢨¨ ¢ë஦¤¥­¨ï. „«ï ¢ëç¨á«¥­¨ï ¢®á¯®«ì§ã¥¬áï ⥬, çâ® ¢¥ªâ®à­ë¥ ä㭪樨 㤮¢«¥â¢®àïîâ ãà ¢­¥­¨ï¬ Œ ªá¢¥«« : rot (E~ m
e; 0 z ) rot (H~ m
e; 0 z ) rot (E~
e; z ) rot (H~
e; z )

= = = =

;j!0 H~ m e; 0 z ; j!"0 E~ m e; 0 z ; ;j!0 H~ e; z ; j!"0 E~ e; z ;

®«ì§ãïáì ⮦¤¥á⢮¬ rot (
F~ ) = 
rot F~ + grad   F~ ; ¯®á«¥ ᮪à é¥­¨ï íªá¯®­¥­â ¯®«ã稬 rot E~ m + 0
~z0  E~ m rot H~ m + 0
~z0  H~ m rot E~ ;
~z0  E~ rot H~ ;
~z0  H~

= = = =

j!0H~ m ; ;j!"0E~ m ; ~ ;j!0H; ~ j!"0E:

‡¤¥áì 0 = ; 0 , â ª ª ª 0 { ¬­¨¬ ï ¢¥«¨ç¨­ . ‘®áâ ¢¨¬ ¢ëà ¦¥­¨¥ div (E~ m  H~ ) + div (E~  H~ m ) =

= H~
rot E~ m ; E~ m
rot H~ + H~ m
rot E~ ; E~
rot H~ m = = H~
(j!0 H~ m ; 0~z0  E~ m ) + E~ m
(j!"0E~ + ~z0  H~ ) ; + H~ m
(;j!0 H~ + ~z0  E~ ) ; E~
(;j!"0 E~ m ; 0
~z0  H~ m ) = = ; 0
H~
(~z0 E~ m ) ;
E~ m
(~z0 H~ ) +
H~ m
(~z0 E~ ) + 0
E~
(~z0 H~ m ) = = ; 0
~z0
(E~ m H~ ) +
~z0
(E~ m H~ ) +
~z0
(E~ H~ m ) ; 0
~z0
(E~ H~ m ) = = ( ; 0 )
(E~ m  H~ + E~  H~ m )
~z0 : 99

I C

à®¨­â¥£à¨à㥬 ¯®«ã祭­®¥ à ¢¥­á⢮ ¯® á¥ç¥­¨î ¢®«­®¢®¤ :

Z

(E~ m  H~ + E~  H~ m )
~n dC = ( ; 0 ) (E~ m  H~ + E~  H~ m )
~z0 dS; S

®âªã¤ 

H (E~   H~ m RC (E~   H~

= 0 +

m

S

+ E~  H~ m )
~n dC

: + E~  H~ m )
~z0 dS

‡¤¥áì ~n { ¥¤¨­¨ç­ë© ¢¥ªâ®à ­®à¬ «¨, ­ ¯à ¢«¥­­ë© ¢ á⥭ªã, C { ª®­âãà á¥ç¥­¨ï ¢®«­®¢®¤ , S { á¥ç¥­¨¥ ¢®«­®¢®¤ . ˆ­â¥£à «

I

C

(E~ m  H~ )~n dC = 0

¢á«¥¤á⢨¥ £à ­¨ç­ëå ãá«®¢¨© ¤«ï E~ m . „ «¥¥, ᮣ« á­® ãá«®¢¨î ‹¥®­â®¢¨ç ,

E~ t = 
(H~  ~n)  
(H~ m  ~n) ­  ª®­âãॠ¯®¯¥à¥ç­®£® á¥ç¥­¨ï C . ®á«¥¤­¥¥ ¯à¨¡«¨¦¥­­®¥ à ¢¥­á⢮ á¢ï§ ­® á ¯à¥¤¯®«®¦¥­¨¥¬, çâ® ­ «¨ç¨¥ ¯®â¥àì ¬ «® ¨§¬¥­ï¥â ¯®«¥. ‚â®à®© ¨­â¥£à « ¢ ç¨á«¨â¥«¥ à ¢¥­

I

C

(E~  H~ m )~n dC =

I

C

h i 
(H~ m  ~n)  H~ m ~n dC

= 


I

C

jH~m j2 dC:

‚ §­ ¬¥­ â¥«¥ E~ ¨ H~ ¬®£ãâ ¡ëâì ¯à¨¡«¨¦¥­­® § ¬¥­¥­ë ­  E~ m ¨

H~ m :

Z S

Z

(E~ m  H~ + E~  H~ m )
~z0 dS  (E~ m  H~ m + E~ m  H~ m )
~z0 dS =

Z

S

= 2 Re (E~ m  H~ m )
~z0 dS : S

ˆâ ª, ®ª®­ç â¥«ì­® ¨¬¥¥¬

100

H


jH~ m j2 dC :

 0 + R C 2 Re (E~ m  H~ m )
~z0 dS S

‚ १ã«ìâ â¥ ¬ë ¬®¦¥¬ ¢ëç¨á«¨âì ¬­¨¬ãî ¨ ¢¥é¥á⢥­­ãî ¤®¡ ¢ª¨ ª ¢á«¥¤á⢨¥ ¯®â¥àì ¢ á⥭ª å ¢®«­®¢®¤ :

H

Re  jH~ m j2 dC

 = 12 R ~ C ~  ; Re (Em  Hm )
~z0 dS S H Im  jH~ m j2 dC 1 :
= 2 R ~ C ~  Re (Em  Hm )
~z0 dS S

“ç¨â뢠ï, çâ®   â ª¦¥ çâ®

+ j;  = 1 

I 1 P1¯®â¥àì = 2 jH~ m j2 dC | C

¬®é­®áâì ¯®â¥àì ­  ¥¤¨­¨æã ¤«¨­ë ¢®«­®¢®¤  ¨ Z 1 P = 2
Re (E~ m  H~ m )
~z0 dS | S

¬®é­®áâì ¢®«­ë ¢ ¢®«­®¢®¤¥, ®ª®­ç â¥«ì­® ¯®«ã稬  =
= P21¯®â P : ‡ âãå ­¨¥ ¬®¦¥â ¡ëâì â ª¦¥ ¯à¥¤áâ ¢«¥­® ¢ ¢¨¤¥ P1¯®â ;  = P21¯®â = P 2v£à W £¤¥ 101

Z W = 20 jH~ m j2 dS | S

§ ¯ á í­¥à£¨¨ ­  ¥¤¨­¨æã ¤«¨­ë ¢ ¢®«­®¢®¤¥;

s

2 v£à = c
1 ; 2 | ªà

£à㯯®¢ ï ᪮à®áâì ¢ ¢®«­®¢®¤¥. ®¤áâ ¢«ïï í⨠ᮮ⭮襭¨ï ¢ ¢ëà ¦¥­¨¥ ¤«ï , ¯®«ã稬: H jH~ j2 dC 1 H ~ 2 m 2 C jHm j dC 1 s 1 CR s  = = 2 ~ m j2 dS : c 2 2 0 R j H    0 2 2 jH~ m j2 dSc 1 ; 2 1 ; 2 S ªà ªà S Ž¤­ ª®

 =  1!2 : ¬

®¤áâ ¢«ïï  ¢ ¯®á«¥¤­¥¥ ¢ëà ¦¥­¨¥ ¤«ï , ¯®«ã稬 H jH~ j2 dC m CR :  =
= 2  ¬ s 1 2 jH~ m j2 dS 0  1 ; 2 S ªà  ¯®¬­¨¬ ãá«®¢¨ï, ¯à¨ ª®â®àëå ¯®«ã祭  íâ  ä®à¬ã« . à¥¤¯®« £ ¥âáï, çâ® ¢ ¢®«­®¢®¤¥ ¯®ï¢«¥­¨¥ ¯®â¥àì ¬ «® ¨§¬¥­ï¥â ¯®«¥. â® ãá«®¢¨¥ ¬®¦¥â ¡ëâì ­ àã襭® ¢ á«¥¤ãîé¨å á«ãç ïå:  ) ¡«¨§®áâì ª ªà¨â¨ç¥áª®© ¢®«­¥, ¡) ­ «¨ç¨¥ ¢ë஦¤¥­¨ï. ‹¥£ª® ¯®«ãç¨âì ä®à¬ã«ã, ª®â®à ï ®áâ ¥âáï ª®à४⭮© ¢ ®ªà¥áâ­®á⨠ªà¨â¨ç¥áª®© ç áâ®âë. „«ï í⮣® à áᬮâਬ ¯®«ã祭­ãî à ­¥¥ â®ç­ãî ä®à¬ã«ã 102

= 0 +

H (E~   H~ m RC (E~   H~ m

S

+ E~  H~ m )
~n dC : + E~  H~ m )
~z0 dS

„«ï ®¯à¥¤¥«¥­­®á⨠à áᬮâਬ á«ãç © ¬ £­¨â­ëå ¬®¤. ‚ í⮬ á«ãç ¥ ¯®¯¥à¥ç­ ï ç áâì ¬ £­¨â­®£® ¯®«ï ¨ í«¥ªâà¨ç¥áª®¥ ¯®«¥ á¢ï§ ­ë ᮮ⭮襭¨¥¬

0 ~z  E~ : H~ m¯®¯ = j! 0 m 0

„«ï ªà¨â¨ç¥áª®© ç áâ®âë 0 ®¡à é ¥âáï ¢ ­ã«ì. ®í⮬㠯ਠ§ ¬¥­¥ ᮣ« á­® ⥮ਨ ¢®§¬ã饭¨© ¢ §­ ¬¥­ â¥«¥ H~ ¯®¯ ­  H~ m §­ ¬¥­ â¥«ì ®¡à é ¥âáï ¢ ­ã«ì, çâ® ¨ ¯à¨¢®¤¨â ª ­¥ª®à४⭮áâ¨. â® ¬®¦­® ¨§¬¥­¨âì, ¢ëà §¨¢ ¯¥à¢®­ ç «ì­® ¢ §­ ¬¥­ â¥«¥ H~ ç¥à¥§ E~ ¯® ä®à¬ã«¥,  ­ «®£¨ç­®© ¢ë襭 ¯¨á ­­®©: ~ H~ ¯®¯ = j!
 ~z0  E: 0 ®á«¥ í⮣® § ¬¥­¨¬ E~ ­  E~ m ¢ ᮮ⢥âá⢨¨ á ⥮ਥ© ¢®§¬ã饭¨©: H j!
0 (E~ m  H~ + E~  H~ m )
~n dC

; 0 = = RC( jE~ j2 + jE~ j2 ) dS m 0 m S

=

H j!
0 (E~ m  H~

+ E~  H~ m )
~n dC C : R ( + 0 ) jE~ m j2 dS S

“¬­®¦ ï ®¡¥ áâ®à®­ë à ¢¥­á⢠ ­  + 0 , ¯®«ã稬 H j!
0 (E~ m  H~ + E~  H~ m )
~n dC C

2 ; 02 = : R jE~ j2 dS m S

„ ­­®¥ ᮮ⭮襭¨¥ ®áâ ¥âáï ª®à४â­ë¬ ¨ ¤«ï ªà¨â¨ç¥áª®© ç áâ®âë.   ªà¨â¨ç¥áª®© ç áâ®â¥ 0 = 0. ®í⮬㠯®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï ¯®«ãç ¥âáï ¯ã⥬ ¨§¢«¥ç¥­¨ï ª®à­ï ª¢ ¤à â­®£®. ‚ १ã«ìâ â¥ § âãå ­¨¥ ­  í⮩ ç áâ®â¥ १ª® 㢥«¨ç¨¢ ¥âáï, ­® ®áâ ¥âáï ª®­¥ç­ë¬. €­ «®£¨ç­® à¥è ¥âáï § ¤ ç  ¨ ¤«ï í«¥ªâà¨ç¥áª¨å ¬®¤. ‚ëç¨á«¨¬ § âãå ­¨¥ ¤«ï ­¥ª®â®àëå á«ãç ¥¢. 103

3.8.1. àאַ㣮«ì­ë© ¢®«­®¢®¤.

H10 {¬®¤ 

Š®¬¯®­¥­âë ¬ £­¨â­®£® ¯®«ï í⮩ ¬®¤ë à ¢­ë  2 Hz = a cos x a;

x Hx = j a sin a ; Hy = 0:

Žâá

jH j2 =

  4 a

cos2 x +

a

  2 a

sin2 x a:

ˆ­â¥£à « ¯® á¥ç¥­¨î ¢®«­®¢®¤  à ¢¥­ (  4  2)  Z   2   2 a  a  ab 2 2 jH j dS = b 2 a + 2 a = 2 a a + = S

â ª ª ª

I C

  2 42 = ab 2 a 2 ;   2

2 = g2 + 2 = k2 = 42 : + a 2 ˆ­â¥£à « ¯® ª®­âãàã á¥ç¥­¨ï " 4  2#  4   4 3 a    4 a 2 jH j dC = 2 a b + 2
2 a + a = a 2b + 2 =

=

  4 a

!

2 2b + a ªà2 :

®¤áâ ¢«ïï í⨠¨­â¥£à «ë ¢ ¢ëà ¦¥­¨¥ ¤«ï , ¯®«ã稬   4  2ªà   2b + a 2 a =  = 2

¬ s 1  2 4  2 2 0 ab   1 ; 2 2 a 2 ªà 104

2 1 + 2ab 2 ¬ ªà : =  b 0 s 2 1 ; 2 ªà

à¨¬¥à: àאַ㣮«ì­ë© ¢®«­®¢®¤ 1023 ¬¬2 , ¬¥¤ì.  = 3 á¬, ¬ = 0 , b = 1 á¬, a = 2:3 á¬, ªà = 4:6 á¬,   0:64
10;4 á¬, f = 1010 ƒæ. ;4 1 + 2
; 3
2 
0 : 64
10  =
q 2:3 ; 4:6
2  1:14
10;4 ­¥¯=ᬠ= 1
3 1 ; 43:6 = 9:9
10;4 ¤=ᬠ= 9:9
10;2 ¤=¬: ˆ§¬¥­¥­¨¥ § âãå ­¨ï  ¢ § ¢¨á¨¬®á⨠®â ç áâ®âë ¯à¥¤áâ ¢«¥­® ­  à¨á. α

ω

ω kp

3.8.2. Šàã£«ë© ¢®«­®¢®¤. Œ®¤ 

H01

0 2
J0 (g0 r); Hz = g01 01 0 0 r ); Hr = j g01
J1 (g01 0 0 r ): E' = ;j!g01
J1 (g01

  ¯®¢¥àå­®á⨠á⥭ª¨ ¬ £­¨â­®¥ ¯®«¥ ¨¬¥¥â ⮫쪮 z -î ª®¬¯®­¥­âã (Hz 6= 0), ¯®í⮬ã I jH j2 dC = 2a
g010 4 J02 (g010 a): C

105

„ «¥¥, ¢ëç¨á«¥­¨¥ jH j2 dS 㤮¡­® § ¬¥­¨âì ¢ëç¨á«¥­¨¥¬ ¨­â¥£à S R «  jE j2 dS , â ª ª ª E ¨¬¥¥â ⮫쪮 ®¤­ã á®áâ ¢«ïîéãî: S

Z S

Z

Z

jH j2 dS = "0 jE j2 dS = "0 !2 20 g010 2 J12 (g010 r)2r dr = 0 0 S S

2 0 2Z 2 0 = 2! c2 g01 J1 (g01 r)r dr: 0 ˆ­â¥£à « ¬®¦¥â ¡ëâì ¢ëç¨á«¥­ ¨ ®ª §ë¢ ¥âáï à ¢­ë¬ a

Za

0 r)r dr = a2 J 2 (g0 a): J12 (g01 2 0 01

0 ®¤áâ ¢«ïï í⨠¨­â¥£à «ë ¢ ¢ëà ¦¥­¨¥ ¤«ï , ¯®«ã稬 Z 2 0 2 2 2 0 jH j2 dS = ! c2 g01 a J0 (g01 a): S

’¥¯¥àì ¬®¦­® ¢ëç¨á«¨âì  (¯à¨ ¬ = 0 ): 0 4 J02 (g01 0 a) 2a
g01 =  = 2 s 1 2 !2 g0 2 a2 J 2 (g0 a)  0 01 2c 1 ; 2 c2 01 ªà 2  s 2ªà : = a 2 1 ; 2 ªà ˆ§ ¤ ­­®£® ᮮ⭮襭¨ï á«¥¤ã¥â, çâ® á 㬥­ì襭¨¥¬ ¤«¨­ë ¢®«­ë § âãå ­¨¥ ã¡ë¢ ¥â ª ª 3=2 (â ª ª ª   1=2 ). â® ã¡ë¢ ­¨¥ ï¥âáï á«¥¤á⢨¥¬ ⮣®, çâ® ®âáãâáâ¢ãî⠯த®«ì­ë¥ ⮪¨ ¢ á⥭ª å,   ¯®¯¥à¥ç­ë¥ ⮪¨ ã¡ë¢ îâ á à®á⮬ ç áâ®âë. Š®à®âª® ®áâ ­®¢¨¬áï ¥é¥ ­  ¯®â¥àïå ¢ á।¥, § ¯®«­ïî饩 ¢®«­®¢®¤, ¯à¥¤¯®« £ ï, çâ® ¯®â¥à¨ ¬ «ë. 106

Š ª ¨ ¯à¥¦¤¥,

 = 2Pv¯®â1 : £àW à¨ í⮬

Z P¯®â1 = 2 jE j2 dS; S

£¤¥  { ¯à®¢®¤¨¬®áâì á।ë. ‚ â® ¦¥ ¢à¥¬ï

Z W = 2" E 2 dS; S

¯®í⮬ã

 = 2v " = £à

s! tg 

2 2c 1 ; 2 ªà

= 

s

tg 

2 1 ; 2 ªà

;

 = tg . £¤¥ !" 4.

4.1.

’…ŽˆŸ –……‰ ‚ €‘—…’€• ‚Ž‹Ž‚Ž„Ž‚

‚®«­®¢®¤­ ï «¨­¨ï ¯¥à¥¤ ç¨

„® á¨å ¯®à ­ á ¨­â¥à¥á®¢ «  «¨èì áâàãªâãà  ¯®«¥© à §«¨ç­ëå ¬®¤ ¨ ªà¨â¨ç¥áª¨¥ ¤«¨­ë ¢®«­. ˆáá«¥¤ã¥¬ ⥯¥àì à á¯à®áâà ­¥­¨¥ ¢®«­ ¢ ¢®«­®¢®¤¥.  áᬮâਬ ¢®«­®¢®¤, à ¡®â î騩 ¢ ®¡« á⨠ç áâ®â, £¤¥ ¬®¦¥â à á¯à®áâà ­ïâìáï «¨èì ®¤­  ®á­®¢­ ï ¬®¤ . ‚ëá訥 ¬®¤ë ¯à¥¤¯®« £ îâáï ­¥à á¯à®áâà ­ïî騬¨áï ¨ ¡ëáâà® § âãå î騬¨. Š ª ¡ë«® ᪠§ ­® à ­¥¥, à¥è¥­¨¥ á®á⮨⠨§ ¤¢ãå ¢®«­, à á¯à®áâà ­ïîé¨åáï ¢ ¯à®â¨¢®¯®«®¦­ëå ­ ¯à ¢«¥­¨ïå ¨ ®â«¨ç îé¨åáï ¬­®¦¨â¥«ï¬¨ ej z .

107

‘ãé¥á⢮¢ ­¨¥ ⮩ ¨«¨ ¨­®© ¨§ íâ¨å ¢®«­ ®¯à¥¤¥«ï¥âáï ãá«®¢¨ï¬¨ ¢®§¡ã¦¤¥­¨ï (ãá«®¢¨¥ ¨§«ã祭¨ï). …᫨ ¯®«ã¡¥áª®­¥ç­ë© ¢®«­®¢®¤ ¢®§¡ã¦¤ ¥âáï ¢ ­ ç «¥ £¥­¥à â®à®¬, â® ¢ ­¥¬ áãé¥áâ¢ã¥â «¨èì ®¤­  ¢®«­ , à á¯à®áâà ­ïîé ïáï ¢ ­ ¯à ¢«¥­¨¨ ®â £¥­¥à â®à  (á¬. à¨á.). à¥¤¯®«®¦¨¬, çâ® ¢ ¢®«­®¢®¤¥ áãé¥áâ¢ã¥â ­¥®¤­®à®¤­®áâì, ­ ¯à¨¬¥à, ¯à®¢®¤­¨ª ¯®¯¥à¥ª ¢®«­®¢®¤ . ’®£¤  ¢¡«¨§¨ ¯à®¢®¤­¨ª  ®á­®¢­®© ¬®¤ë ­¥¤®áâ â®ç­® ¤«ï 㤮¢«¥â¢®à¥­¨ï £à ­¨ç­ë¬ ãá«®¢¨ï¬, â ª ª ª ¤®¡ ¢«ï¥âáï ¥é¥ ãá«®¢¨¥ ­  ¯à®¢®¤­¨ª¥. ‚ ®¡é¥¬ á«ãç ¥ £à ­¨ç­ë¬ ãá«®¢¨ï¬ ¬®¦­® 㤮¢«¥â¢®à¨âì, ¥á«¨ § ¯¨á âì à¥è¥­¨¥ ¢ ¢¨¤¥ áã¬¬ë ¯àאַ© ¨ ®¡à â­®© ¢®«­ ®á­®¢­®© ¬®¤ë ¨ ¢ëáè¨å ¬®¤ á ᮮ⢥âá⢥­­® ¯®¤®¡à ­­ë¬¨ ª®íää¨æ¨¥­â ¬¨. ’ ª ª ª ¯®«ï ¢ëáè¨å ¬®¤ ¡ëáâà® § âãå îâ, â® ­  ­¥ª®â®à®¬ à ááâ®ï­¨¨ ®â ­¥®¤­®à®¤­®á⨠¢®«­  ¢­®¢ì ¡ã¤¥â ¯à¥¤áâ ¢«ïâì ᮡ®© ®á­®¢­ãî ¬®¤ã. ® ¯à¨ í⮬ á«¥¢  ®â ­¥®¤­®à®¤­®á⨠⥯¥àì ¡ã¤¥â ¤¢¥ ¢®«­ë: ¯àï¬ ï ¨ ®âà ¦¥­­ ï,   á¯à ¢  ¯®-¯à¥¦­¥¬ã ⮫쪮 ¯à®è¥¤è ï ¯ ¤ îé ï ¢®«­ . ’ ª¨¬ ®¡à §®¬, ®¡à â­ ï ¢®«­  ¢®§­¨ª ¥â ¢á«¥¤á⢨¥ ®âà ¦¥­¨ï ®â ­¥®¤­®à®¤­®á⥩ ¢ ¢®«­®¢®¤¥. Žâà ¦¥­¨¥ ¬®¦­® å à ªâ¥à¨§®¢ âì ª®íää¨æ¨¥­â®¬ ®âà ¦¥­¨ï ;, à ¢­ë¬ ®â­®è¥­¨î ª®¬¯«¥ªá­ëå  ¬¯«¨â㤠®âà ¦¥­­®© ¨ ¯ ¤ î饩 ¢®«­: ; = ab ; £¤¥ a ¨ b {  ¬¯«¨âã¤ë ¯ ¤ î饩 ¨ ®âà ¦¥­­®© ¢®«­. Š®¬¡¨­ æ¨ï ¯ ¤ î饩 ¨ ®âà ¦¥­­ëå ¢®«­ ¤ ¥â áâ®ïçãî ¢®«­ã, å à ªâ¥à¨§ã¥¬ãî ª®íää¨æ¨¥­â®¬ áâ®ï祩 ¢®«­ë ­ ¯à殮­¨ï (Š‘‚):

jbj  = jjaajj + ; jbj :  áᬮâਬ ­¥®¤­®à®¤­®áâì, ®¡à §ã¥¬ãî ¬¥â ««¨ç¥áª®© ¯« á⨭®©, § ªà뢠î饩 ¢á¥ á¥ç¥­¨¥ ¢®«­®¢®¤ .   í⮩ ¯« á⨭¥ ¯®¯¥à¥ç­®¥ í«¥ªâà¨ç¥áª®¥ ¯®«¥ à ¢­® ­ã«î. ’ ª®¬ã £à ­¨ç­®¬ã ãá«®¢¨î ¬®¦­® 㤮¢«¥â¢®à¨âì á ¯®¬®éìî áã¬¬ë ¯ ¤ î饩 ¨ ®âà ¦¥­­®© ¢®«­ ®¤¨­ ª®¢®©  ¬¯«¨âã¤ë, ¢§ïâëå á ®¡à â­ë¬ §­ ª®¬ ¤«ï ¯®¯¥à¥ç­®£® í«¥ªâà¨ç¥áª®£® ¯®«ï. ¥âà㤭® ¢¨¤¥âì, çâ® ¯®¯¥à¥ç­ë¥ á®áâ ¢«ïî騥 ¬ £­¨â­®£® ¯®«ï 108

¯à¨ í⮬ ­  ¯« á⨭¥ ᪫ ¤ë¢ îâáï (Ey = ;j! @@x , Hx = ;j @@y ). à¨ í⮬ ª®íää¨æ¨¥­â ®âà ¦¥­¨ï à ¢¥­ ;1. ’ ªãî ­¥®¤­®à®¤­®áâì ¥áâ¥á⢥­­® ­ §¢ âì \ª®à®âª¨¬ § ¬ëª ­¨¥¬" ¢®«­®¢®¤ .  §®¬ª­ãâë© ¢®«­®¢®¤ ¯®«ãç¨âì ¤®áâ â®ç­® âà㤭®, â ª ª ª ®âªàëâë© ª®­¥æ ¢®«­®¢®¤  ᨫ쭮 ¨§«ã砥⠢ ¯à®áâà ­á⢮. ˆ¤¥ «ì­ë© 宫®á⮩ 室 ¬®¦­® ¡ë«® ¡ë ¯®«ãç¨âì, § ¬ëª ï ª®­¥æ ¢®«­®¢®¤  ¯« á⨭®© ¨§ ¨¤¥ «ì­®£® ¬ £­¥â¨ª  á  = 1. ’®£® ¦¥ ¬®¦­® ¤®áâ¨çì, § ª®à ç¨¢ ï ¢®«­®¢®¤ ¯« á⨭®© ­  à ááâ®ï­¨¨ =4 ®â ⮣® á¥ç¥­¨ï, £¤¥ ­¥®¡å®¤¨¬® ¯®«ãç¨âì ãá«®¢¨¥ 宫®á⮣® 室 . €­ «¨§¨àãï ¢®«­ë ¢ ¢®«­®¢®¤¥, ¬ë ¢áâà¥â¨¬áï á ¯®­ïâ¨ï¬¨, ª®â®àë¥ ¯à¥¦¤¥ ¡ë«¨ ¢¢¥¤¥­ë ¤«ï TEM-«¨­¨©, ®¯¨á뢠¥¬ëå ⥫¥£à ä­ë¬¨ ãà ¢­¥­¨ï¬¨. â   ­ «®£¨ï ¬®¦¥â ¡ëâì ¯à®¢¥¤¥­  §­ ç¨â¥«ì­® ¤ «ìè¥. à®¢¥¤¥­¨¥ â ª®©  ­ «®£¨¨ ¯®§¢®«ï¥â ¨á¯®«ì§®¢ âì ¤«ï à áç¥â®¢ å à ªâ¥à¨á⨪ ¢®«­®¢®¤­ëå ãáâனá⢠१ã«ìâ âë, ¯®«ã祭­ë¥ ¢ ⥮ਨ 楯¥© ¨ TEM-«¨­¨©. 4.2.

’¥«¥£à ä­ë¥ ãà ¢­¥­¨ï ¤«ï H-¢®«­ ¢ ¢®«­®¢®¤¥

®«ï ¢ ¢®«­®¢®¤¥ 㤮¢«¥â¢®àïîâ ãà ¢­¥­¨ï¬ Œ ªá¢¥«« 

~ rot E~ = ;j!H; ~ rot H~ = j!"E: ã⥬ à §¤¥«¥­¨ï ¯¥à¥¬¥­­ëå ¨§ íâ¨å ãà ¢­¥­¨© ¬®¦­® ¯®«ãç¨âì ⥫¥£à ä­ë¥ ãà ¢­¥­¨ï. „«ï í⮣® § ¯¨è¥¬ ¯®«ï H-¬®¤ ¢ ¢¨¤¥ (ãç¨â뢠ï, çâ® Ez = 0 ¤«ï H ¢®«­)

E~ = E~ ¯®¯ = V (z )
~e(x; y); H~ = H~ ¯®¯ + H~ z = I (z )
~h(x; y) + H~ z ; ¯à¨ç¥¬ ¢¥ªâ®à­ë¥ ä㭪樨 ~e ¨ ~h ¨¬¥îâ ⮫쪮 ¯®¯¥à¥ç­ë¥ á®áâ ¢«ïî騥 ¨ § ¢¨áïâ ⮫쪮 ®â ¯®¯¥à¥ç­ëå ª®®à¤¨­ â. ®¤áâ ¢«ïï í⨠¯®«ï ¢ ¯¥à¢®¥ ãà ¢­¥­¨¥ Œ ªá¢¥««  ¨ ¢ë¤¥«ïï ¯®¯¥à¥ç­ë¥ á®áâ ¢«ïî騥, ¯®«ã稬 [rot (V
~e )]¯®¯ = ; j!
I
~h: 109

‡ ¬¥â¨¬, çâ® rot (V
~e ) = V
rot ~e + grad V  ~e. ¥à¢®¥ á« £ ¥¬®¥ ­¥ ¨¬¥¥â ¯®¯¥à¥ç­ëå á®áâ ¢«ïîé¨å. ’ ª ª ª grad V = dV dz
~z0 , â®

dV
~z  ~e = ; j!I
~h: dz 0 “¬­®¦¨¬ íâ® ãà ¢­¥­¨¥ ­  ~h: dV
(~e  ~h)
~z = ; j!
(~h)2
I: 0 dz

®¤áâ ¢¨¬ ⥯¥àì ¢ëà ¦¥­¨ï ¤«ï ¯®«¥© ¢® ¢â®à®¥ ãà ¢­¥­¨¥ Œ ªá¢¥«« :

dI
(~z  ~h) + [rot H~ ]¯®¯ = j!"V
~e; z dz 0 â ª ª ª ~h = ;
grad , â® rot ~h = 0. ‡ ¬¥â¨¬, çâ® rot H~ z ¨¬¥¥â ⮫쪮 ¯®¯¥à¥ç­ë¥ á®áâ ¢«ïî騥. Šà®¬¥

⮣®,

H~ z = gm2
~z0
e; z ; E~ = ;j!
rot (
~z0
e; z ); ¯®í⮬ã

~ E~ = ; j! gm2 rot Hz ;

®âªã¤ 

gm2 E~ = ; gm2 V
~e: rot H~ z = ; j! j! ®¤áâ ¢«ïï íâ® ¢ ᮮ⢥âáâ¢ãî饥 ãà ¢­¥­¨¥ ¨ 㬭®¦ ï ­  ~e, ¯®«ã稬   dI (~e  ~h)
~z = ; gm2 + j!" ~e 2
V: 0 dz

j!

à®¨­â¥£à¨à㥬 ¯®«ã祭­ë¥ ãà ¢­¥­¨ï ¯® á¥ç¥­¨î ¢®«­®¢®¤  S . ‚¥«¨ç¨­ë ¨­â¥£à «®¢ ®¯à¥¤¥«ïîâáï ãá«®¢¨ï¬¨ ­®à¬¨à®¢ª¨. 110

Œ®é­®áâì ¯ ¤ î饩 ¢®«­ë ¢ ¢®«­®¢®¤¥ à ¢­  Z Z P = 12 Re (E~  H~  )
~z0 dS = 12 Re V I  (~e  ~h) ~z0 dS: S

S

ˆáå®¤ï ¨§ ¯®á«¥¤­¥£® ¢ëà ¦¥­¨ï ¤«ï ¬®é­®á⨠¥áâ¥á⢥­­® ¯à¨­ïâì ­®à¬¨à®¢ªã ~e ¨ ~h â ª®©, ç⮡ë

Z S

„ «¥¥,

(~e  ~h) ~z0 dS = 1:

1 jV j2 Z "~e 2 dS = W = C1 jV j2 | e 2 2 S

í«¥ªâà¨ç¥áª ï í­¥à£¨ï ­  ¥¤¨­¨æã ¤«¨­ë ¢®«­®¢®¤ . ‡¤¥áì

C1 =

Z

"~e 2 dS |

S

íª¢¨¢ «¥­â­ ï ¥¬ª®áâì ­  ¥¤¨­¨æã ¤«¨­ë «¨­¨¨. €­ «®£¨ç­® Z ¯à®¤ 2 Wh¯®¯ = 21 jI j2 ~h2 dS = L1 2 jI j | S

¬ £­¨â­ ï í­¥à£¨ï ­  ¥¤¨­¨æã ¤«¨­ë ¢®«­®¢®¤ , á¢ï§ ­­ ï á ¯®¯¥à¥ç­ë¬¨ ª®¬¯®­¥­â ¬¨ ¬ £­¨â­®£® ¯®«ï; §¤¥áì

L¯à®¤ 1 =

Z S

~h2 dS |

¯à®¤®«ì­ ï ¨­¤ãªâ¨¢­®áâì ­  ¥¤¨­¨æã ¤«¨­ë. ®¤áâ ¢«ïï C1 ¨ L¯à®¤ 1 ¢ ᮮ⢥âáâ¢ãî騥 ãà ¢­¥­¨ï, ¯®«ãç ¥¬

dV = ; j!L¯à®¤I; dI = ; j!C V ; V 1 1 dz dz j!"=gm2 C1 : ® 111

" 1 ¯®¯ gm2 C1 = !m2 C1 = L1 |

¯®¯¥à¥ç­ ï ¨­¤ãªâ¨¢­®áâì ­  ¥¤¨­¨æã ¤«¨­ë. ‚ íâ¨å ®¡®§­ ç¥­¨ïå ãà ¢­¥­¨ï ¯à¨®¡à¥â îâ ä®à¬ã

dV = ;j!L¯à®¤I; 1 dz dI = ;(j!C + 1 ) V: 1 dz j!L¯®¯ 1 â® ⥫¥£à ä­ë¥ ãà ¢­¥­¨ï ¤«ï ¢®«­ H-⨯  ¢ ®¤­®à®¤­®¬ ¢®«­®¢®¤¥. €­ «®£¨ç­ë¬ ®¡à §®¬ ¬®¦­® ¯®«ãç¨âì ãà ¢­¥­¨ï ¤«ï E-¢®«­ (¢¬¥áâ® ¯à®¤ L¯®¯ 1 ­ã¦­® ¢¢¥á⨠C1 ). ˆ§ ¯®«ã祭­ëå ãà ¢­¥­¨© á«¥¤ã¥â, çâ® 1 Z1 = j!L¯à®¤ 1 ; Y1 = j!C1 + j!L¯®¯ : 1 ¥®¡å®¤¨¬® ®â¬¥â¨âì, çâ® ¯à¨¢¥¤¥­­®¥ ¢ëè¥ ®¯à¥¤¥«¥­¨¥ ⮪  ¨ ­ ¯à殮­¨ï ¢ ¢®«­®¢®¤¥ ­¥ ®¯à¥¤¥«ï¥â ¨å ®¤­®§­ ç­®. ‚ á ¬®¬ ¤¥«¥, 㬭®¦¨¬ ­ ¯à殮­¨¥ ­  ¯à®¨§¢®«ì­ë© ¯®áâ®ï­­ë© ¬­®¦¨â¥«ì,   ⮪ à §¤¥«¨¬ ­  â ª®© ¦¥ ¬­®¦¨â¥«ì. ’®£¤  ­®à¬¨à®¢ª  ¯® ¬®é­®á⨠á®åà ­ï¥âáï. ‚ ⥫¥£à ä­ëå ãà ¢­¥­¨ïå Z1 ¨ Y1 ¨§¬¥­ïâáï â ª, çâ® ¨å ¯à®¨§¢¥¤¥­¨¥ ®áâ ¥âáï ¯à¥¦­¨¬ (â.¥. ¯®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï á®åà ­ï¥âáï),   ®â­®è¥­¨¥, â.¥. ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥, ¨§¬¥­ï¥âáï. ‘«¥¤®¢ â¥«ì­®, ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ ®¯à¥¤¥«¥­® ­¥®¤­®§­ ç­®.  ¯à¨¬¥à, ¤«ï ¯àאַ㣮«ì­®£® ¢®«­®¢®¤  (á ¬®¤®© H10 ) ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ ®¯à¥¤¥«ïîâ á«¥¤ãî騬¨ ᯮᮡ ¬¨. Ž¯à¥¤¥«¨¬ ⮪ I ª ª ¨­â¥£à « ®â ¯à®¤®«ì­®© á®áâ ¢«ïî饩 ¯«®â­®á⨠⮪  ­  è¨à®ª®© á⥭ª¥. ’®£¤  ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ ®¯à¥¤¥«ï¥âáï á«¥¤ãî饩 ä®à¬ã«®©: P = 12 jI j2
ZI : Œ®é­®áâì à ¢­  Z Z x P = 12 Ex
Hy dxdy = 12 A! sin x a
A sin a dxdy = S

S

112

2 = ab 4 A ! :

’®ª à ¢¥­

I =

Za

Hx dx =

Za

2a A sin x a dx =  A :

0 0 Žâá ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ à ¢­® 2 2 b = ZI = j2IPj2 = 8 ab ! 8 a

q

p="

:

1 ; 2 =2ªà €­ «®£¨ç­® ¬®¦­® ¢¢¥á⨠¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ ç¥à¥§ ­ ¯à殮­¨¥ ¢ 業âॠ¢®«­®¢®¤ : p=" 2 j U j b q ZU = 2P = 2 a : 1 ; 2 =2ªà 4.3.

Žª®­¥ç­®¥ ãáâனá⢮ (¤¢ã寮«îá­¨ª)

à¥¤¯®«®¦¨¬, çâ® ª ª®­æã ¢®«­®¢®¤  ¯à¨á®¥¤¨­¥­® § ¬ª­ã⮥ í«¥ªâத¨­ ¬¨ç¥áª®¥ ãáâனá⢮ (á¬. à¨á.). „«ï § ¬ª­ã⮩ ¯®¢¥àå­®áâ¨, ®ªà㦠î饩 ®ª®­¥ç­®¥ ãáâனá⢮, á¯à ¢¥¤«¨¢  ⥮६  ®©­â¨­£  1 I (E~ H~  )~n dS = P + 2j! (W ; W ): h e 2 S

’ ª ª ª ¯®«ï ¢¥§¤¥, ªà®¬¥ á¥ç¥­¨ï ¢®«­®¢®¤ , à ¢­ë ­ã«î, â® ¯®¢¥àå­®áâ­ë© ¨­â¥£à « ®¡à é ¥âáï ¢ ¨­â¥£à « ¯® á¥ç¥­¨î ¢®«­®¢®¤ . ®¤áâ ¢«ïï E~ ¨ H~ ¨ ãç¨â뢠ï ãá«®¢¨¥ ­®à¬¨à®¢ª¨, ¯®«ã稬 1 U
I  = P + 2j!(W ; W ): h e 2 „ ­­®¥ ᮮ⭮襭¨¥  ­ «®£¨ç­® ¯®«ã祭­®¬ã ¢ ⥮ਨ 楯¥© ¨ ¯®§¢®«ï¥â ¢¢¥á⨠¯®«­®¥ ᮯà®â¨¢«¥­¨¥ ¨ ¯à®¢®¤¨¬®áâì. 113

4.3.1. ®«­®¥ ᮯà®â¨¢«¥­¨¥ ¨ ¯à®¢®¤¨¬®áâì

‚¢¥¤¥¬ ¯®«­®¥ ᮯà®â¨¢«¥­¨¥ ¨ ¯®«­ãî ¯à®¢®¤¨¬®áâì á«¥¤ãî騬 ®¡à §®¬:

Z = UI ; Y = UI :

®«ã祭­®¥ ᮮ⭮襭¨¥ ¯®§¢®«ï¥â ­ ©â¨ í­¥à£¥â¨ç¥áª¨¥ ¢ëà ¦¥­¨ï ¤«ï íâ¨å ¢¥«¨ç¨­. € ¨¬¥­­®, ¯®¤áâ ¢«ïï U = Z I , ¯®«ã稬 1 jI j2 Z = P + 2j! (W ; W ); h e 2 ®âªã¤  Z = P + 2j!1(Wh ; We ) = R + jX: 2 2 jI j €­ «®£¨ç­® í⮬ã, ¯®¤áâ ¢«ïï I = Y U , ¯®«ã稬 Y = P + 2j!1 (We ; Wh ) = G + jB: 2 2 jU j ˆ§ íâ¨å ᮮ⭮襭¨© ¨¬¥¥¬

R = Re Z = Z (j!) +2 Z (;j!) ; X = Im Z = Z (j!) ;2j Z (;j!) ; Y  (j!) = Y (;j!); G = Re Y = Y (j!) +2 Y (;j!) ; B = Im Y = Y (j!) ;2j Y (;j!) : Žâá á«¥¤ã¥â, çâ® R; G { ç¥â­ë¥ ä㭪樨 ç áâ®âë, X; B { ­¥ç¥â­ë¥ Z  (j!) = Z (;j!);

ä㭪樨 ç áâ®âë. Šà®¬¥ ⮣®: 1. R  0, â ª ª ª P  0. 2. …᫨ P = 0, â® Z { ç¨áâ® ¬­¨¬ ï ¢¥«¨ç¨­ . 3. …᫨ Wh ; We = 0, â® X = 0 ¨«¨ B = 0, ç⮠ᮮ⢥âáâ¢ã¥â १®­ ­áã. 114

4.3.2. ‚®«­ë ¢ ®ª®­¥ç­®¬ ãáâனá⢥

 áᬮâਬ ¯ ¤ îéãî ¨ ®âà ¦¥­­ãî ¢®«­ë ­  § ¦¨¬ å ®ª®­¥ç­®£® ãáâனá⢠. Ž¡®§­ ç¨¬ ª®¬¯«¥ªá­ë¥  ¬¯«¨âã¤ë ¯ ¤ î饩 ¨ ®âà ¦¥­­®© ¢®«­ ç¥à¥§ a ¨ b. Š®íää¨æ¨¥­â ®âà ¦¥­¨ï ®¯à¥¤¥«¨¬ ª ª ; = ab : à¨ í⮬ ¬®é­®á⨠¯ ¤ î饩 ¨ ®âà ¦¥­­®© ¢®«­ à ¢­ë P¯ ¤ = 12 jaj2 ; P®âà = 21 jbj2 : ’ ª ª ª í«¥ªâà¨ç¥áª¨¥ ¯®«ï ¯ ¤ î饩 ¨ ®âà ¦¥­­®© ¢®«­ ᪫ ¤ë¢ îâáï,   ¬ £­¨â­ë¥ { ¢ëç¨â îâáï, â®

U = p(a + b) = pa(1 + ;); I = 1p (a ; b) = 1p a(1 ; ;): Œ­®¦¨â¥«¨ p ¨ 1=p ®¡à â­ë ¢ ᨫ㠭®à¬¨à®¢ª¨ ¯® ®â­®è¥­¨î ª ¬®é­®áâ¨. Žâá ᮯà®â¨¢«¥­¨¥ ®ª®­¥ç­®£® ãáâனá⢠ à ¢­® + ;: Z = UI = p2 11 ; ; à¨ ; = 0 ¤®«¦­® ¡ëâì Z = Z0 , ®âªã¤  p2 = Z0. ¥è ï ®â­®á¨â¥«ì­® ;, ¯®«ã稬

Z0 ; = ZZ ; + Z0 : Š ª 㪠§ ­® ¢ëè¥, ¢ë¡®à ¢®«­®¢®£® ᮯà®â¨¢«¥­¨ï ­¥®¤­®§­ ç¥­. ‚ à拉 á«ãç ¥¢ 㤮¡­® ¢ë¡¨à âì ⮪ ¨ ­ ¯à殮­¨¥ â ª, çâ®¡ë ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ ¡ë«® à ¢­® ¥¤¨­¨æ¥, â.¥., ¯® áãé¥áâ¢ã, ­®à¬¨à®¢ âì ᮯà®â¨¢«¥­¨¥ ­ £à㧪¨ ­  ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥. ’®£¤  + ; ; ; = z ; 1: z = 11 ; ; z +1 €­ «®£¨ç­® { ¤«ï ¯à®¢®¤¨¬®áâ¨. 115

à¨ «î¡®¬ p, ¯®¤áâ ¢«ïï ­ ¯à殮­¨¥ ¨ ⮪ ¢ ¢ëà ¦¥­¨¥ ¤«ï U
I  , ¯®«ã稬 (1 + ;)(1 ; ; ) 12 jaj2 = P + 2j!(We ; Wh ); ®âªã¤  (1 + ;)(1 ; ; ) = P + 2j!1(We ; Wh ) ; 2 2 jaj  §¤¥«ïï ¢¥é¥á⢥­­ë¥ ¨ ¬­¨¬ë¥ ç áâ¨, ¯®«ã稬 1 ; ;
; = 1 P ; 2 2 jaj (; ; ; ) = 2j!(W1e ; Wh ) : 2 2 jaj ’ ª ª ª P  0, â® ;
;  1 ¨ j;j  1. ®«ã祭­ë¥ ¢ëè¥ á®®â­®è¥­¨ï ¯®§¢®«ïîâ ­ ©â¨ ;: ; =

p

1 ; P 0
e j' ;

¯à¨ç¥¬ ;; P0 = P : sin ' = Im 1 jaj2 j;j 2 4.4.

‘®¥¤¨­¥­¨¥ ­¥áª®«ìª¨å ¢®«­®¢®¤®¢

®«ã祭­ë¥ १ã«ìâ âë ¬®£ãâ ¡ëâì ®¡®¡é¥­ë ­  á«ãç ©, ª®£¤  ᮥ¤¨­¥­ë ­¥áª®«ìª® ¢®«­®¢®¤­ëå «¨­¨© ¯¥à¥¤ ç¨, ¯à¨ç¥¬ ¢ ®¡é¥¬ á«ãç ¥

116

¢®«­®¢®¤ë ¬®£ãâ ¨¬¥âì à §«¨ç­ë¥ á¥ç¥­¨ï. ãáâì ¤ ­® ᮥ¤¨­¥­¨¥ N ¢®«­®¢®¤®¢. â® ᮥ¤¨­¥­¨¥ ¯®¬¥á⨬ ¢ § ¬ª­ãâãî ¯®¢¥àå­®áâì S , ¯¥à¥á¥ª îéãî ¢á¥ ¢®«­®¢®¤ë ¯® ¯«®áª®áâï¬, ¯¥à¯¥­¤¨ªã«ïà­ë¬ ¨å ®áï¬ (á¬. à¨á.). ˆá¯®«ì§®¢ ­¨¥ ª®¬¯«¥ªá­®© â¥®à¥¬ë ®©­â¨­£  á ãç¥â®¬ ­®à¬¨à®¢ª¨ ¯®¯¥à¥ç­ëå ¢¥ªâ®à­ëå ä㭪権 ¤ ¥â ¢ í⮬ á«ãç ¥

S

1 X U
I  = P + 2j! (W ; W ); h e 2 n n n £¤¥ P { á।­ïï à áᥨ¢ ¥¬ ï ¢ ᮥ¤¨­¥­¨¨ ¬®é­®áâì, Wh ¨ We { á।­¨¥ § ¯ áë í­¥à£¨¨ ¬ £­¨â­®£® ¨ í«¥ªâà¨ç¥áª®£® ¯®«¥©, n { ­®¬¥à ¢å®¤ . 4.4.1. Œ âà¨æë ᮯà®â¨¢«¥­¨ï ¨ ¯à®¢®¤¨¬®áâ¨

ˆ§ «¨­¥©­®á⨠ãà ¢­¥­¨© Œ ªá¢¥««  á«¥¤ã¥â «¨­¥©­ ï § ¢¨á¨¬®áâì ­ ¯à殮­¨© ®â ⮪®¢

Un =

X m

Znm Im :

Š®íää¨æ¨¥­âë Znm ®¡à §ãîâ ¬ âà¨æã

Z = fZnm g ; ­ §ë¢ ¥¬ãî ¬ âà¨æ¥© ¯®«­®£® ¢å®¤­®£® ᮯà®â¨¢«¥­¨ï ¬­®£®¯®«îá­¨ª . “ª § ­­®¥ ¢ëè¥ á®®â­®è¥­¨¥ ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¬ âà¨ç­®© ä®à¬¥

U = Z I; £¤¥ U (U1 ; U2 ; : : : ; UN ) ¨ I (I1 ; I2 ; : : : ; IN ) { ᮢ®ªã¯­®á⨠­ ¯à殮­¨© ¨

⮪®¢ ¢å®¤®¢. â® ᮮ⭮襭¨¥ ¬®¦¥â ¡ëâì ®¡à é¥­®: 117

I = Y U; £¤¥ Y = fYnm g { ¬ âà¨æ  ¯®«­®© ¯à®¢®¤¨¬®áâ¨. Žç¥¢¨¤­®, çâ® Y = Z ;1 . ®«ã祭­®¥ ¢ëè¥ á®®â­®è¥­¨¥ ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¢¨¤¥ 1 U I  = P + 2j! (W ; W ) h e 2 ¨«¨, ¯®¤áâ ¢«ïï U = Z I , 1 I  ZI = P + 2j! (W ; W ): h e 2 4.4.2. ‘¨¬¬¥âà¨ï ¬ âà¨æë ᮯà®â¨¢«¥­¨ï (¯à®¢®¤¨¬®áâ¨)

Œ âà¨æë ¯®«­®£® ᮯà®â¨¢«¥­¨ï ¨ ¯®«­®© ¯à®¢®¤¨¬®á⨠®¡« ¤ îâ ¢ ¦­ë¬ ᢮©á⢮¬ ᨬ¬¥âਨ. â® ᢮©á⢮ ¬®¦¥â ¡ëâì ¤®ª § ­® á ¯®¬®éìî «¥¬¬ë ‹®à¥­æ . à¥¤¯®«®¦¨¬, çâ® ¢ ­¥ª®â®à®© ®¡« á⨠V ¨§¢¥áâ­ë ¤¢  à¥è¥­¨ï ®¤­®à®¤­ëå ãà ¢­¥­¨© Œ ªá¢¥««  ¯à¨ ®¤­®© ¨ ⮩ ¦¥ ç áâ®â¥. ‚ᥠ¢¥«¨ç¨­ë, ᮮ⢥âáâ¢ãî騥 í⨬ à¥è¥­¨ï¬, ¡ã¤¥¬ ®¡®§­ ç âì ¨­¤¥ªá ¬¨ 1 ¨ 2 ᢥàåã.  áᬮâਬ ¢ëà ¦¥­¨¥ div (E~ (1)  H~ (2) ) ; div (E~ (2)  H~ (1) ) = H~ (2) rot E~ (1) ; E~ (1) rot H~ (2) ;

; H~ (1) rot E~ (2) + E~ (2) rot H~ (1)

à ¢ ï ç áâì ¢ ᨫã ãà ¢­¥­¨© Œ ªá¢¥««  à ¢­  ;j!H~ (2)H~ (1) ; j!E~ (1) "E~ (2) + j!H~ (1) H~ (2) + j!E~ (2)"E~ (1) ¨ ®¡à é ¥âáï ¢ ­ã«ì, ¥á«¨  ¨ " { ᪠«ïàë ¨«¨ ᨬ¬¥âà¨ç­ë¥ ⥭§®àë1 . ˆâ ª, ¯®«ãç ¥¬ à ¢¥­á⢮ div (E~ (1)  H~ (2) ) ; div (E~ (2)  H~ (1)) = 0; 1 ‚¥«¨ç¨­ë  ¨ " ¬®£ãâ ¡ëâì ª®¬¯«¥ªá­ë¬¨, ¥á«¨ ¯à¨áãâáâ¢ãîâ ¯®â¥à¨ ¢ á⥭ª å ¨ á।¥, § ¯®«­ïî饩 ¯à®áâà ­á⢮.

118

ª®â®à®¥ ï¥âáï ¢ëà ¦¥­¨¥¬ «¥¬¬ë ‹®à¥­æ  ¢ ¤¨ää¥à¥­æ¨ «ì­®© ä®à¬¥. ˆ­â¥£à¨àãï ¯® ®¡ê¥¬ã V , ®£à ­¨ç¥­­®¬ã ¯®¢¥àå­®áâìî S , ¯¥à¥¢¥¤¥¬ íâã «¥¬¬ã ¢ ¨­â¥£à «ì­ãî ä®à¬ã (¯à¥®¡à §ãï ®¡ê¥¬­ë© ¨­â¥£à « ¢ ¯®¢¥àå­®áâ­ë©):

I S

fE~ (1)  H~ (2) ; E~ (2)  H~ (1) g ~n dS = 0:

‚롥६ ¯®¢¥àå­®áâì S â ª, çâ®¡ë ®­  ®å¢ â뢠«  ¢®«­®¢®¤­®¥ á®ç«¥­¥­¨¥ á­ à㦨, ¯¥à¥á¥ª ï ¢®«­®¢®¤ë ¯® ¯«®áª®áâï¬, ¯¥à¯¥­¤¨ªã«ïà­ë¬ ®á¨ ¢®«­®¢®¤®¢. ‚ëà ¦ ï ¯®«¥ ¢ ¢®«­®¢®¤ å ç¥à¥§ ­ ¯à殮­¨ï ¨ ⮪¨ ¨ ãç¨âë¢ ï ­®à¬¨à®¢ªã ¢¥ªâ®à­ëå ä㭪権, ¯®«ã稬 á«¥¤ãî饥 à ¢¥­á⢮: X (1) (2) [Un
In ; Un(2)
In(1) ] = 0: n

‡¤¥áì n { ­®¬¥à ¢®«­®¢®¤ . „ ­­®¥ ãà ¢­¥­¨¥ á¯à ¢¥¤«¨¢® ¤«ï «î¡ëå ¤¢ãå ᮢ®ªã¯­®á⥩ ­ ¯à殮­¨© ¨ ᮮ⢥âáâ¢ãîé¨å ⮪®¢, ¯à¨«®¦¥­­ëå ª § ¦¨¬ ¬ ᮥ¤¨­¥­¨ï. ®«ã祭­®¥ ãà ¢­¥­¨¥ ¬®¦­® § ¯¨á âì ¢ ¬ âà¨ç­®© ä®à¬¥: U (1) I (2) ; U (2) I (1) = 0: ‡¤¥áì U ¨ I { ᮢ®ªã¯­®á⨠(¬ âà¨æë) ­ ¯à殮­¨© ¨ ⮪®¢. ‘ ¤à㣮© áâ®à®­ë, ­ ¯à殮­¨ï ¨ ⮪¨ á¢ï§ ­ë ¬ âà¨ç­ë¬ ᮮ⭮襭¨¥¬

U = Z I: ®¤áâ ¢«ïï íâ® ¢ ­ ¯¨á ­­®¥ ¢ëè¥ ãà ¢­¥­¨¥, ¯®«ã稬 I (2) Z I (1) ; I (1) Z I (2) = 0: ¥à¥áâ ¢«ïï ᮬ­®¦¨â¥«¨ ¢ ¯¥à¢®¬ á« £ ¥¬®¬, ᮣ« á­® ¨§¢¥áâ­®¬ã ᢮©áâ¢ã ¬ âà¨æ ¯®«ã稬 I (1) Z T I (2) ; I (1) Z I (2) = 0; £¤¥ Z T { â࠭ᯮ­¨à®¢ ­­ ï ¬ âà¨æ . ’ ª ª ª ¯®«ã祭­®¥ ᮮ⭮襭¨¥ ¢ë¯®«­ï¥âáï ¤«ï ¯à®¨§¢®«ì­ëå ⮪®¢ I (1) ¨ I (2) , â® ¨§ í⮣® á«¥¤ã¥â, çâ® 119

Z T = Z , â.¥. ¬ âà¨æ  Z ¤¥©á⢨⥫쭮 ᨬ¬¥âà¨ç­ . “á«®¢¨¥¬ í⮣®

ï¥âáï á¯à ¢¥¤«¨¢®áâì «¥¬¬ë ‹®à¥­æ , ¤«ï 祣® ­¥®¡å®¤¨¬® ¨ ¤®áâ â®ç­®, ç⮡ë  ¨ " áà¥¤ë ¡ë«¨ «¨¡® ᪠«ïà ¬¨, «¨¡® ᨬ¬¥âà¨ç­ë¬¨ ⥭§®à ¬¨. ‘¨¬¬¥âà¨ï ¬ âà¨æë ¯®«­®£® ᮯà®â¨¢«¥­¨ï Z ï¥âáï ¢ëà ¦¥­¨¥¬ ¨§¢¥áâ­®£® ¯à¨­æ¨¯  ¢§ ¨¬­®áâ¨. €­ «®£¨ç­® ¬®¦¥â ¡ëâì ¤®ª § ­® ᢮©á⢮ ᨬ¬¥âਨ ¬ âà¨æë ¯à®¢®¤¨¬®á⨠Y . 4.4.3. Œ­®£®¯®«îá­®¥ á®ç«¥­¥­¨¥ ¡¥§ ¯®â¥àì

‚® ¬­®£¨å á«ãç ïå á®ç«¥­¥­¨¥ ®¡« ¤ ¥â ­ á⮫쪮 ¬ «ë¬¨ ¯®â¥àﬨ, çâ® ¨¬¨ ¬®¦­® ¯à¥­¥¡à¥çì ¨ à áᬠâਢ âì á®ç«¥­¥­¨¥ ¡¥§ ¯®â¥àì. ‚ í⮬ á«ãç ¥, ª ª 㢨¤¨¬ ¤ «ìè¥, ¢á¥ í«¥¬¥­âë ¬ âà¨æë ᮯà®â¨¢«¥­¨ï (¨ ¯à®¢®¤¨¬®áâ¨) ¯à¥¤áâ ¢«ïîâ ᮡ®© ç¨áâ® ¬­¨¬ë¥ ¢¥«¨ç¨­ë. ã¤¥¬ ¨á室¨âì ¨§ ­ ©¤¥­­®£® ¢ëè¥ á®®â­®è¥­¨ï 1 I  ZI = 1 X I  Z I = P + 2j! (W ; W ): h e 2 2 n;m n nm m ‚ á«ãç ¥ ãáâனá⢠ ¡¥§ ¯®â¥àì P = 0 ¨ 1 X I  Z I = 2j! (W ; W ); h e 2 n;m n nm m â.¥. á㬬  ï¥âáï ç¨áâ® ¬­¨¬®©. ®ª ¦¥¬, çâ® ¢ í⮬ á«ãç ¥ ¢á¥ í«¥¬¥­âë Znm ç¨áâ® ¬­¨¬ë¥. „«ï í⮣® à áᬮâਬ â ª®© ०¨¬, ª®£¤  ¢á¥ § ¦¨¬ë, ªà®¬¥ ®¤­®£®, à §®¬ª­ãâë, â.¥. ¢á¥ In = 0, ªà®¬¥ ®¤­®£® ⮪  Ik 6= 0. ’®£¤  ¯à¨¢¥¤¥­­ ï ¢ëè¥ á㬬  ᢮¤¨âáï ª ®¤­®¬ã á« £ ¥¬®¬ã: 1 Z I  I = 2j! (W ; W ); h e 2 kk k k ®âªã¤  á«¥¤ã¥â, çâ® ¢á¥ ¤¨ £®­ «ì­ë¥ í«¥¬¥­âë Zkk { ç¨áâ® ¬­¨¬ë¥. ’¥¯¥àì à áᬮâਬ ०¨¬, ª®£¤  ¢á¥ In à ¢­ë ­ã«î, ªà®¬¥ ¤¢ãå ⮪®¢ Ik 6= 0 ¨ Im 6= 0. ’®£¤  ¢ á㬬¥ ®áâ ¥âáï ç¥âëॠ童­ : 1 fI I  Z + I I  Z + I  Z I + I  Z I g = 2j! (W ; W ): m m mm h e m mk k k km m 2 k k kk …᫨ Zmk = Zkm , â® ¤«ï ¢ë¯®«­¥­¨ï í⮣® à ¢¥­á⢠ ¤®«¦­® ¡ëâì 120

Re (Im Ik + Ik Im )Zmk = 0: ’ ª ª ª á㬬  Im Ik + Ik Im ¢¥é¥á⢥­­ , â® ¨§ í⮣® à ¢¥­á⢠ á«¥¤ã¥â, çâ® Re Zmk = 0; çâ® ¨ âॡ®¢ «®áì ¤®ª § âì. €­ «®£¨ç­® ¤®ª §ë¢ ¥âáï, çâ® ¢ ¤ ­­®¬ á«ãç ¥ ¢á¥ í«¥¬¥­âë ¬ âà¨æë ¯à®¢®¤¨¬®á⨠⠪¦¥ ¬­¨¬ë. 4.4.4. ‚ à¨ æ¨ï ¬ âà¨æë ᮯà®â¨¢«¥­¨ï (¯à®¢®¤¨¬®áâ¨) á®ç«¥­¥­¨ï ¡¥§ ¯®â¥àì

«¥¬¥­âë ¬ âà¨æë ¯®«­®£® ᮯà®â¨¢«¥­¨ï (¯à®¢®¤¨¬®áâ¨) ïîâáï ä㭪樮­ « ¬¨ ¢¥ªâ®à­ëå ä㭪権 í«¥ªâà¨ç¥áª®£® ¨ ¬ £­¨â­®£® ¯®«¥©. …᫨ ¢ àì¨à®¢ âì ä㭪樨 ¯®«¥©, â® í«¥¬¥­âë ¬ âà¨æë ᮯà®â¨¢«¥­¨ï â ª¦¥ ¡ã¤ãâ ¢ àì¨à®¢ âìáï. „«ï ¢ëç¨á«¥­¨ï ¨å ¢ à¨ æ¨¨ ¡ã¤¥¬ ¨á室¨âì ¨§ ᮮ⭮襭¨ï

I  ZI = 4j! (Wh £¤¥

Z

; We ) = j! (jH j2 ; "jE j2 ) dV; V

Z 2 Z 2 Wh = 12 jH2 j dV; We = 21 "jE2 j dV: V

V

‡¤¥áì ¨­â¥£à « ¡¥à¥âáï ¯® ®¡ê¥¬ã V , ®£à ­¨ç¥­­®¬ã ¯®¢¥àå­®áâìî S , ᮢ¯ ¤ î饩 á ¢­ãâ७­¥© ¯®¢¥àå­®áâìî á®ç«¥­¥­¨ï ¨ ¯¥à¥á¥ª î饩

¢®«­®¢®¤ë ¯® ¯«®áª®áâï¬, ¯¥à¯¥­¤¨ªã«ïà­ë¬ ®áï¬ ¢®«­®¢®¤®¢. à¥¤¯®«®¦¨¬, çâ® ¢ àì¨àã¥âáï ⮫쪮 ¬ £­¨â­®¥ ¯®«¥,   í«¥ªâà¨ç¥áª®¥ ¨§¬¥­ï¥âáï ¢ § ¢¨á¨¬®á⨠®â ¬ £­¨â­®£® ᮣ« á­® ¢â®à®¬ã ãà ¢­¥­¨î Œ ªá¢¥««  ~ E~ = 1 rot H:

j!"

®¤áâ ¢¨¬ íâ® ¢ ¯à¨¢¥¤¥­­®¥ ¢ëè¥ á®®â­®è¥­¨¥ 121

I  ZI = j!

V

j Z

= !"

V

Z

 jH~ j2 ; !2""2 jrot H~ j2 dV =

 j Z k2 H~ H~  ; rot H~
rot H~   dV: k2 jH~ j2 ; jrot H~ j2 dV = !" V

‚®§ì¬¥¬ ⥯¥àì ¢ à¨ æ¨î ¯à ¢®© ¨ «¥¢®© ç á⥩:

j Z k2 H~
H~  + k2 H~
H~  ; I  ZI + I  ZI + I  ZI = !" V



; rot H~
rot H~  ; rot H~
rot H~  dV: ‚®á¯®«ì§ã¥¬áï ¢¥ªâ®à­ë¬ ⮦¤¥á⢮¬ div (rot H~  H~  ) = H~ 
rot rot H~ ; rot H~
rot H~  ; ¨§ ª®â®à®£® á«¥¤ã¥â, çâ®

~ ;rot H~
rot H~  = div (rot H~  H~  ) ; H~ 
rot rot H; ;rot H~ 
rot H~ = div (rot H~   H~ ) ; H~
rot rot H~  :

®¤áâ ¢«ïï íâ® ¢ ãà ¢­¥­¨¥ ¢ à¨ æ¨©, ¯®«ã稬

j Z h(k2 H~ ; rot rot H~ )
H~  + I  ZI + I  ZI + I  ZI = !"

i ~

+ (k2 H~  ; rot rot H~  )
H dV ;

V

Ih S

i

(rot H~  H~  ) ~n + (rot H~   H~ ) ~n dS:

‡¤¥áì ~n { ¢­ãâ७­ïï ¯® ®â­®è¥­¨î ª ¯®¢¥àå­®á⨠¥¤¨­¨ç­ ï ­®à¬ «ì. ~ ¯à ¢ãî ç áâì í⮣® à ¢¥­á⢠ ¬®¦­® ¯à¥“ç¨â뢠ï, çâ® rot H~ = j!"E; ®¡à §®¢ âì:

j Z h(k2 H~ ; rot rot H~ )
H~  + I  ZI + I  ZI + I  ZI = !" V

122

+ (k2 H~  ; rot rot H~  )
H~ dV +

S

E~  H~  + E~   H~ ~n dS:

®ª ¦¥¬ ⥯¥àì, çâ® ¥á«¨ H~ 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î k2 H~ ; rot rot H~ = 0 ¨ £à ­¨ç­®¬ã ãá«®¢¨î rot H~  ~n = 0; ¨«¨ E~  ~n = 0 (çâ® ®¤­® ¨ â® ¦¥), â® ¢ à¨ æ¨ï Z = 0. „¥©á⢨⥫쭮, ¢ í⮬ á«ãç ¥ ®¡ê¥¬­ë© ¨­â¥£à « ¢ ¯à ¢®© ç á⨠­ ¯¨á ­­®£® ¢ëè¥ à ¢¥­á⢠ ®¡à é ¥âáï ¢ ­ã«ì. — áâì ¯®¢¥àå­®áâ­®£® ¨­â¥£à « , ¢§ïâ ï ¯® ¢­ãâ७­¥© ¯®¢¥àå­®á⨠á®ç«¥­¥­¨ï, ®¡à é ¥âáï ¢ ­ã«ì ¢ ᨫ㠣࠭¨ç­ëå ãá«®¢¨©, ®áâ «ì­ ï ç áâì í⮣® ¨­â¥£à «  ¯à¥¢à é ¥âáï ¢ á㬬㠨­â¥£à «®¢ ¯® ¯®¯¥à¥ç­ë¬ á¥ç¥­¨ï¬ ¢®«­®¢®¤®¢. ‚ëà ¦ ï ¯®«¥ ¢ ¢®«­®¢®¤ å ç¥à¥§ ­ ¯à殮­¨ï ¨ ⮪¨, ¯®«ã稬 ¢ ¨â®£¥ á«¥¤ãî饥 à ¢¥­á⢮: I  ZI + I  ZI + I  ZI = I  ZI ; I  Z I: ® ¤«ï á®ç«¥­¥­¨ï ¡¥§ ¯®â¥àì Z  = ;Z . ®¤áâ ¢«ïï ¢ ¯à ¢ãî ç áâì à ¢¥­á⢠ ¨ ¯à¨¢®¤ï ¯®¤®¡­ë¥ ç«¥­ë ¢ ¯à ¢®© ¨ «¥¢®© ç áâ¨, ¯®«ãç ¥¬ ¨áª®¬®¥ à ¢¥­á⢮ I  ZI = 0: ‚á«¥¤á⢨¥ ¯à®¨§¢®«ì­®á⨠⮪®¢ ®âá á«¥¤ã¥â Z = 0. ˆâ ª, ¢ à¨ æ¨ï ¬ âà¨æë ᮯà®â¨¢«¥­¨ï ®¡à é ¥âáï ¢ ­ã«ì ¯à¨ á«¥¤ãîé¨å ãá«®¢¨ïå: 1. ®«¥ H~ 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î ƒ¥«ì¬£®«ìæ  k2 H~ ; rot rot H~ = 0: 2. ’ ­£¥­æ¨ «ì­ ï á®áâ ¢«ïîé ï rot H~ ¨«¨, çâ® à ¢­®á¨«ì­®, â ­£¥­æ¨ «ì­ ï á®áâ ¢«ïîé ï í«¥ªâà¨ç¥áª®£® ¯®«ï E~ à ¢­  ­ã«î. ® íâ® §­ ç¨â, çâ® H~ ï¥âáï à¥è¥­¨¥¬ ãà ¢­¥­¨© Œ ªá¢¥««  ¯à¨ áâ ­¤ àâ­ëå £à ­¨ç­ëå ãá«®¢¨ïå. ‘«¥¤®¢ â¥«ì­®, à¥è¥­¨¥ í«¥ªâத¨­ ¬¨ç¥áª®© § ¤ ç¨ á®®¡é ¥â ¬ âà¨æ¥ ᮯà®â¨¢«¥­¨ï Z , à áᬠâਢ ¥¬®© ª ª ä㭪樮­ «, áâ æ¨®­ à­®¥ §­ ç¥­¨¥. ‡ ¬¥â¨¬, çâ® í«¥ªâத¨­ ¬¨ç¥áª¨¥ £à ­¨ç­ë¥ ãá«®¢¨ï ¢ ¤ ­­®¬ á«ãç ¥ ïîâáï ¥áâ¥á⢥­­ë¬¨ £à ­¨ç­ë¬¨ ãá«®¢¨ï¬¨ ᮮ⢥âáâ¢ãî饩 ¢ à¨ æ¨®­­®© § ¤ ç¨. 123

€­ «®£¨ç­® ¬®¦¥â ¡ëâì à áᬮâ७ á«ãç © ¢ à¨ æ¨¨ í«¥ªâà¨ç¥áª®£® ¯®«ï. à¨ í⮬ ¯®«ãç ¥âáï ᮮ⭮襭¨¥, ª®â®à®¥ ®¯à¥¤¥«ï¥â ¢ à¨ æ¨î ¬ âà¨æë ᮯà®â¨¢«¥­¨©:

j Z h(k2 E~ ; rot rot E~ )
E~  + I  ZI + I  ZI + I  ZI = ! V

i

+ (k2 E~  ; rot rot E~ )
E~  dV +

Ih i H~  E~  + H~   E~ ~n dS: S

„«ï ⮣® çâ®¡ë ¢ í⮬ á«ãç ¥ ¢ à¨ æ¨ï Z ®¡à â¨« áì ¢ ­ã«ì, í«¥ªâà¨ç¥áª®¥ ¯®«¥ E~ ¤®«¦­® 㤮¢«¥â¢®àïâì ãà ¢­¥­¨î ƒ¥«ì¬£®«ìæ . Šà®¬¥ ⮣®, â ­£¥­æ¨ «ì­ ï á®áâ ¢«ïîé ï ¢ à¨ æ¨¨ E~ ¤®«¦­  ®¡à é âìáï ¢ ­ã«ì ­  ¯®¢¥àå­®á⨠á®ç«¥­¥­¨ï. à¨ í⮬, ª ª ¢¨¤¨¬, ­¥ ­ ª« ¤ë¢ ¥âáï £à ­¨ç­®¥ ãá«®¢¨¥ ­  á ¬® ¯®«¥ E~ . â®¬ã ãá«®¢¨î äã­ªæ¨ï E~ ¤®«¦­  㤮¢«¥â¢®àïâì ¢ ᨫã âॡ®¢ ­¨© í«¥ªâத¨­ ¬¨ª¨, ­® ­¥ ®¡à é¥­¨ï ¢ ­ã«ì ¢ à¨ æ¨¨ ¬ âà¨æë ᮯà®â¨¢«¥­¨ï. ‚ í⮬ á®á⮨â à §«¨ç¨¥ á ¯à¥¤ë¤ã騬 á«ãç ¥¬. „ «¥¥ ¯à®æ¥¤ãà  â  ¦¥, çâ® ¨ ¢ ¯à¥¤ë¤ã饬 á«ãç ¥. ˆâ ª, ¢ ¤ ­­®¬ á«ãç ¥ ¢ à¨ æ¨ï ¬ âà¨æë ᮯà®â¨¢«¥­¨ï ®¡à é ¥âáï ¢ ­ã«ì ¯à¨ á«¥¤ãîé¨å ãá«®¢¨ïå: 1. «¥ªâà¨ç¥áª®¥ ¯®«¥ E~ 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î ƒ¥«ì¬£®«ìæ . 2. ‚ à¨ æ¨ï E~ ¨¬¥¥â à ¢­ãî ­ã«î ­  ¢­ãâ७­¥© ¯®¢¥àå­®á⨠S á®ç«¥­¥­¨ï â ­£¥­æ¨ «ì­ãî á®áâ ¢«ïîéãî. ’ ª¨¬ ®¡à §®¬, 㤮¢«¥â¢®à¥­¨¥ £à ­¨ç­ë¬ ãá«®¢¨ï¬ ä㭪樥© E~ ­¥ ¢å®¤¨â ¢ ç¨á«® ãá«®¢¨©, ­¥®¡å®¤¨¬ëå ¤«ï ®¡à é¥­¨ï ¢ ­ã«ì ¢ à¨ æ¨¨ ¬ âà¨æë ᮯà®â¨¢«¥­¨ï. ‚ ¤ ­­®¬ á«ãç ¥ í«¥ªâத¨­ ¬¨ç¥áª¨¥ £à ­¨ç­ë¥ ãá«®¢¨ï ­¥ ïîâáï ¥áâ¥á⢥­­ë¬¨ £à ­¨ç­ë¬¨ ãá«®¢¨ï¬¨ ¢ à¨ æ¨®­­®© § ¤ ç¨ (  ïîâáï â ª ­ §ë¢ ¥¬ë¬¨ £« ¢­ë¬¨ £à ­¨ç­ë¬¨ ãá«®¢¨ï¬¨). 4.4.5. Ž¡®¡é¥­¨¥ ­  á«ãç © ¢®«­®¢®¤®¢, ¯® ª®â®àë¬ ¬®£ãâ à á¯à®áâà ­ïâìáï ¢®«­ë ­¥áª®«ìª¨å ¬®¤

„® á¨å ¯®à à áᬮâ७¨¥ ®£à ­¨ç¨¢ «®áì ¢®«­®¢®¤ ¬¨, ¯® ª®â®àë¬ à á¯à®áâà ­ï¥âáï «¨èì ®¤­  (®á­®¢­ ï) ¬®¤ . ‚®§¬®¦­® ®¡®¡é¥­¨¥ ­  á«ãç © «î¡®£® ç¨á«  à á¯à®áâà ­ïîé¨åáï ¬®¤. 124

„ ­­®¥ ®¡®¡é¥­¨¥ ¯à®¨§¢®¤¨âáï ¯ã⥬ ¢¢¥¤¥­¨ï ­ ¯à殮­¨© ¨ ⮪®¢, ᮮ⢥âáâ¢ãîé¨å ª ¦¤®© à á¯à®áâà ­ïî饩áï ¬®¤¥. à¨ í⮬ ®á­®¢­ë¥ ᮮ⭮襭¨ï ­¥ ¨§¬¥­ïîâáï ¡« £®¤ àï ⮬ã, çâ® ¯®«ï à §«¨ç­ëå ¬®¤ ¢ ¢®«­®¢®¤¥ ®à⮣®­ «ì­ë, çâ® ¯à¨¢®¤¨â ª ¨á祧­®¢¥­¨î ¯¥à¥ªà¥áâ­ëå ¨­â¥£à «®¢ ⨯ 

Z S

(E~ m  H~ k ) ~z0 dS ¯à¨ m 6= k;

£¤¥ S { ¯®¯¥à¥ç­®¥ á¥ç¥­¨¥ ¢®«­®¢®¤ , ~z0 { ¥¤¨­¨ç­ë© ¢¥ªâ®à ¢ ­ ¯à ¢«¥­¨¨ ®á¨ z . ‚®«­®¢®¤, ¯® ª®â®à®¬ã ¬®¦¥â à á¯à®áâà ­ïâìáï N à §«¨ç­ëå ¬®¤, íª¢¨¢ «¥­â¥­ N ¯ à ¬ § ¦¨¬®¢. 4.4.6. Œ âà¨æ  à áá¥ï­¨ï

Šà®¬¥ ®¯¨á ­¨ï ¯®«ï ¢ ¬­®£®¯®«îá­¨ª¥ á ¯®¬®éìî ­ ¯à殮­¨© ¨ ⮪®¢, ¢®§¬®¦­® â ª¦¥ ¢®«­®¢®¥ ®¯¨á ­¨¥, â.¥. ®¯¨á ­¨¥ á ¯®¬®éìî  ¬¯«¨â㤠¢å®¤ïé¨å ¨ ¢ë室ïé¨å ¢®«­.  ¯à殮­¨ï ¨ ⮪¨ ¬®£ãâ ¡ëâì ¢ëà ¦¥­ë ç¥à¥§ á㬬㠨 à §­®áâì  ¬¯«¨â㤠¢å®¤ï饩 ¨ ¢ë室ï饩 ¢®«­:

Un = pn (an + bn ); In = p1 (an ; bn ): n ‚¥«¨ç¨­  pn á¢ï§ ­  á ¢®«­®¢ë¬ ᮯà®â¨¢«¥­¨¥¬ ¢®«­®¢®¤ . „¥©á⢨⥫쭮, ª ª ¡ë«® ¯®ª § ­® à ­¥¥, p2n = Z0n ;

p

®âªã¤  pn = Z0n . ‚ ¤ «ì­¥©è¥¬ ¬ë ¡ã¤¥¬ ¯à¥¤¯®« £ âì, çâ® ¢á¥ ¢®«­®¢ë¥ ᮯà®â¨¢«¥­¨ï ¢ë¡à ­ë à ¢­ë¬¨ ¥¤¨­¨æ¥. ’®£¤ 

Un = an + bn ; In = an ; bn ; ®âªã¤  125

an = 12 (Un + In ); bn = 12 (Un ; In ): Œ¥¦¤ã ­ ¯à殮­¨ï¬¨ ¨ ⮪ ¬¨ ¨¬¥¥â ¬¥á⮠ᮮ⭮襭¨¥

Un =

X m

ZnmIm :

®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨ï ¤«ï an ¨ bn, ­ ©¤¥¬

an = 21 bn = 21

X m

X m

Znm Im + In Znm Im ; In

£¤¥

nm =

! !

X = 12 (Znm + nm ) Im ; m X = 12 (Znm ; nm ) Im ; m

 0 ¯à¨ n 6= m; 1 ¯à¨ n = m:

‚ ¬ âà¨ç­®© ä®à¬¥ í⨠ᮮ⭮襭¨ï § ¯¨á뢠îâáï á«¥¤ãî騬 ®¡à §®¬:

a = 12 (Z + 1) I; b = 12 (Z ; 1) I:  ©¤¥¬ ¨§ ¯¥à¢®£® ᮮ⭮襭¨ï ⮪ I :

I = 2(Z + 1);1 a: ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï b, ¯®«ã稬 b = (Z ; 1)(Z + 1);1 a: Œ âà¨æ  126

S = (Z ; 1)(Z + 1);1 ; á¢ï§ë¢ îé ï ¢ë室ï騥 ¢®«­ë á ¢å®¤ï騬¨, ­ §ë¢ ¥âáï ¬ âà¨æ¥© à áá¥ï­¨ï. Œ®¦­® ¯®ª § âì, çâ® ¬ âà¨æ  à áá¥ï­¨ï ¬®¦¥â ¡ëâì ¢ëà ¦¥­  â ª¦¥ ç¥à¥§ ¬ âà¨æ㠯஢®¤¨¬®áâ¨: S = (1 ; Y )(1 + Y );1 : „¨ £®­ «ì­ë¥ í«¥¬¥­âë ¬ âà¨æë à áá¥ï­¨ï { íâ® ª®íää¨æ¨¥­âë ®âà ¦¥­¨ï ¯à¨ ®âáãâá⢨¨ ¢å®¤ïé¨å ¢®«­ ¢® ¢á¥å ¢®«­®¢®¤ å, ªà®¬¥ ¤ ­­®£®, â.¥. ª®£¤  ¢á¥ ¢®«­®¢®¤ë, ªà®¬¥ ¯¨â ¥¬®£®, ­ £à㦥­ë ­  ᮣ« á®¢ ­­ë¥ ­ £à㧪¨. ¥¤¨ £®­ «ì­ë¥ í«¥¬¥­âë { íâ® ª®íää¨æ¨¥­âë ¯¥à¥¤ ç¨ ¯à¨ â¥å ¦¥ ãá«®¢¨ïå. 4.4.7. ‘¢®©á⢠ ¬ âà¨æë à áá¥ï­¨ï

‘¨¬¬¥âà¨ï “¬­®¦¨¬ ¬ âà¨æã à áá¥ï­¨ï á«¥¢  ­  (Z + 1): (Z + 1)
S = (Z + 1)(Z ; 1)(Z + 1);1 : ¥âà㤭® ¢¨¤¥âì, çâ® ¬ âà¨æë Z + 1 ¨ Z ; 1 ª®¬¬ãâ â¨¢­ë. â® ¬®¦­® ¯à®¢¥à¨âì 㬭®¦¥­¨¥¬. ¥à¥áâ ¢«ïï ᮬ­®¦¨â¥«¨ ¢ ¯à ¢®© ç á⨠ࠢ¥­á⢠, ¯®«ã稬 (Z + 1) S = (Z ; 1)(Z + 1)(Z + 1);1 ¨«¨, ®¡ê¥¤¨­ïï ¤¢  ¯®á«¥¤­¨å ¬­®¦¨â¥«ï, (Z + 1) S = Z ; 1: “¬­®¦ ï ⥯¥àì á«¥¢  ­  (Z + 1);1 , ­ å®¤¨¬ S = (Z + 1);1 (Z ; 1): ’ ª¨¬ ®¡à §®¬, ¬ âà¨æë Z ; 1 ¨ (Z + 1);1 ª®¬¬ãâ â¨¢­ë. Žâ¬¥â¨¬, çâ® í⨠¬ âà¨æë ᨬ¬¥âà¨ç­ë. ˆ§ ⥮ਨ ¬ âà¨æ ¨§¢¥áâ­®, çâ® ¬ âà¨æ , ïîé ïáï ¯à®¨§¢¥¤¥­¨¥¬ ¤¢ãå ᨬ¬¥âà¨ç­ëå ¬ âà¨æ, ª®¬¬ãâ¨àãîé¨å ¬¥¦¤ã ᮡ®©, ᨬ¬¥âà¨ç­ . ‘«¥¤®¢ â¥«ì­®, ¬ âà¨æ  à áá¥ï­¨ï â ª¦¥ ᨬ¬¥âà¨ç­ , çâ® § ¯¨á뢠¥âáï ¢ ¢¨¤¥ ᮮ⭮襭¨ï 127

S T = S; £¤¥ S T { â࠭ᯮ­¨à®¢ ­­ ï ¬ âà¨æ . ‘¨¬¬¥âà¨ï ¬ âà¨æë à áá¥ï­¨ï ï¥âáï á«¥¤á⢨¥¬ ᨬ¬¥âਨ ¬ âà¨æë ᮯà®â¨¢«¥­¨ï (¨«¨ ¬ âà¨æë ¯à®¢®¤¨¬®áâ¨). 4.4.8. ­¥à£¥â¨ç¥áª¨¥ ᮮ⭮襭¨ï

®¤áâ ¢«ïï ¢ ­ ©¤¥­­®¥ à ­¥¥ ᮮ⭮襭¨¥ ¢ëà ¦¥­¨ï ­ ¯à殮­¨© ¨ ⮪®¢ ç¥à¥§  ¬¯«¨âã¤ë ¢®«­, ¯®«ã稬 1 X(a + b )(a ; b ) = P + 2j! (W ; W ) n n h e n 2 n n ¨«¨, à áªà뢠ï ᪮¡ª¨, 1 X(a a ; b b + a b ; a b ) = P + 2j! (W ; W ): h e 2 n n n nn nn nn  §¤¥«ïï ¢¥é¥á⢥­­ãî ¨ ¬­¨¬ãî ç áâ¨, ¯®«ã稬

X

(an an ; bn bn ) = 2P;

n X n

(an bn ; an bn ) = 4j! (Wh ; We ):

„ ­­ë¥ ᮮ⭮襭¨ï ¬®£ãâ ¡ëâì § ¯¨á ­ë ¢ ¬ âà¨ç­®© ä®à¬¥:

a
a ; b
b = 2P; a
b ; a
b = 4j! (Wh ; We ): …᫨ ¢ á®ç«¥­¥­¨¨ ®âáãâáâ¢ãîâ ¯®â¥à¨, â.¥. P = 0, â® ¯¥à¢®¥ ᮮ⭮襭¨¥ ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¢¨¤¥ jaj2 = jbj2 ; çâ® ¢ëà ¦ ¥â § ª®­ á®åà ­¥­¨ï í­¥à£¨¨: ¬®é­®áâì ¢å®¤ïé¨å ¢®«­ à ¢­  ¬®é­®á⨠¢ë室ïé¨å. 128

‚ ¯¥à¢®¬ ᮮ⭮襭¨¨  ¬¯«¨âã¤ë ¢ë室ïé¨å ¢®«­ ¬®£ãâ ¡ëâì ¢ëà ¦¥­ë ç¥à¥§  ¬¯«¨âã¤ë ¢å®¤ïé¨å á ¯®¬®éìî ¬ âà¨æë à áá¥ï­¨ï:

a
a ; Sa
S  a = 2P: ¥à¥áâ ­®¢ª  ¢® ¢â®à®¬ á« £ ¥¬®¬ á § ¬¥­®© S  ­  S T , â.¥. ­  âà ­á-

¯®­¨à®¢ ­­ãî ¬ âà¨æã, ¤ ¥â

a (1 ; S T
S )a = 2P: ‡¤¥áì 1 { ¥¤¨­¨ç­ ï ¬ âà¨æ , í«¥¬¥­âë ª®â®à®© à ¢­ë ik (ᨬ¢®« Šà®­¥ª¥à ). …᫨ P = 0, â® 1 ; S T
S = 0; ¨«¨

S T
S = 1: Œ âà¨æã, 㤮¢«¥â¢®àïîéãî â ª®¬ã ᮮ⭮襭¨î, ­ §ë¢ îâ ã­¨â à­®©. ¥à¥ç¨á«¨¬ ®á­®¢­ë¥ ᢮©á⢠ ã­¨â à­®© ¬ âà¨æë. 1. Š ª ¯®ª § ­® ¢ëè¥, ¬ âà¨æ  á®åà ­ï¥â á㬬㠪¢ ¤à â®¢ ¢å®¤ïé¨å ¨ ¢ë室ïé¨å ¢®«­: X 2 X 2 jan j = jbn j : n

n

2. ˆ§ à ¢¥­á⢠ S T
S = 1 á«¥¤ã¥â (S T
S )ik =

X m

T
Smk = Sim

X m


Smk = ik : Smi

€­ «®£¨ç­® ¬®¦­® ¯®ª § âì, çâ®

X m


Skm = ik : Sim

3. Œ®¤ã«ì ®¯à¥¤¥«¨â¥«ï à ¢¥­ 1. 4. Ž¡à â­ ï ¬ âà¨æ  à ¢­  íନ⮢® ᮯà殮­­®©. 129

‡ ¬¥â¨¬, çâ® í⨠ᮮ⭮襭¨ï ¯®«ãç¥­ë ¡¥§ ¯à¥¤¯®«®¦¥­¨ï ® ᨬ¬¥âਨ ¬ âà¨æë à áá¥ï­¨ï. Ž­¨ á¯à ¢¥¤«¨¢ë, ¢ ç áâ­®áâ¨, ¤«ï ä¥àà¨â®¢ëå æ¨àªã«ïâ®à®¢. ‚â®à®¥ ᮮ⭮襭¨¥ ¤ ¥â

a
Sa ; a
S  a = 4j! (Wh ; We ) ¨«¨, ¯®á«¥ ¯¥à¥áâ ­®¢ª¨ ¢® ¢â®à®¬ á« £ ¥¬®¬ ¢ ¯à ¢®© ç áâ¨

a
(S ; S T )a = 4j! (Wh ; We ): ‚ ç áâ­®áâ¨, ¥á«¨ S { ᨬ¬¥âà¨ç­ ï ¬ âà¨æ , â® ¯®«ã稬 a
(S ; S )a = 4j! (Wh ; We ): Œ âà¨æ  S ; S  { ç¨áâ® ¬­¨¬ ï. 4.4.9. à¥®¡à §®¢ ­¨¥ ¬ âà¨æë à áá¥ï­¨ï ¯à¨ ¯¥à¥­®á¥ ®âáç¥â­ëå ¯«®áª®á⥩

an

, bn

 áᬮâਬ ¯¥à¥­®á ®âáç¥â­ëå ¯«®áª®á⥩ ¢® ¢á¥å ¢®«­®¢®¤ å ­  à ááâ®ï­¨ï ln ¢ áâ®à®­ã ®â á®ç«¥­¥­¨ï ª £¥­¥à â®àã (á¬. à¨á.). „® ¯¥à¥­®á  ¬ë ¨¬¥¥¬ ᮮ⭮襭¨¥ b = Sa. „«ï ­®¢ëå ¯«®áª®á⥩ ®âáç¥â   ­ «®£¨ç­®¥ ᮮ⭮襭¨¥ ¯à¨¬¥â ¢¨¤ b0 = S 0 a0 . „«ï an , bn , a0n , b0n ¨¬¥¥¬

an , b n

ln

®âªã¤ 

a0n = an e j n ln ; b0n = bn e;j n ln ;

a0n = an e j n ln ; bn = b0n e j n ln : ‚¢¥¤¥¬ ¤¨ £®­ «ì­ãî ¬ âà¨æã L: L =

0 e j l @


1 1

0

130

0 e j n ln

1 A

‘ ¯®¬®éìî í⮩ ¬ âà¨æë § ¯¨è¥¬ ¯à¨¢¥¤¥­­ë¥ ¢ëè¥ á®®â­®è¥­¨ï:

a0 = La; b = Lb0 ; ¨«¨

a = L;1 a0 ; b = Lb0: ®¤áâ ¢«ïï íâ® ¢ ᮮ⭮襭¨¥, á¢ï§ë¢ î饥  ¬¯«¨âã¤ë ¢å®¤ïé¨å ¨ ¢ë室ïé¨å ¢®«­, ¯®«ã稬 Lb0 = S
L;1 a0 ¨«¨ b0 = L;1 S
L;1 a0 : Žâá á«¥¤ã¥â, çâ®

S 0 = L;1 SL;1: „ ­­ ï ä®à¬ã«  ¤ ¥â ¯à¥®¡à §®¢ ­¨¥ ¬ âà¨æë à áá¥ï­¨ï ¯à¨ ¯¥à¥­®á¥ ®âáç¥â­ëå ¯«®áª®á⥩. 4.4.10. „¢®©­®© âன­¨ª

‚ ¨§¬¥à¨â¥«ì­®© â¥å­¨ª¥ ç áâ® ¯à¨¬¥­ï¥âáï ¤¢®©­®© âன­¨ª, á®áâ ¢«¥­­ë© ¨§ ¯àאַ㣮«ì­ëå ¢®«­®¢®¤®¢ á ®á­®¢­®© ¬®¤®© H10 (á¬. à¨á.). ’ ª®© âன­¨ª ®¡« ¤ ¥â ­¥ª®â®à묨 ¨­â¥à¥á­ë¬¨ ᢮©á⢠¬¨. 4

„«ï ¢ëïá­¥­¨ï íâ¨å ᢮©á⢠¢®á¯®«ì§ã¥¬áï ¬ âà¨æ¥© à áá¥ï­¨ï á ãç¥â®¬ ᢮©á⢠ᨬ¬¥âਨ ¤¢®©­®£® âன­¨ª . „¢®©­®© âன­¨ª ï¥âáï ¢®á쬨¯®«îá­¨ª®¬ ¨ ¨¬¥¥â, á«¥¤®¢ â¥«ì­®, ¬ âà¨æã à áá¥ï­¨ï ç¥â¢¥à⮣® ¯®à浪 . ’ ª ª ª ¯à¥¤¯®« £ ¥âáï, çâ® á⥭ª¨ ¢®«­®¢®¤®¢ ¨¤¥ «ì­® ¯à®¢®¤ï騥, â® ¬ âà¨æ  à áá¥ï­¨ï ¤¢®©­®£® âன­¨ª  ã­¨â à­ .

2

1

3

ˆ§ ᨬ¬¥âਨ ¤¢®©­®£® âன­¨ª  á«¥¤ã¥â

S11 = S22 ; S13 = S23 ; S14 = ; S24 ; S34 = S43 = 0: 131

Šà®¬¥ ⮣®, ¨§ ᨬ¬¥âਨ ¬ âà¨æë à áá¥ï­¨ï á«¥¤ã¥â, çâ®

S12 = S21 ; S13 = S31 ; S14 = ;S41 ; S23 = S32 ; S24 = S42 ; S34 = ;S43 : ’ ª¨¬ ®¡à §®¬, ¬ âà¨æ  à áá¥ï­¨ï ¨¬¥¥â ¢¨¤

S S S S

S11 S12 S13 ;14S

S12 S11 S13 0 14

S13 ;13S 0 33 S

14 14 44 ’ ª ª ª ¬ âà¨æ  à áá¥ï­¨ï ã­¨â à­ , â®

X i

Sik
Sil = kl ;

X i

Ski
Sli = kl :

“áâ ­®¢¨¬ ­¥ª®â®àë¥ á¢®©á⢠ ¤¢®©­®£® âன­¨ª . 1. …᫨ âன­¨ª ᮣ« á®¢ ­ á® áâ®à®­ë ¯«¥ç 3 ¨ 4, â® ¬¥¦¤ã ¡®ª®¢ë¬¨ ¯«¥ç ¬¨ ®âáãâáâ¢ã¥â ­¥¯®á।á⢥­­ ï á¢ï§ì. ‚ í⮬ á«ãç ¥ ¡®ª®¢ë¥ ¯«¥ç¨ â ª¦¥ ᮣ« á®¢ ­ë. —â®¡ë ¯®ª § âì íâ®, á®áâ ¢¨¬ áã¬¬ë ¯à®¨§¢¥¤¥­¨© í«¥¬¥­â®¢ ¯¥à¢®© ¨ âà¥â쥩 áâப,   â ª¦¥ ¯à®¨§¢¥¤¥­¨© í«¥¬¥­â®¢ ¢â®à®© ¨ ç¥â¢¥à⮩ áâப:  + S12 S  + S13 S  = 0; S11 S13 13 33  ; S11 S  ; S14 S  = 0: S12 S14 14 44

Žâá ¯®«ãç ¥¬ ¤¢  ãà ¢­¥­¨ï

®âªã¤ 

 S11 + S12 = ; SS13 13 S33 ;  S11 ; S12 = ; SS14 14 S44 ;





S14 S  ;  S11 = ; 12 SS13 13 S33 + S14  44    ; S14 S  ; S S12 = ; 12 SS13  33  44 S14 13 132

…᫨ S33 = S44 = 0, â® ¨§ í⮣® á«¥¤ã¥â, çâ® S11 = 0 ¨ S12 = 0. ‚롮஬ ¯«®áª®á⥩ ®âáç¥â  ¢ ¯«¥ç å 3 ¨ 4 ¢á¥£¤  ¬®¦­® ᤥ« âì S13 ¨ S14 ¢¥é¥á⢥­­ë¬¨ ¢¥«¨ç¨­ ¬¨. ’®£¤   + S  ); S11 = ; 21 (S33 44  ; S  ): S12 = ; 12 (S33 44

Žâá á«¥¤ã¥â, çâ® S11 ¨ S12 ®¤­®¢à¥¬¥­­® à ¢­ë ­ã«î ⮫쪮 ¯à¨ ãá«®¢¨¨, çâ® S33 = 0 ¨ S44 = 0. …᫨ S33 ¨ S44 ®â«¨ç­ë ®â ­ã«ï, â® ¢ ­ã«ì ¬®¦­® ®¡à â¨âì «¨èì ®¤­ã ¨§ 㪠§ ­­ëå ¢¥«¨ç¨­. 2. ‚®§ì¬¥¬ ⥯¥àì áã¬¬ë ª¢ ¤à â®¢ ¬®¤ã«¥© âà¥â쥩 ¨ ç¥â¢¥à⮩ áâப ¬ âà¨æë à áá¥ï­¨ï (¯à¨ ãá«®¢¨¨ S33 = S44 = 0)

jS13 j2 + jS13 j2 = 1; jS14 j2 + jS14 j2 = 1: Žâá ¯®«ãç ¥¬

jS13 j = jS14 j = p1 : 2

â®â १ã«ìâ â ¬®¦­® áä®à¬ã«¨à®¢ âì á«¥¤ãî騬 ®¡à §®¬: ¯à¨ ãá«®¢¨¨ S33 = S44 = 0 ¢ ¤¢®©­®¬ âன­¨ª¥ ¬®é­®áâì ¢®«­ë, ¯®¤ ­­®© ¢ ¡®ª®¢®¥ ¯«¥ç®, ¤¥«¨âáï ¯®à®¢­ã ¬¥¦¤ã ¯«¥ç ¬¨ 3 ¨ 4. „¢®©­®© âன­¨ª, ã ª®â®à®£® S33 = S44 = 0 ¨, á«¥¤®¢ â¥«ì­®, S11 = 0, S22 = 0 ¨ S12 = S21 = 0, ­ §ë¢ îâ ᮣ« á®¢ ­­ë¬. ‚ í⮬ á«ãç ¥ ¬ âà¨æ  à áá¥ï­¨ï ¨¬¥¥â ®á®¡¥­­® ¯à®á⮩ ¢¨¤, â ª ª ª ¨

S11 = S22 = S33 = S44 = S34 = S43 = 0 S13 = p1 ; S14 = p1 : 2

’®£¤  ¨¬¥¥¬

S =

2

0 0

0 0 1 p

1 1 2

1 ;1 133

1 1 0 0



;1

: 0

0 1

‘®£« á®¢ ­­ë© ¤¢®©­®© âன­¨ª ­ å®¤¨â ¯à¨¬¥­¥­¨¥ ¢ ¨§¬¥à¨â¥«ì­®© â¥å­¨ª¥, â ª ª ª ¯®§¢®«ï¥â ¯à®¨§¢®¤¨âì ¨§¬¥à¥­¨¥ ª®íää¨æ¨¥­â  ®âà ¦¥­¨ï ®ª®­¥ç­ëå ãáâனáâ¢. à¥¤¯®«®¦¨¬, çâ® ¢®«­   ¬¯«¨âã¤ë a3 ¯®¤ ¥âáï ¢ ¯«¥ç® 3. Š ¯«¥çã 2 ¯à¨á®¥¤¨­¨¬ ᮣ« á®¢ ­­ãî ­ £à㧪ã,   ª ¯«¥çã 1 { ¨§¬¥à塞ãî.  ©¤¥¬  ¬¯«¨âã¤ã ¢®«­ë, ¯®áâ㯠î饩 ¢ ¯«¥ç® 4: b4 = p1 (a1 ; a2 ): 2 ‚ â® ¦¥ ¢à¥¬ï a2 = 0, â ª ª ª ª ¯«¥çã 2 ¯à¨á®¥¤¨­¥­  ᮣ« á®¢ ­­ ï ­ £à㧪 . ‚¥«¨ç¨­  a1 ®¯à¥¤¥«ï¥âáï ª®íää¨æ¨¥­â®¬ ®âà ¦¥­¨ï ®â ­ £à㧪¨ ;:

a1 = ; b1 : ®

b1 = p1 (a3 ; a4 ):

2 à¥¤¯®«®¦¨¬, çâ® ¢ ¯«¥ç¥ 4 ãáâ ­®¢«¥­ ᮣ« á®¢ ­­ë© ¨­¤¨ª â®à. ’®£¤  a4 = 0 ¨ b1 = p1 a3 : 1 ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï a1 , ¯®«ã稬 a1 = ; p1 a3 ; ¨ b4 = 21 ; a3 : 2 ’ ª¨¬ ®¡à §®¬,  ¬¯«¨â㤠 ¢®«­ë ­  ¢ë室¥ 4-£® ¯«¥ç  ¯à®¯®à樮­ «ì­  ª®íää¨æ¨¥­âã ®âà ¦¥­¨ï ¢ ¯«¥ç¥ 1. Œ®¦­® ¯®ª § âì, çâ® ¥á«¨ ª ¯«¥ç ¬ 1 ¨ 2 ¯à¨á®¥¤¨­¥­ë ®¤¨­ ª®¢ë¥ (å®âï ¡ë ¨ ­¥ ᮣ« á®¢ ­­ë¥) ­ £à㧪¨, â® ¢®«­  ­  ¢ë室¥ ¯«¥ç  4 à ¢­  ­ã«î. â® ᢮©á⢮ ¨á¯®«ì§ã¥âáï ¤«ï ¨§¬¥à¥­¨ï ¯®«­ëå ᮯà®â¨¢«¥­¨©. 4.5.

— áâ®â­ë¥ ᢮©á⢠

4.5.1. — áâ®â­ ï § ¢¨á¨¬®áâì ¬ âà¨æë ᮯà®â¨¢«¥­¨ï (¯à®¢®¤¨¬®áâ¨) á®ç«¥­¥­¨ï ¡¥§ ¯®â¥àì

„«ï á®ç«¥­¥­¨ï ¡¥§ ¯®â¥àì ãà ¢­¥­¨ï Œ ªá¢¥««  ¨¬¥îâ ¢¨¤ 134

~ rot H~ = j!"E: ~ rot E~ = ; j!H; à¥¤¯®« £ ¥âáï, çâ®  ¨ " ­¥ § ¢¨áïâ ®â ç áâ®âë. à®¤¨ää¥à¥­æ¨à㥬 í⨠ãà ¢­¥­¨ï ¯® ç áâ®â¥: ~

~

E = ;jH~ ; j! @ H ; rot @@! @! ~ ~ @ H @ E rot @! = j"E~ + j!" @! :

“¬­®¦¨¬ ¯¥à¢®¥ à ¢¥­á⢮ ᪠«ïà­® ­  H~  ,   ¢â®à®¥ { ­  E~  ¨ ¢ëç⥬ ¯¥à¢®¥ ¨§ ¢â®à®£®. ’®£¤  (¯à¥¤¯®« £ ï  ¨ " ᪠«ïà ¬¨ ¨«¨ ᨬ¬¥âà¨ç­ë¬¨ ⥭§®à ¬¨) ¯®«ã稬

~ H~ = ; j (jH~ j2 + "jE~ j2 ) ; H~ 
rot @@!E ; E~ 
rot @@! ~

~

; rot E~  @@!H + rot H~  @@!E :

„ «¥¥, ¯¥à¥­®áï ¯®á«¥¤­¨¥ ¤¢  á« £ ¥¬ëå ¨§ ¯à ¢®© ç á⨠¢ «¥¢ãî, ¯®«ã稬

E~  H~  div @@!

!

H~ + div E~   @@!

!

= ; j (jH~ j2 + "jE~ j2 ):

à®¨­â¥£à¨à㥬 ⥯¥àì íâ® à ¢¥­á⢮ ¯® ®¡ê¥¬ã ¢­ãâਠ¯®¢¥àå­®áâ¨, ®å¢ â뢠î饩 ¢á¥ á®ç«¥­¥­¨¥ ¨ ¯¥à¥á¥ª î饩 ¢®«­®¢®¤ë ¯® ¯«®áª®áâï¬, ¯¥à¯¥­¤¨ªã«ïà­ë¬ ¨å ®áï¬. à¨ í⮬ ®¡ê¥¬­ë© ¨­â¥£à « ®â ¤¨¢¥à£¥­æ¨¨ ¯¥à¥å®¤¨â ¢ ¯®¢¥àå­®áâ­ë©. Œ¥­ïï ¢­¥è­îî ­®à¬ «ì ­  ¢­ãâ७­îî, ¯®«ã稬

! I ( @ E~  ~ @!  H S

+

H~ E~   @@!

!)


~n dS =

Z

V

j (jH~ j2 + "jE~ j2 ) dV:

‡¤¥áì ~n { ¢­ãâ७­ïï ­®à¬ «ì. …᫨ ⥯¥àì ¢ëà §¨âì ¯®«ï ç¥à¥§ ­ ¯à殮­¨ï ¨ ⮪¨, â® ­ ©¤¥¬, çâ®

@U I  + U  @I = 4j (W + W ): H E @! @! 135

‡¤¥áì U ¨ I { ᮢ®ªã¯­®á⨠­ ¯à殮­¨© ¨ ⮪®¢ ¢® ¢á¥å ¢®«­®¢®¤ å, R 2 R 2 WH = 12 jH2 j dV ¨ WE = 21 "jE2 j dV { á।­¨¥ § ¯ áë ¬ £­¨â­®© ¨ V V í«¥ªâà¨ç¥áª®© í­¥à£¨¨ ¢ ®¡ê¥¬¥ á®ç«¥­¥­¨ï.  ¯à殮­¨ï ¢ëà ¦ îâáï ç¥à¥§ ⮪¨ á ¯®¬®éìî ¬ âà¨æë ᮯà®â¨¢«¥­¨ï

U = Z I; ®âªã¤ 

@U = @Z I + Z @I : @! @! @!

®¤áâ ¢«ïï íâ® ¢ § ¯¨á ­­ë¥ ¢ëè¥ ãà ¢­¥­¨ï, ¯®«ã稬

@I    @I I  @Z @! I + I Z @! + @! Z I = 4j (WH + WE ): ¨«¨, ãç¨â뢠ï çâ® Z  = ;Z ¤«ï á®ç«¥­¥­¨ï ¡¥§ ¯®â¥àì, @I   @I I  @Z @! I + I Z @! ; @! ZI = 4j (WH + WE ):

‘ ãç¥â®¬ ᨬ¬¥âਨ ¬ âà¨æë ᮯà®â¨¢«¥­¨© ¤¢  ¯®á«¥¤­¨¥ á« £ ¥¬ë¥ ¢ «¥¢®© ç á⨠ࠢ¥­á⢠ ¢§ ¨¬­® ã­¨ç⮦ îâáï, ¢ १ã«ìâ â¥ 祣® ¯®«ãç ¥¬

I  @Z @! I = 4j (WH + WE ):

€­ «®£¨ç­®¥ ᮮ⭮襭¨¥ ¬®¦­® ¯®«ãç¨âì ¤«ï ¬ âà¨æë ¯à®¢®¤¨¬®á⥩

U  @Y @! U = 4j (WH + WE ):

@Y @Zmn ‡ ¬¥â¨¬, çâ® ¬ âà¨æë @Z @! ¨ @! á®áâ ¢«¥­ë ¨§ í«¥¬¥­â®¢ @! ¨ @Ymn . @!

136

4.5.2. — áâ®â­ ï § ¢¨á¨¬®áâì ¬ âà¨æë à áá¥ï­¨ï

ã¤¥¬ ¨á室¨âì ¨§ ¯®«ã祭­®£® ¢ëè¥ á®®â­®è¥­¨ï:

@U I  + U  @I = 4j (W + W ): H E @! @! ’ ª ª ª U = a + b, I = a ; b, â®, ¯®¤áâ ¢«ïï íâ®, ¯®«ãç ¥¬  @a @b     ; b ) + (a + b ) @a ; @b = 4j (WH + WE ): + ( a @! @! @! @!  áªà뢠ï ᪮¡ª¨ ¨ ᮪à é ï, ­ ©¤¥¬

@a ; b @b = 2j (W + W ): a @! H E @! @b = @S a + S @a . ®¤áâ ¢«ïï íâ®, ¯®«ã稬 „ «¥¥, b = S  a , @! @! @!   @a ; S  a @S a + S @a = a @a ; @S a S a ; S  a S @a : a @! @! @! @! @! @! ’ ª ª ª S ¨ @S @! { ᨬ¬¥âà¨ç­ë¥ ¬ âà¨æë, â® ¢ ¯®á«¥¤­¨å ¤¢ãå á« -

£ ¥¬ëå ¢ ¯à ¢®© ç á⨠¬®¦­® ᮢ¥àè¨âì ¯¥à¥áâ ­®¢ªã:

@a ; a S  @S a ; @a SS a = ; a S  @S a; a @! @! @! @! â ª ª ª S
S  = 1 ¤«ï á®ç«¥­¥­¨ï ¡¥§ ¯®â¥àì.

ˆâ ª, ¤«ï á®ç«¥­¥­¨ï ¡¥§ ¯®â¥àì ¨¬¥¥¬ ᮮ⭮襭¨¥, ®¯à¥¤¥«ïî饥 ç áâ®â­ãî § ¢¨á¨¬®áâì ¬ âà¨æë à áá¥ï­¨ï:

;a S  @S @! a = 2j (WH + WE ): 4.5.3. Žª®­¥ç­®¥ ãáâனá⢮

„«ï ®ª®­¥ç­®£® ãáâனá⢠ (¤¢ã寮«îá­¨ª ) ¨§ ¯à¥¤ë¤ã饣® ­ å®¤¨¬

@Z = 2j W ; @Y = 2j W : 1 II  @! 1 UU  @! 2 2 137

‘®®â­®è¥­¨ï ¤«ï ¬ âà¨æë à áá¥ï­¨ï ¯¥à¥å®¤ïâ ¢ á«¥¤ãî騥:

S = ; = e j' ; ¥á«¨ ®ª®­¥ç­®¥ ãáâனá⢮ ¡¥§ ¯®â¥àì;   @;  ;j' j' d' ;a S  @S @! a = ; a ; @! a = ; a e j e d! a = 2j (WH + WE ): Žâá ¨¬¥¥¬

d' = ; W ; d! P £¤¥ P = 12 aa { ¬®é­®áâì ¯ ¤ î饩 ¢®«­ë. ®á«¥¤­¥¥ ᮮ⭮襭¨¥ ¨¬¥¥â ¯à®á⮩ 䨧¨ç¥áª¨© á¬ëá«. à¥¤¯®«®¦¨¬, çâ® ­  ¢å®¤ ®ª®­¥ç­®£® ãáâனá⢠ ¯®¤ ­  á㬬  ¤¢ãå ¢®«­ ¡«¨§ª¨å ç áâ®â e j!1 t ¨ e j!2 t . ’®£¤  ®âà ¦¥­­ ï ¢®«­  â ª¦¥ ¡ã¤¥â á®áâ®ïâì ¨§ ¤¢ãå ¢®«­ íâ¨å ç áâ®â, ­® á à §«¨ç­ë¬¨ ᤢ¨£ ¬¨ ä §ë: b = e j(!1 t+'1 ) + e j(!2 t+'2 ) = e j(!1 t+'1 f1 + + e j[(!2 ;!1 )t+('2 ;'1 )]

o

(


' = e j(!1 t+'1 ) 1 + e j
!(t+
! )

)

:

’ ª¨¬ ®¡à §®¬, ¬ ªá¨¬ã¬ ®âà ¦¥­­®© ¢®«­ë ᤢ¨­ãâ ¯® ¢à¥¬¥­¨ ®â­®á¨â¥«ì­® ¯ ¤ î饩 ¢®«­ë ­  ¢à¥¬ï

d' :  = d!

â  ¢¥«¨ç¨­ , ª ª ¢¨¤­® ¨§ ¯à¥¤ë¤ã饣®, ®âà¨æ â¥«ì­ , â.¥. ¯à¥¤áâ ¢«ï¥â ᮡ®© § ¤¥à¦ªã. âã § ¤¥à¦ªã ­ §ë¢ îâ £à㯯®¢®©. ‘«¥¤®¢ â¥«ì­®, ¢à¥¬ï £à㯯®¢®© § ¤¥à¦ª¨ à ¢­®

W £à = d' d! = ; P :

’ ª¨¬ ®¡à §®¬, ¨¬¯ã«ìá, ¯®¤ ­­ë© ­  § ¦¨¬ë ®ª®­¥ç­®£® ãáâனá⢠, ®âà ¦ ¥âáï ç¥à¥§ ¯à®¬¥¦ã⮪ ¢à¥¬¥­¨, à ¢­ë© § ¯ á¥­­®© í­¥à£¨¨ ­  ¥¤¨­¨æã ¬®é­®á⨠( ­ «®£¨ï á á®á㤮¬, § ¯®«­ï¥¬ë¬ ¦¨¤ª®áâìî ç¥à¥§ âàã¡ã).

138

4.6.

¥®¤­®à®¤­®á⨠¢ ¢®«­®¢®¤ å

4.6.1. à¥¤áâ ¢«¥­¨¥ ­¥®¤­®à®¤­®á⨠¢ ¢¨¤¥ ç¥âëà¥å¯®«îá­¨ª 

l

1

l

…᫨ ¢ ॣã«ïà­®¬ ¢®«­®¢®¤¥ ¨¬¥¥âáï ­¥®¤­®à®¤­®áâì, â® ¢®§­¨ª ¥â ®âà ¦¥­­ ï ¢®«­ . ‚®«­®¢®¤, ¢ ª®â®à®¬ ¨¬¥¥âáï ­¥®¤­®à®¤­®áâì (á¬. à¨á.), ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­ ¢ ¢¨¤¥ ç¥âëà¥å¯®«îá­¨ª , å à ªâ¥à¨§ã¥¬®£® ¬ âà¨æ¥© à áá¥ï­¨ï

2

S =

S S

S11 S12

; S12 21 22

= S21 :

à¨ à áᬮâ७¨¨ ­¥®¤­®à®¤­®á⥩ ¡ã¤¥¬ ¯à¥­¥¡à¥£ âì ¯®â¥àﬨ. ’ ª ª ª ¢ í⮬ á«ãç ¥ ¬ âà¨æ  à áá¥ï­¨ï ã­¨â à­ , â® jS11 j2 + jS12 j2 = 1; jS12 j2 + jS22 j2 = 1:

Žâá á«¥¤ã¥â, çâ® jS11 j = jS22 j. ‚롥६ á¥ç¥­¨ï ®âáç¥â  ¢ ¢®«­®¢®¤¥ â ª, ç⮡ë S11 = S22 . ’à¥âì¥ á®â­®è¥­¨¥, á«¥¤ãî饥 ¨§ ã­¨â à­®áâ¨, ¡ã¤¥â  + S12 S  = 0: S11 S12 22 …᫨, ª ª ¢ë¡à ­®, S11 = S22 , â®  + S12 S  = 0; S11 S12 11

 ) = 0. ®âªã¤  Re (S11 S12 …᫨, ­ ¯à¨¬¥à, á¥ç¥­¨ï ®âáç¥â  ¢ë¡à ­ë â ª, çâ® S11 = S22 ¢¥é¥á⢥­­ë, â® Re S12 = 0, â.¥. S12 = S21 { ç¨áâ® ¬­¨¬ë¥. ‚ १ã«ìâ â¥ ¯®«ãç ¥¬, çâ®

q

p

2: S12 = j 1 ; jS11 j2 = j 1 ; S11

’ ª¨¬ ®¡à §®¬, çâ®¡ë ¯®«­®áâìî ®å à ªâ¥à¨§®¢ âì ­¥®¤­®à®¤­®áâì ¢ ¢®«­®¢®¤¥, ¤®áâ â®ç­® § ¤ âì jS11 j ¨ à ááâ®ï­¨ï l1 ¨ l2 , ®¯à¥¤¥«ïî騥 139

á¥ç¥­¨ï, ¢ ª®â®àëå S11 = S22 ¢¥é¥á⢥­­ë, â.¥. ¢á¥£® âॡã¥âáï âਠ¯ à ¬¥âà . …᫨ à ááâ®ï­¨ï ­¥ ¯à¥¤áâ ¢«ïîâ ¨­â¥à¥á , â® ¤®áâ â®ç­® ®¤­®£® ¯ à ¬¥âà  S11 . ‚¬¥áâ® í⮣® ¯ à ¬¥âà  ¬®¦¥â ¡ëâì ¢¢¥¤¥­  èã­â¨àãîé ï ¯à®¢®¤¨¬®áâì B , á¢ï§ ­­ ï á S11 ᮮ⭮襭¨¥¬ jB ) = ;jB S11 = 11 ;+ (11 + + jB 2 + jB ¨«¨ jB = ; 1 2+S11S : 11 ‡¤¥áì S11 { ª®íää¨æ¨¥­â ®âà ¦¥­¨ï ¢ ¯«®áª®á⨠¢ª«î祭¨ï ¯à®¢®¤¨¬®á⨠B . …᫨ ç¥âëà¥å¯®«îá­¨ª ᨬ¬¥âà¨ç¥­, â® ç¨á«® ¯ à ¬¥â஢ 㬥­ìè ¥âáï ¤® ¤¢ãå.  ª®­¥æ, ¯à¨ ®¯à¥¤¥«¥­­®¬ à á¯®«®¦¥­¨¨ ¯«®áª®á⥩ ®âáç¥â  ­ã¦¥­ «¨èì ®¤¨­ ¯ à ¬¥âà. ¥¤­®à®¤­®áâ¨, ¢áâà¥ç î騥áï ¢ ¢®«­®¢®¤ å, ¬®¦­® à §¤¥«¨âì ­  ¤¢  ª« áá : ᪠窮®¡à §­ë¥ ­¥®¤­®à®¤®á⨠¨ ¯à®â殮­­ë¥ ­¥®¤­®à®¤­®áâ¨. ¥à¢ë¥ { íâ® â ª¨¥, ã ª®â®àëå ¤«¨­  ­¥à¥£ã«ïà­®£® ãç á⪠ ¡«¨§ª  ª ­ã«î, â.¥. £à ­¨æ¥© ï¥âáï ¯«®áª®áâì, ¯¥à¯¥­¤¨ªã«ïà­ ï ­ ¯à ¢«¥­¨î ®á¨ ¢®«­®¢®¤ . “ ¯à®â殮­­ëå ­¥®¤­®à®¤­®á⥩ ­¥à¥£ã«ïà­ë© ãç á⮪ ¨¬¥¥â ­¥ª®â®àãî ¤«¨­ã ¢ ­ ¯à ¢«¥­¨¨ ®á¨ ¢®«­®¢®¤ . „«ï ¯¥à¢ëå £à ­¨ç­ë¥ ãá«®¢¨ï á«¥¤ã¥â ¢ë¯®«­¨âì «¨èì ­  ®¤­®© £à ­¨ç­®© ¯«®áª®áâ¨. ‚® ¢â®à®¬ á«ãç ¥ ­¥®¡å®¤¨¬® ­ ©â¨ ®¡é¥¥ à¥è¥­¨¥ ¤«ï ¯®«ï ­  ­¥à¥£ã«ïà­®¬ ãç á⪥ ¨ 㤮¢«¥â¢®à¨âì £à ­¨ç­ë¬ ãá«®¢¨ï¬ ­  ¤¢ãå £à ­¨ç­ëå ¯«®áª®áâïå. 4.6.2. ‘ª çª®®¡à §­®¥ ¨§¬¥­¥­¨¥ ¯ à ¬¥â஢ ¢¥é¥á⢠, § ¯®«­ïî饣® ¢®«­®¢®¤

à¥¤¯®«®¦¨¬, çâ® § ¤ ­   ¬¯«¨00000000000 II â㤠 ¯ ¤ î饩 ¢®«­ë; ­¥®¡å®¤¨11111111111 I 00000000000 11111111111 ¬® ­ ©â¨ ¯à®è¥¤èãî ¨ ®âà ¦¥­00000000000 11111111111 00000000000 11111111111 ­ãî ¢®«­ë ¢ ¨§®¡à ¦¥­­®¬ ­  à¨11111111111 00000000000 áã­ª¥ ¢®«­®¢®¤¥. à¨ í⮬ ¯à¥¤¯®« £ ¥âáï, çâ® ¯«®áª ï £à ­¨æ  ¯¥à¯¥­¤¨ªã«ïà­  ª ®áï¬ ®¡®¨å ¢®«­®¢®¤®¢,   ¢®«­®¢®¤ II ¯à¥¤áâ ¢«ï¥â ᮡ®© ¯à®¤®«¦¥­¨¥ ¢®«­®¢®¤  I . à¨ íâ¨å ãá«®¢¨ïå ®ª §ë¢ ¥âáï ¢®§¬®¦­ë¬ 㤮¢«¥â¢®à¨âì e1m 1

e2 m 2

140

£à ­¨ç­ë¬ ãá«®¢¨ï¬ ¯à¨ ­ «¨ç¨¨ ⮫쪮 ¢®«­ ⨯  ¯ ¤ î饩 ¡¥§ ¢ëáè¨å ¬®¤. ƒà ­¨ç­ë¥ ãá«®¢¨ï ᢮¤ïâáï ª á®åà ­¥­¨î ¯®¯¥à¥ç­ëå á®áâ ¢«ïîé¨å E~ ¨ H~ . ‚ ᨫ㠮¤¨­ ª®¢®£® á¥ç¥­¨ï ¢®«­®¢®¤®¢ ¨å ᮡá⢥­­ë¥ ä㭪樨 ᮢ¯ ¤ îâ (á â®ç­®áâìî ¤® ¯®áâ®ï­­ëå ¬­®¦¨â¥«¥©). ‚ ¢®«­®¢®¤¥ I ¯®¯¥à¥ç­®¥ í«¥ªâà¨ç¥áª®¥ ¯®«¥ (¯à¥¤¯®« £ ï  ¬¯«¨âã¤ã ¯ ¤ î饩 ¢®«­ë à ¢­®© ¥¤¨­¨æ¥) à ¢­® E~ I¯®¯ = E~ m¯®¯(e;j 1 z + ;e j 1 z );   ¯®¯¥à¥ç­®¥ ¬ £­¨â­®¥ ¯®«¥ à ¢­®

~ ¯®¯ H~ I¯®¯ = (e;j 1 z ; ;e j 1 z )H~ m¯®¯ = (e;j 1 z ; ;e j 1 z ) ~z0 ZEm :

‚ ¢®«­®¢®¤¥ II ¯à¨áãâáâ¢ã¥â «¨èì ¯ ¤ îé ï ¢®«­ :

I

E~ II¯®¯ = T
E~ m¯®¯
e;j 2 z ; ~ ¯®¯ H~ II¯®¯ = T
H~ m¯®¯
e;j 2 z = T
~z0 Z Em e;j 2 z : II

à¨à ¢­¨¢ ï ¯®«ï á«¥¢  ¨ á¯à ¢  ¯à¨ z = 0, (¢® ¢â®à®¬ ãà ¢­¥­¨¨ 㬭®¦ ï ­  ~z0 ¢¥ªâ®à­® «¥¢ãî ¨ ¯à ¢ãî ç áâ¨) ¯®«ã稬

E~ m¯®¯(1 + ;) = T E~ m¯®¯; E~ m¯®¯ (1 ; ;) = T E~ m¯®¯ ; ZI ZII ¨«¨ 1 + ; = T; 1 ;; = T : ZI ZII „¥«ï ®¤­® ãà ¢­¥­¨¥ ­  ¤à㣮¥ ¨ à¥è ï ®â­®á¨â¥«ì­® ;, ¯®«ã稬

ZI : ; = ZZII ; + ZI II Š®íää¨æ¨¥­â ¯à®å®¦¤¥­¨ï T à ¢¥­ 141

T = 1 + ; = Z 2Z+II Z : II I ‚®«­®¢ë¥ ᮯà®â¨¢«¥­¨ï ZI ¨ ZII , ¨á¯®«ì§ã¥¬ë¥ §¤¥áì, ¯à¥¤áâ ¢«ï-

îâ ᮡ®© ®â­®è¥­¨ï ¯®¯¥à¥ç­ëå ª®¬¯®­¥­â í«¥ªâà¨ç¥áª®£® ¨ ¬ £­¨â­®£® ¯®«¥©. „«ï ¬ £­¨â­ëå ¬®¤ í⮠ᮯà®â¨¢«¥­¨¥ à ¢­®

ZH =

! =

pk2 ; g2 = p!2"!; g2 = m m !

=

p="

q

1 ; 2 =2ªà €­ «®£¨ç­® ¤«ï í«¥ªâà¨ç¥áª¨å ¬®¤

= Z E = !"

p="

p1 ; g2 =!2" m

=

:

p q

=" 1 ; 2 =2ªà :

‹¥£ª® à á¯à®áâà ­¨âì íâ® à¥è¥­¨¥ ­  á«ãç © ¤¨í«¥ªâਪ  á ¯®â¥àﬨ. ’®£¤  " = "0 ; j ! { ª®¬¯«¥ªá­ ï ¢¥«¨ç¨­ . 4.6.3. „¨ äà £¬ë ¢ ¢®«­®¢®¤¥

„¨ äà £¬®© ­ §ë¢ îâ â®­ªãî ¬¥â ««¨ç¥áªãî ¯« á⨭ªã á ®â¢¥àá⨥¬, ãáâ ­ ¢«¨¢ ¥¬ãî ¢ ¢®«­®¢®¤¥ ¯¥à¯¥­¤¨ªã«ïà­® ¥£® ®á¨. ƒà ­¨ç­ë¥ ãá«®¢¨ï ­  ¤¨ äà £¬¥ ᢮¤ïâáï ª à ¢¥­áâ¢ã ­ã«î â ­£¥­I 樠«ì­®© á®áâ ¢«ïî饩 í«¥ªâà¨ç¥áª®£® ¯®«ï ­  ¯®¢¥àå­®á⨠¤¨ äà £¬ë S1 ¨ ᮢ¯ ¤¥­¨î ¢ ®¡« S2 á⨠®â¢¥àáâ¨ï S2 ¯®¯¥à¥ç­ëå ª®¬S1 ¯®­¥­â ¯®«ï á«¥¢  ¨ á¯à ¢  (á¬. à¨á.). „«ï ¢ë¯®«­¥­¨ï £à ­¨ç­ëå ãá«®¢¨© ­¥®¡å®¤¨¬® ¯à¨¢«¥ç¥­¨¥ ¢ëáè¨å ¬®¤ ¢ ¤ ­­®¬ ¢®«­®¢®¤¥. „«ï ®¯à¥¤¥«¥­­®á⨠¡ã¤¥¬ ¯à¥¤¯®« £ âì, çâ® ¢ ¢®«­®¢®¤¥ ¬®¦¥â à á¯à®áâà ­ïâìáï «¨èì ®á­®¢­ ï ¬®¤ ; ¢á¥ ¢ëá訥 ¬®¤ë ¯à¥¤¯®« £ îâáï II

z

142

­¥à á¯à®áâà ­ïî騬¨áï. ’®£¤  ­¥®¤­®à®¤­®áâì ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­  ç¥âëà¥å¯®«îá­¨ª®¬. Ž¡à â¨¬ ¢­¨¬ ­¨¥ ­  â®, çâ® ¤®áâ â®ç­® ­ «®¦¨âì £à ­¨ç­ë¥ ãá«®¢¨ï «¨èì ­  ¯®¯¥à¥ç­ë¥ ª®¬¯®­¥­âë ¯®«¥©, â ª ª ª ᮢ¯ ¤¥­¨¥ ¯®¯¥à¥ç­ëå ª®¬¯®­¥­â ®¡¥á¯¥ç¨¢ ¥â ¢ë¯®«­¥­¨¥ £à ­¨ç­ëå ãá«®¢¨© â ª¦¥ ¨ ¤«ï ¯à®¤®«ì­ëå ª®¬¯®­¥­â. ®¯¥à¥ç­®¥ í«¥ªâà¨ç¥áª®¥ ¯®«¥ ¢ ®¡« á⨠I ¬®¦­® § ¯¨á âì ¢ ¢¨¤¥

E~ I = (e;j z + ;e j z ) E~ 1 +   ¢ ®¡« á⨠II

E~ II = T e;j z E~ 1 +

1 X An E~ n e nz ;

n=2

1 X Bn E~ n e; nz ;

n=2

£¤¥ E~ 1 ; E~ 2 ; : : : ; E~ n { ᮡá⢥­­ë¥ ¯®¯¥à¥ç­ë¥ ¢¥ªâ®à­ë¥ ä㭪樨 ¢®«­®¢®¤ ; á㬬¨à®¢ ­¨¥ ¢¥¤¥âáï ¯® ¢á¥¬ ¢ëá訬 ¬®¤ ¬ ¢ ¢®«­®¢®¤¥, ª ª ¬ £­¨â­ë¬, â ª ¨ í«¥ªâà¨ç¥áª¨¬. ®¯¥à¥ç­®¥ ¬ £­¨â­®¥ ¯®«¥ ¬®¦¥â ¡ëâì ­ ©¤¥­® á ¯®¬®éìî ᮮ⢥âáâ¢ãîé¨å ¢®«­®¢ëå ᮯà®â¨¢«¥­¨©:

~ H~ I = (e;j z ; ;e j z ) ~z0 Z E1 + 1

1 X

~ ;An ~z0 Z En e nz ;

n n=2 ~ ~ H~ II = T e;j z ~z0 Z E1 + Bn ~z0 Z En e; n z : 1 n n=2

1 X

®« £ ï z = 0 ¨ ¯à¨à ¢­¨¢ ï í«¥ªâà¨ç¥áª¨¥ ¯®«ï á«¥¢  ¨ á¯à ¢ , ¯®«ã稬 (1 + ;) E~ 1 +

1 X An E~ n

n=2

= T E~ 1 +

1 X Bn E~

n=2

=

 E~ (x; y) 0

­  ®â¢¥àá⨨; ­  ¤¨ äà £¬¥:

‡¤¥áì E~ (x; y) { ­¥¨§¢¥áâ­ ï ¢¥ªâ®à­ ï äã­ªæ¨ï, ¨¬¥îé ï ⮫쪮 ¯®¯¥à¥ç­ë¥ ª®¬¯®­¥­âë. â  äã­ªæ¨ï ¬®¦¥â ¡ëâì à §«®¦¥­  ¯® ¯®¯¥à¥ç­ë¬ ᮡá⢥­­ë¬ äã­ªæ¨ï¬ ¢®«­®¢®¤ :

E~ =

Z 1 X E~ n E~
E~ n dS;

n=1

S

143

¯à¨ç¥¬ ä㭪樨 En ­®à¬¨à®¢ ­ë â ª, çâ®

Z S

E~ n
E~ n dS = 1:

‡¤¥áì ¨ ¢ëè¥ ¨­â¥£à « ¡¥à¥âáï ¯® á¥ç¥­¨î ¢®«­®¢®¤ . ‘à ¢­¨¢ ï ª®íää¨æ¨¥­âë á«¥¢  ¨ á¯à ¢ , ¯®«ãç ¥¬

Z

1 +; = T =

An = Bn =

S2

E~
E~ 1 dS;

Z

S2

E~
E~ n dS:

‡¤¥áì ¨­â¥£à «ë ¢§ïâë ¯® ®â¢¥àáâ¨î, â ª ª ª ¢­¥ ®â¢¥àáâ¨ï äã­ªæ¨ï E~ (x; y)  0. „ «¥¥ á«¥¤ã¥â ¯à¨à ¢­ïâì ¬ £­¨â­ë¥ ¯®«ï á«¥¢  ¨ á¯à ¢  á ãç¥â®¬ ⮣®, çâ® 1+; = T ¨ An = Bn . ‡ ¬¥â¨¬, çâ® ¯à¨à ¢­¨¢ ­¨¥ á¯à ¢¥¤«¨¢® ⮫쪮 ­  ®â¢¥àá⨨ (S2 ):

~ ~ X ~ ~ X (1 ; ;) ~z0 Z E1 + ;An ~z0 Z En = (1 + ;) ~z0 Z E1 + An ~z0 Z En : 1 n 1 n n=2 n=2 …᫨ ⥯¥àì 㬭®¦¨âì «¥¢ãî ¨ ¯à ¢ãî ç á⨠¢¥ªâ®à­® ­  ~z0 ¨ ᮡà âì ¢á¥ á« £ ¥¬ë¥ á«¥¢ , â® ¯®«ã稬 ãà ¢­¥­¨¥ 1

1

1 X

~

~

En = 0 : 1 + ;E A n Z1 n=2 Zn ®¤áâ ¢«ïï á ¨­â¥£à «ì­ë¥ ¢ëà ¦¥­¨ï ¤«ï ; ¨ An , ¯®«ã稬 ¨­â¥£à «ì­®¥ ãà ¢­¥­¨¥ ¤«ï ­¥¨§¢¥áâ­®© ä㭪樨 E~ (x; y) 1 E~ Z X n ~ ~ Zn E
En dS

n=1

S2

~1 = E Z1 :

…é¥ à § ­ ¯®¬­¨¬, çâ® ¤ ­­®¥ ãà ¢­¥­¨¥ á¯à ¢¥¤«¨¢® ­  ®â¢¥àá⨨

S2 . â® { ãà ¢­¥­¨¥ á ¯à ¢®© ç áâìî. ® ¬®¦­® ¯®«ãç¨âì â ª¦¥ ®¤-

­®à®¤­®¥ ¨­â¥£à «ì­®¥ ãà ¢­¥­¨¥, ¢ëà §¨¢ ª®íää¨æ¨¥­â ®âà ¦¥­¨ï ; 144

ç¥à¥§ íª¢¨¢ «¥­â­ãî ¯à®¢®¤¨¬®áâì jB (­®à¬¨à®¢ ­­ãî ®â­®á¨â¥«ì­® ¢®«­®¢®£® ᮯà®â¨¢«¥­¨ï ¢®«­®¢®¤ ),   ¨¬¥­­®, jB ) ;jB ; = 11 ;+ (11 + + jB = 2 + jB ; ®âªã¤  1 ;jB = 1 2; + ; ¨«¨ ; = ; 2 jB (1 + ;): “ç¨â뢠ï, çâ® 1+ ; = ­ å®¤¨¬, çâ®

Z

S2

E~
E~ 1 dS;

Z ; = ; 12 jB E~
E~ 1 dS: S2

®¤áâ ¢«ïï íâ® ¢ëà ¦¥­¨¥ ¤«ï ; ¢ ¨á室­®¥ ¨­â¥£à «ì­®¥ ãà ¢­¥­¨¥, ¯®«ã稬 ®¤­®à®¤­®¥ ãà ¢­¥­¨¥ ¤«ï E~ (x; y)

Z Z 1 2Z X 1 E~ E~
E~  dS ; B E~ E~
E~  dS n 1 n 1 jZn

n=2

S2

S2

= 0:

‚¥ªâ®à­ ï äã­ªæ¨ï E~ (x; y) ªà®¬¥ ᮮ⢥âáâ¢ãî饣® ãà ¢­¥­¨ï ¤®«¦­  㤮¢«¥â¢®àïâì â ª¦¥ £à ­¨ç­®¬ã ãá«®¢¨î ­  ªà î ¤¨ äà £¬ë,   ¨¬¥­­®, ¥¥ â ­£¥­æ¨ «ì­ ï á®áâ ¢«ïîé ï ¤®«¦­  ®¡à é âìáï ¢ ­ã«ì ­  ªà î ¨¤¥ «ì­® ¯à®¢®¤ï饩 ¤¨ äà £¬ë. ®á«¥¤­¥¥ (®¤­®à®¤­®¥) ãà ¢­¥­¨¥ ¨¬¥¥â ­¥âਢ¨ «ì­®¥ à¥è¥­¨¥, 㤮¢«¥â¢®àïî饥 £à ­¨ç­®¬ã ãá«®¢¨î, «¨èì ¯à¨ ­¥ª®â®àëå §­ ç¥­¨ïå B , ª®â®àë¥ ï¢«ïîâáï ᮡá⢥­­ë¬¨ §­ ç¥­¨ï¬¨ ¤«ï ¤ ­­®£® ãà ¢­¥­¨ï. „ «¥¥ ¡ã¤¥â ¯®ª § ­®, çâ® íâ® ãà ¢­¥­¨¥ ®¡« ¤ ¥â «¨èì ®¤­¨¬ ᮡá⢥­­ë¬ §­ ç¥­¨¥¬ (ª ª íâ® á«¥¤ã¥â ¨§ 䨧¨ç¥áª®£® á¬ëá«  § ¤ ç¨). Œ®¦­® ¯®«ãç¨âì ®¥ ¢ëà ¦¥­¨¥ ¤«ï B . „«ï í⮣® 㬭®¦¨¬ ᪠«ïà­® ¯®á«¥¤­¥¥ ãà ¢­¥­¨¥ ­  E~  (x; y) ¨ ¯à®¨­â¥£à¨à㥬 ¯® S2 . ’®£¤ 

Z Z Z 1 2Z Z X 1 E~ 
E~ dS E~
E~  dS ; B E~ 
E~ dS E~
E~  dS n 1 n 1 jZn

n=2

S2

S2

S2

145

S2

= 0:

¥è ï íâ® à ¢¥­á⢮ ®â­®á¨â¥«ì­® B , ¯®«ã稬 2

R E~ 
E~ n dS 1 X 2Z1 S2 B = 2 : n=2 jZn R ~  ~ E
E1 dS S



2

Žâ¬¥â¨¬, çâ® ¤«ï í«¥ªâà¨ç¥áª¨å ¬®¤ ZnE = n =j!", ¤«ï ¬ £­¨â­ëå ¬®¤ ZnH = j!= n, Z1 { ¢¥é¥á⢥­­ ï ¯®«®¦¨â¥«ì­ ï ¢¥«¨ç¨­ . ®í⮬ã á« £ ¥¬ë¥ ¢ ¯®á«¥¤­¥© á㬬¥, ᮮ⢥âáâ¢ãî騥 í«¥ªâà¨ç¥áª¨¬ ¬®¤ ¬, ¤ îâ ¯®«®¦¨â¥«ì­ë© (¥¬ª®áâ­®©) ¢ª« ¤ ¢ B ,   ç«¥­ë, ᮮ⢥âáâ¢ãî騥 ¬ £­¨â­ë¬ ¬®¤ ¬, { ®âà¨æ â¥«ì­ë© (¨­¤ãªâ¨¢­ë©). ‚ ¡®«ì設á⢥ á«ãç ¥¢ â®ç­®¥ à¥è¥­¨¥ ¯®«ã祭­ëå ãà ¢­¥­¨© ¯®«ãç¨âì ­¥¢®§¬®¦­®, ¯®í⮬㠨ᯮ«ì§ãîâ ⥠¨«¨ ¨­ë¥ ¯à¨¡«¨¦¥­­ë¥ ¬¥â®¤ë ¨«¨ ¨áªãáá⢥­­ë¥ ¯à¨¥¬ë ¤«ï à¥è¥­¨ï § ¤ ç¨. 4.6.4. ‚ à¨ æ¨®­­ë¥ ¬¥â®¤ë

®«ã祭­®¥ à ­¥¥ ¢ëà ¦¥­¨¥ ¤«ï B

B =

2 R E~ 
E~ n dS 1 2Z S X 1 2 : n=2 jZn R ~  ~ E
E1 dS S 2

2

®¡« ¤ ¥â ­¥ª®â®àë¬ ¢ à¨ æ¨®­­ë¬ ᢮©á⢮¬. ã¤¥¬ à áᬠâਢ âì í⮠ᮮ⭮襭¨¥ ª ª ä㭪樮­ « ®â ä㭪樨 E~ (x; y). ®ª ¦¥¬, çâ® äã­ªæ¨ï E~ (x; y), 㤮¢«¥â¢®àïîé ï ­ ©¤¥­­®¬ã ¢ à §¤¥«¥ 4.6.3 ¨­â¥£à «ì­®¬ã ãà ¢­¥­¨î, ®¡à é ¥â ¢ ­ã«ì ¢ à¨ æ¨î B , â.¥. á®®¡é ¥â B áâ æ¨®­ à­®¥ §­ ç¥­¨¥. „«ï í⮣® ¯¥à¥¯¨è¥¬ ¯à¨¢¥¤¥­­®¥ ¢ëè¥ ¢ëà ¦¥­¨¥ ¢ ¢¨¤¥

Z Z Z 1 2Z Z X 1 E~ 
E~ dS E~
E~  dS ; B E~ 
E~ dS E~
E~  dS n 1 n 1 jZn

n=2

S2

S2

S2

¨ ¡ã¤¥¬ ¢ àì¨à®¢ âì äã­ªæ¨î E~ : 146

S2

= 0

Z 1 2Z Z X 1 E~ 
E~ dS E~
E~  dS n n

n=2 jZn S2

S2

+

Z 1 2Z Z X 1 E~ 
E~ dS E~
E~  dS ; n n

n=2 jZn S2

S2

Z 2 8Z Z <   ~ ~ ~ ~ ; B E
E1 dS ; B : E
E1 dS E~
E~ 1 dS + S S S 9 Z Z =  ~ ~ + E
E1 dS
E~
E~ 1 dS ; = 0: S S 2

2

2

2

2

¥à¥£à㯯¨àãï á« £ ¥¬ë¥, ¯®«ã祭­®¥ à ¢¥­á⢮ ¬®¦­® ¯¥à¥¯¨á âì á«¥¤ãî騬 ®¡à §®¬:

9 81 Z = <X 2Z1  Z  dS E~
: jZ E~ n E~
E~ n dS ; B E~ 1 E~ 
E~ 1 dS ; + n=2 n S S S 8 9 Z Z Z 1 2Z < = X + dS E~ 
: jZ 1 E~ n E~
E~ n dS ; B E~ 1 E~
E~ 1 dS ; ; n=2 n S S S Z 2 ; B E~
E~ 1 dS = 0: S Z

2

2

2

2

2

2

2

„«ï ⮣® çâ®¡ë ¢ à¨ æ¨ï B ¡ë«  à ¢­  ­ã«î ¯à¨ ¯à®¨§¢®«ì­®© ¢ à¨ æ¨¨ E~ , ­¥®¡å®¤¨¬® ¨ ¤®áâ â®ç­®, çâ®¡ë ¢ëà ¦¥­¨ï ¢ 䨣ãà­ëå ᪮¡ª å ¡ë«¨ à ¢­ë ­ã«î. ‡ ¬¥â¨¬, çâ® ¯à¨ ¢¥é¥á⢥­­®¬ B (ª ª íâ® á«¥¤ã¥â ¨§ 䨧¨ç¥áª®£® á¬ëá«  B ) ᪮¡ª¨ ª®¬¯«¥ªá­® ᮯà殮­ë ®¤­  ¤à㣮© ¨ ®¡à é îâáï ¢ ­ã«ì ®¤­®¢à¥¬¥­­®. ˆâ ª, ãá«®¢¨¥¬ áâ æ¨®­ à­®á⨠B ï¥âáï ãà ¢­¥­¨¥, ª®â®à®¬ã ¤®«¦­  㤮¢«¥â¢®àïâì äã­ªæ¨ï E~ : Z Z 1 2Z X 1 E~ E~
E~  dS ; B E~ E~
E~  dS = 0: n 1 n 1 n=2 jZn S2 S2 â® ãà ¢­¥­¨¥ ¢ â®ç­®á⨠ᮢ¯ ¤ ¥â á ¯®«ã祭­ë¬ ¢ëè¥ (¨§ ãá«®¢¨© á訢 ­¨ï ¯®«¥© ­  ¤¨ äà £¬¥) ãà ¢­¥­¨¥¬, çâ® ¨ âॡ®¢ «®áì ¤®ª § âì. 147

ã¦­® ®â¬¥â¨âì ¯à¨ í⮬, çâ® ¢¥ªâ®à­ ï äã­ªæ¨ï E ¤®«¦­  ¤®¯®«­¨â¥«ì­® 㤮¢«¥â¢®àïâì 㪠§ ­­ë¬ ¢ëè¥ £à ­¨ç­ë¬ ãá«®¢¨ï¬ (â ª ª ª £à ­¨ç­ë¥ ãá«®¢¨ï ­¥ ¢å®¤ïâ ¢ ãá«®¢¨ï áâ æ¨®­ à­®áâ¨). â® ­ ª« ¤ë¢ ¥â ®£à ­¨ç¥­¨ï ­  ¢ë¡®à ¯à®¡­ëå ä㭪権 ¯à¨ à¥è¥­¨¨ ¢ à¨ æ¨®­­®© § ¤ ç¨. 4.6.5. à¨¡«¨¦¥­­®¥ ¢ëç¨á«¥­¨¥ ¯à®¢®¤¨¬®áâ¨

‘â æ¨®­ à­®áâì ä㭪樮­ «  B ¢ ®ªà¥áâ­®á⨠à¥è¥­¨ï ¯®§¢®«ï¥â ¨á¯®«ì§®¢ âì ¤«ï ¢ëç¨á«¥­¨ï B ­¥ ⮫쪮 â®ç­®¥ à¥è¥­¨¥, ª®â®à®¥ ®¡ëç­® § à ­¥¥ ­¥¨§¢¥áâ­®, ­® ¨ ­¥ª®â®à®¥ ¯à¨¡«¨¦¥­¨¥, ¢ë¡à ­­®¥ ¨§ â¥å ¨«¨ ¨­ëå á®®¡à ¦¥­¨©. « £®¤ àï áâ æ¨®­ à­®á⨠ä㭪樮­ «  íâ® ¤ ¥â ¬ «ãî ®è¨¡ªã. à¨¬¥à: ˆ­¤ãªâ¨¢­ ï ¤¨ äà £¬  ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥ (á¬. à¨á.). ‚ í⮬ á«ãç ¥ ¤«ï ⮣®, ç⮡ë 㤮¢«¥â¢®à¨âì £à ­¨ç­ë¬ ãá«®¢¨ï¬, ¢ ᨫã ãá«®¢¨© ᨬ¬¥âਨ, ¤®áâ â®ç­® ¨á¯®«ì§®¢ âì ⮫쪮 ¢ëá訥 ¬®¤ë ⨯  Hn0 . „«ï íâ¨å ¬®¤ ®â«¨ç­  ®â ­ã«ï ⮫쪮 y-ï ª®¬¯®­¥­â  í«¥ªâà¨ç¥áª®£® ¯®«ï E~ :

y

z

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 x 0

a

d

111 000 000 111 000 111 000 111 000 111 000 111 000 111

Eny =

r

2 nx ab sin a |

­®à¬¨à®¢ ­­ ï äã­ªæ¨ï. „«ï ¢ëç¨á«¥­¨ï B ¢ ¯¥à¢®¬ ¯à¨¡«¨¦¥­¨¨ ¬®¦­®  ¯¯à®ªá¨¬¨à®¢ âì äã­ªæ¨î Ey (x) ¯®«ã¯¥à¨®¤®¬ ᨭãᮨ¤ë: x

Ey =

(

x0 ) ; ­  ®â¢¥àá⨨; cos (x ; d 0 ; ¢­¥ ®â¢¥àáâ¨ï:

„«ï ¤ «ì­¥©è¥£® ­ã¦­® ¢ëç¨á«¨âì ¨­â¥£à «

148

In =

x0 + d2

Z

x0 ; d2

x0 ) sin nx dx: cos (x ; d a

â®â ¨­â¥£à « à ¢¥­ 1 nx0 nd 2 In = d 1=d2 ; n2 =a2 sin a cos 2a :

! ®¤áâ ¢«ïï ¢ ¢ëà ¦¥­¨¥ ¤«ï B (ãç¨â뢠ï, çâ® Zn = j!

n , Z1 = ), ¯®«ã稬 B 

0 cos2 nd nX =1 2 (1=d2 ; 1=a2)2 sin2 nx a 2a ; n ; 2 ; n2 =a2 )2 x d (1 =d n=2 sin2 a 0 cos2 2a

p £¤¥ = 2 , n = (n=a)2 ; k2 . „ ­­®¥ ¢ëà ¦¥­¨¥ á¯à ¢¥¤«¨¢®, ¥á«¨ n=a 6= 1=d ­¨ ¤«ï ª ª®£® n. ‚ ¯à®â¨¢­®¬ á«ãç ¥ ­¥®¡å®¤¨¬® à áªàëâì ­¥®¯à¥¤¥«¥­­®áâì ¢¨¤  00 ¤«ï

⮣® ç«¥­ , £¤¥ n=a = 1=d (â ª ª ª ⮣¤  cos nd 2a = 0). „«ï à áç¥â®¢ ¤®áâ â®ç­® ®£à ­¨ç¨âìáï ç¨á«®¬ ç«¥­®¢  da , â ª ª ª ¤ «ìè¥ ç«¥­ë à鸞 ¡ëáâà® ã¡ë¢ îâ. 4.6.6. ‚ à¨ æ¨®­­ë¥ ¬¥â®¤ë. Œ¥â®¤ ¨âæ 

Œ¥â®¤ ¨âæ  ï¢«ï¥âáï ¯à¨¡«¨¦¥­­ë¬ ¯àï¬ë¬ ¬¥â®¤®¬ à¥è¥­¨ï ¢ à¨ æ¨®­­®© § ¤ ç¨ ­  íªáâ६㬠ä㭪樮­ « , ¢ ¤ ­­®¬ á«ãç ¥ B . Ž­ á®á⮨⠢ á«¥¤ãî饬. ¥¨§¢¥áâ­®¥ ¯®¯¥à¥ç­®¥ ¯®«¥ ­  ®â¢¥àá⨨ E~ (x; y) ¯à¥¤áâ ¢«ïîâ ¢ ¢¨¤¥ à §«®¦¥­¨ï ¢ àï¤ ¯® ­¥ª®â®à®© ¯®«­®© ®à⮣®­ «ì­®© á¨á⥬¥ ¢¥ªâ®à­ëå ä㭪権, ®¯à¥¤¥«¥­­®© ­  ®â¢¥àá⨨. „«ï ®¡¥á¯¥ç¥­¨ï 㤮¢«¥â¢®à¥­¨ï £à ­¨ç­ë¬ ãá«®¢¨ï¬ ä㭪樨 ¤®«¦­ë 㤮¢«¥â¢®àïâì £à ­¨ç­ë¬ ãá«®¢¨ï¬ ­  £à ­¨æ¥ ®â¢¥àáâ¨ï. à¨ í⮬ ®£à ­¨ç¨¢ îâáï ­¥ª®149

â®àë¬ ª®­¥ç­ë¬ ç¨á«®¬ ç«¥­®¢ ¢ í⮬ àï¤ã. ‡ â¥¬ á㬬ã, ¯à¥¤áâ ¢«ïîéãî E~ , ¯®¤áâ ¢«ïîâ ¢ ¢ëà ¦¥­¨¥ ¤«ï B , ¯à¨ç¥¬ B ®ª §ë¢ ¥âáï ä㭪樥© ª®íää¨æ¨¥­â®¢ à §«®¦¥­¨ï. Š®íää¨æ¨¥­âë ¯®¤¡¨à îâáï â ª, ç⮡ë äã­ªæ¨ï B ¨¬¥«  áâ æ¨®­ à­®¥ §­ ç¥­¨¥. „«ï í⮣® ¯à¨à ¢­¨¢ îâ ­ã«î ¯à®¨§¢®¤­ë¥ ®â B ¯® ª®íää¨æ¨¥­â ¬. ®«ã祭­ ï á¨á⥬  ãà ¢­¥­¨© ¯®§¢®«ï¥â ­ ©â¨ ª®íää¨æ¨¥­âë à §«®¦¥­¨ï. ®«­®© ®à⮣®­ «ì­®© á¨á⥬®© ä㭪権, 㤮¢«¥â¢®àïî饩 £à ­¨ç­ë¬ ãá«®¢¨ï¬ ­  £à ­¨æ å ®â¢¥àáâ¨ï, ï¥âáï á¨á⥬  ᮡá⢥­­ëå ¯®¯¥à¥ç­ëå ¢¥ªâ®à­ëå ä㭪権 ¢®«­®¢®¤ , á¥ç¥­¨¥ ª®â®à®£® ᮢ¯ ¤ ¥â á ®â¢¥àá⨥¬ ¤¨ äà £¬ë. Ž¡®§­ ç¨¬ í⨠ä㭪樨 ç¥à¥§ ~en (x; y). ’®£¤  N X

E~  ‚ëç¨á«¨¬ ¨­â¥£à «

In = £¤¥

Z

S2

E~
E~  dS =

m=1

am~em :

Z N X

N X

m=1

m=1

bmn =

am ~em E~ n dS = S2

Z S2

am bmn ;

~em E~ n dS:

®¤áâ ¢«ïï ¢ ¢ëà ¦¥­¨¥ ¤«ï B , ¯®«ã稬

B =

2 1 2Z1 P N P am bmn n=2 jZn m=1 P 2 : N ambm1 m=1

¥à¥¯¨è¥¬ íâ® à ¢¥­á⢮ ¢ á«¥¤ãî饬 ¢¨¤¥: N X

N X

N 1 2Z X X 1

N X

as bsn : jZ n m=1 s=1 m=1 s=1 n=2 ’¥¯¥àì ¡ã¤¥¬ ¢ àì¨à®¢ âì ª®íää¨æ¨¥­âë am , as (¢ ¯à¥¤¯®«®¦¥­¨¨ B

am bm1

as bs1 =

B = 0):

150

am bmn

B =

(X N m=1

am bm1

1 2Z ( X N X 1

N X s=1

as bs1 + N X

¨«¨, ¨­ ç¥

N X

m=1 N X

(

am B bm1

(

N X s=1

a b

s s1

X as B bs1 am bm1 N

m=1

am bm1 N X

as bsn +

jZn m=1 am bmn s=1

n=2

N X

;

1 2Z X 1

n=2

s=1

as bs1

am bmn N X

)

N X s=1

a b 

=

a b

)

s sn

)

jZn bmn s=1 s sn +

1 2Z X 1

b

N X

)

jZn sn
m=1 am bmn = 0: ‚á«¥¤á⢨¥ ¯à®¨§¢®«ì­®á⨠¢ à¨ æ¨© am ¨ as íâ® à ¢¥­á⢮ ¬®¦¥â +

s=1

m=1

;

m=1

N X

n=2

¢ë¯®«­ïâìáï, «¨èì ¥á«¨ à ¢­ë ­ã«î ¢ëà ¦¥­¨ï ¢ 䨣ãà­ëå ᪮¡ª å. ’ ª ª ª Z1 { ¢¥é¥á⢥­­®,   Zn { ç¨áâ® ¬­¨¬ë¥ ¢¥«¨ç¨­ë (n > 1), â® ¢ëà ¦¥­¨ï ¢ 䨣ãà­ëå ᪮¡ª å ª®¬¯«¥ªá­® ᮯà殮­ë ¤à㣠¤àã£ã ¨ ¤®áâ â®ç­®, çâ®¡ë ¢ ­ã«ì ®¡à é «®áì ®¤­® ¨§ ­¨å, ­ ¯à¨¬¥à,

B bs1

N X

m=1

am bm1 ;

1 2Z X 1

N X

bsn am bmn = 0 ¤«ï s = 1; 2; : : :; N: jZ n n=2 m=1

„ ­­ë¥ à ¢¥­á⢠ ¬®£ãâ ¡ëâì ¯¥à¥¯¨á ­ë ¢ ¢¨¤¥ (¯à¨ ¯¥à¥áâ ­®¢ª¥ ¯®à浪  á㬬¨à®¢ ­¨ï) N X

m=1

am

(X 1

n=2

2Z1   jZ bsn bmn ; B bs1 bm1 n

)

= 0 ¯à¨ s = 1; 2; : : : ; N:

Œë ¯®«ã稫¨ á¨á⥬㠨§ N ®¤­®à®¤­ëå «¨­¥©­ëå  «£¥¡à ¨ç¥áª¨å ãà ¢­¥­¨© ¤«ï N ­¥¨§¢¥áâ­ëå ª®íää¨æ¨¥­â®¢ am . â  á¨á⥬  ¨¬¥¥â ­¥âਢ¨ «ì­®¥ à¥è¥­¨¥, ¥á«¨ ¥¥ ®¯à¥¤¥«¨â¥«ì à ¢¥­ ­ã«î. à¨à ¢­¨¢ ï ­ã«î ®¯à¥¤¥«¨â¥«ì, ¯®«ã稬  «£¥¡à ¨ç¥áª®¥ ãà ¢­¥­¨¥ ¤«ï ®¯à¥¤¥«¥­¨ï ­¥¨§¢¥áâ­®© ¯®ª  ¯à®¢®¤¨¬®á⨠B . ® á¬ëá«ã § ¤ ç¨ íâ  ¢¥«¨ç¨­  ¤®«¦­  ¡ëâì ¢¥é¥á⢥­­®©,   à¥è¥­¨¥ { ¥¤¨­á⢥­­ë¬. Œ®¦­® ¯®ª § âì, çâ® íâ® ¤¥©á⢨⥫쭮 â ª. „«ï í⮣® ¢­ ç «¥ ¯à¥¤¯®«®¦¨¬, çâ® ­¨ ®¤¨­ ¨§ ª®íää¨æ¨¥­â®¢ bs1 ­¥ à ¢¥­ ­ã«î.  §¤¥«¨¬ ª ¦¤®¥ ¨§ ãà ¢­¥­¨© á¨áâ¥¬ë ­  ᮮ⢥âáâ¢ãî饥 bs1 . ’®£¤  ¯®«ã稬 á¨á⥬ã 151

N X

)

(X 1

2Z1 bsn bmn ; B b am m1 = 0 ¯à¨ s = 1; 2; : : : ; N:  m=1 n=2 jZn bs1 “à ¢­¥­¨¥ ¤«ï B ¢ í⮬ á«ãç ¥ ¯à¨®¡à¥â ¥â ¢¨¤

) sn bmn b 2 Z 1 ; B bm1 = 0: Det cms = Det  n=2 jZn bs1 ‚ ®¯à¥¤¥«¨â¥«¥ ¢ ¯à ¢®© ç á⨠᫠£ ¥¬®¥ B bm1 ­¥ § ¢¨á¨â ®â ­®¬¥à  áâப¨ s. …᫨ ¢ëç¥áâì ¯¥à¢ãî áâப㠨§ ¢á¥å ®áâ «ì­ëå áâப, â® B ®áâ ­¥âáï ⮫쪮 ¢ ¯¥à¢®© áâப¥.  §« £ ï ®¯à¥¤¥«¨â¥«ì ¯® í«¥¬¥­â ¬ ¯¥à¢®© áâப¨, ¯®«ãç ¥¬ «¨­¥©­®¥ ®â­®á¨â¥«ì­® B ãà ¢­¥­¨¥ á ¢¥é¥á⢥­­ë¬¨ ª®íää¨æ¨¥­â ¬¨, çâ® ¨ âॡ®¢ «®áì ¤®ª § âì. (X 1

4.6.7. ‚ à¨ æ¨®­­ë¥ ¬¥â®¤ë. Œ¥â®¤ ƒ «¥àª¨­ 

 áᬮâਬ ­¥®¤­®à®¤­®¥ ¨­â¥£à «ì­®¥ ãà ¢­¥­¨¥ ¤«ï ¯®«ï ­  ®â¢¥àá⨨ ¤¨ äà £¬ë E~ : 1 E~ Z X n E~
E~  dS ; E~ 1 n Z1 n=1 Zn S

= 0:

2

¥è¥­¨¥ í⮣® ãà ¢­¥­¨ï ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­® ¢ ¢¨¤¥ à §«®¦¥­¨ï ¯® ­¥ª®â®à®© ¯®«­®© á¨á⥬¥ ®à⮣®­ «ì­ëå ¢¥ªâ®à­ëå ä㭪権, 㤮¢«¥â¢®àïîé¨å £à ­¨ç­ë¬ ãá«®¢¨ï¬ ­  ª®­âãॠ®â¢¥àáâ¨ï. ’ ª¨¬¨ äã­ªæ¨ï¬¨ ïîâáï ᮡá⢥­­ë¥ ¢¥ªâ®à­ë¥ ä㭪樨 ~em ¢®«­®¢®¤ , á¥ç¥­¨¥ ª®â®à®£® ᮢ¯ ¤ ¥â á ®â¢¥àá⨥¬ ¤¨ äà £¬ë:

E =

1 X

m=1

am~em ;

£¤¥ am { ­¥¨§¢¥áâ­ë¥ ª®íää¨æ¨¥­âë. Œ¥â®¤ ƒ «¥àª¨­  ¯®§¢®«ï¥â ¯à¨¡«¨¦¥­­® ¢ëç¨á«¨âì «î¡®¥ ç¨á«® íâ¨å ª®íää¨æ¨¥­â®¢. „«ï í⮣® ®áâ ¢¨¬ ¢ á㬬¥ N ç«¥­®¢

E 

N X

m=1

152

am~em

¨ ¯®¤áâ ¢¨¬ íâ® ¢ëà ¦¥­¨¥ ¢ ¨­â¥£à «ì­®¥ ãà ¢­¥­¨¥. ’ ª ª ª íâ  á㬬  ¯à¥¤áâ ¢«ï¥â à¥è¥­¨¥ ¯à¨¡«¨¦¥­­®, â® ¯à ¢ ï ç áâì ãà ¢­¥­¨ï ¯à¨ ¯®¤áâ ­®¢ª¥ ­¥ ®¡à é ¥âáï ¢ ­ã«ì:

Z 1 E~ X N X n

~ n dS ; E~ 1 =
: a ~ e
E m m Z1 n=1 Zn m=1 S 2

‘®£« á­® ¬¥â®¤ã ƒ «¥àª¨­  ª®íää¨æ¨¥­âë am ¢ë¡¨à îâ â ª, ç⮡ë
¡ë«  ®à⮣®­ «ì­  ¢á¥¬ äã­ªæ¨ï¬ e~s (s = 1; 2; : : :; N ) ­  S2 :

9 Z8 Z 1 E~ X N <X =  ~ E n 1 ~ n dS ; a ~ e
E m m : Zn m=1 S Z1 ;
~es
dS S n=1 2

= 0; s = 1; 2; : : : ; N:

2

Ž¡®§­ ç¨¬

bmn =

Z S2

~emE~ n dS ¨ bsn =

Z S2

~es E~ n dS:

’®£¤  ãà ¢­¥­¨ï ¯à¨®¡à¥â îâ ¢¨¤ N 1 b X X sn

¨«¨

bs1 ; a b = m mn Z1 n=1 Zn m=1 N X m=1

am

1 b b X sn mn

n=1

Zn

 = bZs1 ; s = 1; 2; : : : ; N: 1

Œë ¯®«ã稫¨ á¨á⥬ã N «¨­¥©­ëå ãà ¢­¥­¨© ®â­®á¨â¥«ì­® N ª®íää¨æ¨¥­â®¢ am . ¥è ï ¥¥, ¬ë ­ ©¤¥¬ ¯à¨¡«¨¦¥­­®¥ ¢ëà ¦¥­¨¥ ¤«ï E~ , çâ® ¯®§¢®«ï¥â ¢ëç¨á«¨âì B , ¨á¯®«ì§ãï à ­¥¥ ¯®«ã祭­ãî ä®à¬ã«ã. €­ «®£¨ç­ë¬ ᯮᮡ®¬ ¬®¦­® à¥è âì â ª¦¥ ®¤­®à®¤­®¥ ãà ¢­¥­¨¥

Z Z 1 2Z X 1 E~ E~
E~  dS ; B E~ E~
E~  dS 1 n 1 jZn n

n=2

S2

S2

„«ï í⮣® ¢­®¢ì § ¯¨è¥¬ 153

= 0:

E 

N X m=1

am~em :

®¤áâ ¢«ïï íâ® ¢ ¨­â¥£à «ì­®¥ ãà ¢­¥­¨¥, ¯®«ã稬 1 2Z N X 1~ X

~ 1 X am bm1 =
: E a b ; B E n m mn n=2 jZn m=1 m=1 “¬­®¦ ï íâ® ãà ¢­¥­¨¥ ­  ~es ¨ ¨­â¥£à¨àãï ¯® S2 (á ãç¥â®¬ âॡ®¢ N

­¨ï ®à⮣®­ «ì­®áâ¨) ¯®«ã稬 1 2Z X 1

¨«¨

N X

m=1

N X

N X

bsn am bmn ; B bs1 am bm1 = 0 jZ n n=2 m=1 m=1 am

(X 1 n=2

2Z1   jZn bsn bmn ; B
bs1 bm1

)

= 0 ¯à¨ s = 1; 2; : : :; N:

„ ­­®¥ ãà ¢­¥­¨¥ ¢ â®ç­®á⨠ᮢ¯ ¤ ¥â á  ­ «®£¨ç­ë¬ ãà ¢­¥­¨¥¬, ¯®«ã祭­ë¬ á ¯®¬®éìî ¬¥â®¤  ¨âæ , çâ® ¤ ¥â ®á­®¢ ­¨ï ®â­¥á⨠¬¥â®¤ ƒ «¥àª¨­  ª ª« ááã ¢ à¨ æ¨®­­ëå ¬¥â®¤®¢. 4.6.8. „¨ äà £¬ë ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥

‚ëè¥ ¬ë 㦥 à áᬠâਢ «¨ ¨­¤ãªâ¨¢­ãî ¤¨ äà £¬ã ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥. ’ ¬ ¡ë«® ­ ©¤¥­® ᮮ⭮襭¨¥ ¤«ï ¯à®¢®¤¨¬®áâ¨ á ¯®¬®éìî ¯à¨¡«¨¦¥­­®£® ¬¥â®¤ , ®á­®¢ ­­®£® ­  áâ æ¨®­ à­®á⨠ä㭪樮­ «  ¯à®¢®¤¨¬®áâ¨. ‘ ¯®¬®éìî ª¢ §¨áâ â¨ç¥áª®£® ¬¥â®¤  ¬®¦¥â ¡ëâì ¯®«ã祭® ¡®«¥¥ ª®¬¯ ªâ­®¥ ¢ëà ¦¥­¨¥ (å®âï ¨ ­¥ ¡®«¥¥ â®ç­®¥) 2 d ctg2 x0 ): B = ; a ctg2 d (1 + sec 2a 2a 2a

154

111111111111 000000000000 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 111111111111 000000000000

’¥¯¥àì à áᬮâਬ ¥¬ª®áâ­ãî ¤¨ äà £¬ã (á¬. à¨á.). ’ ª ï ¤¨ äà £¬  ¨¬¥¥â ¯®«®¦¨â¥«ì­ãî ॠªâ¨¢­ãî ¯à®¢®¤¨¬®áâì ¨, á«¥¤®¢ â¥«ì­®, íª¢¨¢ «¥­â­  ¥¬ª®áâ¨. ‚ á«ãç ¥ ᨬ¬¥âà¨ç­®£® ®â¢¥àáâ¨ï ¯à®¢®¤¨¬®áâì ®¯à¥¤¥«ï¥âáï ¯à¨¡«¨¦¥­­ë¬ ᮮ⭮襭¨¥¬

d

B = 4b ln csc d 2b :

ƒà ä¨ª í⮩ ¯à®¢®¤¨¬®á⨠ª ª ä㭪樨 ®â b= ¯à¨¢¥¤¥­ ­  à¨áã­ª¥. B

b Λ

‘®ç¥â ­¨¥ ¨­¤ãªâ¨¢­®© ¨ ¥¬ª®áâ­®© ¤¨ äà £¬ ¯®§¢®«ï¥â ¯®«ãç¨âì èã­â¨àãî騩 १®­ ­á­ë© ª®­âãà (á¬. à¨á.). ë«® ­ ©¤¥­®, ç⮠१®­ ­á ¨¬¥¥â ¬¥áâ® ¤«ï ¢®«­ë, ®¯à¥¤¥«ï¥¬®© ãà ¢­¥­¨¥¬ s   s   a 1 ;  2 = a0 1 ;  2 : b 2a b0 2a0 b

111111111111 000000000000 000000000000 111111111111 000000000000 111111111111 a‘ b‘ 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 a

155

L

C

Žâá, ¢ ç áâ­®áâ¨, á«¥¤ã¥â, çâ® ¥á«¨ ãáâ६¨âì b0 ª 0, ⮠१®­ ­á ¨¬¥¥â ¬¥áâ® ¯à¨  = 2a0 , â.¥. ¯à¨ a0 = =2. ‚ í⮬ á«ãç ¥ é¥«ì ¬®¦­® à áᬠâਢ âì ª ª ª®à®âª®§ ¬ª­ãâãî á ¤¢ãå ª®­æ®¢ 饫¥¢ãî «¨­¨î. à®¢®¤¨¬®áâì १®­ ­á­®£® ª®­âãà  á ¯à¨á®¥¤¨­¥­­ë¬ ª ­¥¬ã ¢®«­®¢®¤®¬ ¨¬¥¥â ¢¨¤ ! Y = 1 + jQ 2
!0 ; £¤¥ !0 { १®­ ­á­ ï ç áâ®â . ‡­ ç¥­¨ï Q ¤«ï ¤¨ äà £¬ ­¥¢¥«¨ª¨ (¯®à浪  10) ¨ ¢®§à áâ îâ á 㬥­ì襭¨¥¬ b0 . Œ¥â ««¨ç¥áª¨© èâëàì ¬ «®£® ¤¨ ¬¥âà  ¯à¨¬¥­ï¥âáï ¤«ï ᮣ« á®¢ ­¨ï ¢®«­®¢®¤®¢. ¥âà㤭® ¢¨¤¥âì, çâ® èâëàì ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­ ¯®á«¥¤®¢ â¥«ì­® ᮥ¤¨­¥­­ë¬¨ ¥¬ª®áâìî ¨ ¨­¤ãªâ¨¢­®áâìî (á¬. à¨á.).

11 00 00 11 11 00

L C

à¨ ¬ «®© £«ã¡¨­¥ ¯®£à㦥­¨ï ¯à¥®¡« ¤ ¥â ¥¬ª®áâ­ ï ¯à®¢®¤¨¬®áâì. à¨ £«ã¡¨­¥ ¯®£à㦥­¨ï, ¯à¨¬¥à­® à ¢­®© =4, ¨¬¥¥â ¬¥á⮠१®­ ­á. à¨ í⮬ ¢®«­®¢®¤ ®ª §ë¢ ¥âáï § ª®à®ç¥­­ë¬ ¨ ¯¥à¥á⠥⠯யã᪠âì ¢®«­ë. à¨ ¥é¥ ¡®«ì襬 ¯®£à㦥­¨¨ ¯à¥®¡« ¤ ¥â ¨­¤ãªâ¨¢­®áâì, ¨, å®âï ¨¬¥¥âáï ®âà ¦¥­¨¥, ¢®«­  ç áâ¨ç­® ¯à®å®¤¨â. ‚®§¬®¦­ë â ª¦¥ ¤à㣨¥ १®­ ­á­ë¥ ª®¬¡¨­ æ¨¨, ­ ¯à¨¬¥à, ¨­¤ãªâ¨¢­ ï ¤¨ äà £¬  á ¥¬ª®áâ­ë¬ èâë६, ¯®§¢®«ïî騬 ®áãé¥á⢫ïâì ­ áâனªã १®­ ­á­®© ¤¨ äà £¬ë. 4.6.9. ‘®¥¤¨­¥­¨¥ ¢®«­®¢®¤®¢ à §«¨ç­®£® á¥ç¥­¨ï

 áᬮâਬ á®ç«¥­¥­¨¥ ¤¢ãå ¢®«­®¢®¤®¢ à §«¨ç­®£® á¥ç¥­¨ï S1 ¨ S2 , ¯à¨ç¥¬ ¢ ®¡®¨å ¢®«­®¢®¤ å ¬®£ãâ à á¯à®áâà ­ïâìáï ⮫쪮 ¯® ®¤­®© ¬®¤¥. Ž¡é¥¥ ®â¢¥àá⨥ ¢®«­®¢®¤®¢ ®¡®§­ ç¨¬ ç¥à¥§ A. ’®£¤  ¤«ï ¯®¯¥à¥ç­ëå á®áâ ¢«ïîé¨å í«¥ªâà¨ç¥áª®£® ¯®«ï ­  £à ­¨æ¥ ¨¬¥îâ ¬¥áâ® ãá«®¢¨ï (1 + ;) E~ 1 + á«¥¢  ¨

1 X An E~ n

n=2

=

 E~ (x; y) ;

156

0

­  A;

; ­  S1 ; A

T
E~ 10 +

1 X Bn E~ n0

n=2

=

 E~ (x; y) ; 0

­  A;

; ­  S2 ; A

á¯à ¢ . ‡¤¥áì E~ n { ᮡá⢥­­ë¥ ¯®¯¥à¥ç­ë¥ ­®à¬¨à®¢ ­­ë¥ ¢¥ªâ®à­ë¥ ä㭪樨 ¢®«­®¢®¤  I , E~ n0 { â® ¦¥ ¤«ï ¢®«­®¢®¤  II , E~ (x; y) { ­¥¨§¢¥áâ­ ï ¯®ª  äã­ªæ¨ï à á¯à¥¤¥«¥­¨ï ¯®¯¥à¥ç­®£® í«¥ªâà¨ç¥áª®£® ¯®«ï ­  ®â¢¥àá⨨. …᫨ 㪠§ ­­ë¥ ᮡá⢥­­ë¥ ä㭪樨 ­®à¬¨à®¢ ­ë ª ¥¤¨­¨æ¥, â®

R

R

1 + ; = E~
E~ 1 dS; An = E~
E~ n dS; AR R A T = E~
E~ 0 dS; Bn = E~
E~ n0  dS: A

A

®¯¥à¥ç­ë¥ á®áâ ¢«ïî騥 ¬ £­¨â­®£® ¯®«ï ­  ®â¢¥àá⨨ á«¥¢  ¨ á¯à ¢  à ¢­ë ¤à㣠¤àã£ã: (1 ; ;)
H~ 1 ; ¨«¨

1 X An H~ n

n=2

= T
H~ 10 +

1 X Bn H~ n0 ;

n=2

X ~z0  E~ n X ~z0  E~ n0 ~ ~0 (1 ; ;) ~z0 Z E1 ; An Z = T
~z0 Z0 E1 + Bn Z 0 : 1 n n 1 n=2 n=2 ‡¤¥áì Z1 , Zn , Z10 , Zn0 { ®â­®è¥­¨ï ¯®¯¥à¥ç­ëå á®áâ ¢«ïîé¨å í«¥ªâà¨ç¥áª®£® ¨ ¬ £­¨â­®£® ¯®«¥©. “¬­®¦ ï íâ® ãà ¢­¥­¨¥ ¢¥ªâ®à­® ­  ~z0 ¨ à áªàë¢ ï ¤¢®©­ë¥ ¢¥ªâ®à­ë¥ ¯à®¨§¢¥¤¥­¨ï, ¯®«ã稬 ãà ¢­¥­¨¥ 1

1

X E~ n ~1 E~ 10 + X B E~ n0 : (1 ; ;) E ; A = T n n 0 Z1 Z10 n=2 Zn n=2 Zn …᫨ ⥯¥àì § ¬¥­¨âì ;, T , An ¨ Bn ¨å ¢ëà ¦¥­¨ï¬¨ ¢ ¢¨¤¥ ¨­â¥£à «®¢, â® ¬®¦­® ¯®«ãç¨âì ¨­â¥£à «ì­®¥ ãà ¢­¥­¨¥ ®â­®á¨â¥«ì­® ­¥¨§¢¥áâ­®© ä㭪樨 E~ (x; y): 1

1 E~ Z X n ~ ~ Zn E
En dS

n=1

A

1

+

1 E~ 0 Z X n ~ ~0  Z 0 E
En dS

n=1 n A

157

~ = 2ZE1 : 1

‚®§¬®¦¥­, ®¤­ ª®, ¤à㣮© ¯ãâì ¯à¥®¡à §®¢ ­¨ï ãà ¢­¥­¨ï. „«ï í⮣® 㬭®¦¨¬ «¥¢ãî ¨ ¯à ¢ãî ç á⨠ãà ¢­¥­¨ï ­  E~  ¨ ¯à®¨­â¥£à¨à㥬 ¯® ®â¢¥àáâ¨î. ®«ã稬 1 jA j2 1 1 ; ; Z E~ 
E~ dS ; X n = jT j2 1 + X jBn j2 : 1 0 Z1 Z10 n=2 Zn n=2 Zn A

R“¬­®¦¨¬ ¨ à §¤¥«¨¬ ¯¥à¢®¥ á« £ ¥¬®¥ ¢ «¥¢®© ç á⨠­  1 + ; = = E~ E~ 1 dS : A

1 1

Z 2 1 ; ;
1 E~
E~  dS ; X jAn j2 1 + ; Z1 n=2 Zn A

®âªã¤ 

1;; = 1+;

1 jB j2 X 1 n ; 2 = jT j Z 0 + 0 Z 1 n=2 n

1 1 jT j2 ZZ10 + P ZZn1 jAn j2 + P ZZ10 jBn j2

1

n=2

R E~
E~  dS 2 A 1

n=2 n

:

; ® 11 ; + ; = Y { ¢å®¤­ ï ¯à®¢®¤¨¬®áâì á®ç«¥­¥­¨ï, ­ £à㦥­­®£® á ¤à㣮© áâ®à®­ë ­  ᮣ« á®¢ ­­ãî ­ £à㧪ã. ’ ª¨¬ ®¡à §®¬, ¯®«ãç ¥¬ ¢ëà ¦¥­¨¥ ¤«ï ¯à®¢®¤¨¬®á⨠Y :

Y =

jT j2 ZZ10

1 R 2 E~
E~ 1 dS A

+

1 Z1 1 Z1 P 2 + P j A j n 0 jBn j2 jZ jZ n n n=2 j n=2 : R 2 E~
E~ 1 dS A

n E  ¯®¬­¨¬, çâ® ¤«ï H-¬®¤ ZnH = j!

n , ¤«ï E-¬®¤ Zn = j!" { ¢¥é¥á⢥­­ë¥ ¤«ï à á¯à®áâà ­ïî饩áï ¨ ç¨áâ® ¬­¨¬ë¥ ¤«ï ­¥à á¯à®áâà ­ïîé¨åáï ¬®¤. Žâá, ¥á«¨ S1 ! S2 , ¬®¦­® ¯®«ãç¨âì ¢ëà ¦¥­¨¥ ¤«ï ¯à®¢®¤¨¬®á⨠¤¨ äà £¬ë (¯à¨ E~ n0 ! E~ n ¨ Zn0 ! Zn). ¥à¢®¥ á« £ ¥¬®¥ ¢ ¯®«ã祭­®© ä®à¬ã«¥ { ¢¥é¥á⢥­­ ï ç áâì, ¢â®à®¥ { ¬­¨¬ ï. ¥à¢®¥ á« £ ¥¬®¥ ¯à¥¤áâ ¢«ï¥â ¯à®¢®¤¨¬®áâì ¢â®à®£® 158

¢®«­®¢®¤ , ­ £à㦥­­®£® ­  ᮣ« á®¢ ­­ãî ­ £à㧪ã, ¢â®à®¥ { ॠªâ¨¢­ãî ¯à®¢®¤¨¬®áâì á®ç«¥­¥­¨ï. ª¢¨¢ «¥­â­ ï á奬  á®ç«¥­¥­¨ï ¨¬¥¥â ¢¨¤ n:1 Z0 =1 jB

Z 0=1

à¨ á®ç«¥­¥­¨¨ ¤¢ãå ¢®«­®¢®¤®¢ ®á®¡ë© ¨­â¥à¥á ¯à¥¤áâ ¢«ï¥â á«ãç ©, ª®£¤  ¢®«­®¢®¤ë ¬ «® ®â«¨ç îâáï ®¤¨­ ®â ¤à㣮£®. ‚ í⮬ á«ãç ¥ ¢ ª ç¥á⢥ ¯¥à¢®£® ¯à¨¡«¨¦¥­¨ï ¢¬¥áâ® E~ ¬®¦­® ¯®¤áâ ¢¨âì ¯®«¥ E~ 1 ¢ ¯¥à¢®¬ ¢®«­®¢®¤¥.  áᬮâਬ ¢ ª ç¥á⢥ ¯à¨¬¥à  ᮥ¤¨­¥­¨¥ ¤¢ãå ¯àאַ㣮«ì­ëå ¢®«­®¢®¤®¢, ¢ ª®â®àëå ¬®¦¥â à á¯à®áâà ­ïâìáï ⮫쪮 ¬®¤  H10 . ˆ§¬¥­¥­¨¥ á¥ç¥­¨ï ¢ ¯«®áª®á⨠H .  áᬠâਢ ¥âáï ᨬ¬¥âà¨ç­®¥ á®ç«¥­¥­¨¥, ¯à¥¤áâ ¢«¥­­®¥ ­  à¨á. y a‘

a‘

a z

b

x

x a

‚ í⮬ á«ãç ¥ £à ­¨ç­ë¥ ãá«®¢¨ï ¬®£ãâ ¡ëâì 㤮¢«¥â¢®à¥­ë á ¨á¯®«ì§®¢ ­¨¥¬ «¨èì ¬ £­¨â­ëå ¬®¤ Hn0 . „«ï ¨á¯®«ì§®¢ ­¨ï ᨬ¬¥âਨ ®áì y ¯®¬¥á⨬ ¯®á¥à¥¤¨­¥ ¢®«­®¢®¤ . à¨ ¢ëç¨á«¥­¨¨ ¯à®¢®¤¨¬®á⨠á®ç«¥­¥­¨ï ¢ ª ç¥á⢥ ä㭪樨 ¯®«ï E~ (x; y) ¢ ®â¢¥àá⨨ ¯à¨¬¥¬ ¯à¨¡«¨¦¥­­® äã­ªæ¨î E~ 1 í«¥ªâà¨ç¥áª®£® ¯®«ï ®á­®¢­®© ¬®¤ë ¢ ¢®«­®¢®¤¥ I . ’ ª ª ª E~ 1 ᨬ¬¥âà¨ç­  ®â­®á¨â¥«ì­® ®á¨ y, â® ¯à¨ ¢ëç¨á«¥­¨¨ ¨­â¥£à «®¢ ­¥á¨¬¬¥âà¨ç­ë¥ ᮡá⢥­­ë¥ ä㭪樨 ¤ îâ ­ã«ì, ¨ ¢ à §«®¦¥­¨¨ ®áâ îâáï ⮫쪮 ᨬ¬¥âà¨ç­ë¥. ‘¨¬¬¥âà¨ç­ë¥ ᮡá⢥­­ë¥ ä㭪樨 í«¥ªâà¨ç¥áª®£® ¯®«ï ¢ ¢®«­®¢®¤¥ I (­®à¬¨à®¢ ­­ë¥ ª ¥¤¨­¨æ¥) ¨¬¥îâ ¢¨¤

Eym =

r

x 2 ab cos(2m + 1) a ; 159

  ¢ ¢®«­®¢®¤¥ II

E0

ym =

r

2 cos(2m + 1) x : a0 b a0

¥âà㤭® ¢¨¤¥âì, çâ® ¢ í⮬ á«ãç ¥

Z

A

E~
E~ m dS =

0

¯à¨ m 6= 0; 1 ¯à¨ m = 0:

®í⮬ã A2m+1 = 0 ¯à¨ m 6= 0. „ «¥¥,

B2m+1 =

Z

A

E~
E~ m0  dS =

à¨ m = 0 ¨¬¥¥¬

T =

ra

 cos(2m + 1) ;  = a : ; (2m + 1)2 2 2a0

a0 2 =4

r a 
cos  a0 2 =4 ; 2 :

0 Z1 0 = ; n , ®¤áâ ¢«ïï ¢ ä®à¬ã«ã ¤«ï Y ¨ ãç¨â뢠ï, çâ® ZZ10 = , jZ 0 1 n ¯®«ã稬    cos(2m + 1) 2 1 0 0   2 ; ja X 2m+1 Y = aa0 2=4
cos ; 2 a0 m=1 2 =4 ; (2m + 1)2 2 :

®«ã祭­®¥ ᮮ⭮襭¨¥ ¯®ª §ë¢ ¥â, çâ® á®ç«¥­¥­¨¥ ¤¥©áâ¢ã¥â ª ª âà ­áä®à¬ â®à á ª®íää¨æ¨¥­â®¬ âà ­áä®à¬ æ¨¨, à ¢­ë¬ T . ª¢¨¢ «¥­â­ ï á奬  ¯à¥¤áâ ¢«¥­  ­  à¨á. n:1 Y0 =1

L

Y0 =1

s

0 a   cos  2 ‡¤¥áì n = a0 2 =4 ; 2 .

160

”®à¬ã«  ã¯à®é ¥âáï, ¥á«¨ ᪠箪 à §¬¥à  ­¥¢¥«¨ª: aa0 = 1 ; , ¯à¨ç¥¬   1. ’®£¤  0  12 0  12 1 sin sin(2 m + 1) 0 0 X Y  B @ 2 CA ; j 2m +1 B@  m(m + 1)2 CA : m=1 2 ˆ§¬¥­¥­¨¥ á¥ç¥­¨ï ¢ ¯«®áª®á⨠E (á¬. à¨á.). y

b

b‘

y z

b‘ b

x

a

‘®ç«¥­¥­¨¥ ¯à¥¤¯®« £ ¥âáï ᨬ¬¥âà¨ç­ë¬. « £®¤ àï ᨬ¬¥âਨ ¢ à §«®¦¥­¨¨ ¯® ¢ëá訬 ¬®¤ ¬ ®áâ îâáï ⮫쪮 ¬®¤ë, ç¥â­ë¥ ¢ ­ ¯à ¢«¥­¨¨ y. —⮡ë 㤮¢«¥â¢®à¨âì £à ­¨ç­ë¬ ãá«®¢¨ï¬, ¢ ¤ ­­®¬ á«ãç ¥ ­¥®¡å®¤¨¬® ¨á¯®«ì§®¢ âì ª ª í«¥ªâà¨ç¥áª¨¥, â ª ¨ ¬ £­¨â­ë¥ ¬®¤ë. Ž¤­ ª® § ¤ ç  ã¯à®é ¥âáï ¡« £®¤ àï ⮬ã, çâ® ¨áå®¤ï ¨§ ᨬ¬¥âਨ ¬®¦­® ¯®«ãç¨âì ¨­ä®à¬ æ¨î ®â­®á¨â¥«ì­® ­¥¨§¢¥áâ­®© ä㭪樨 E~ . à¥¦¤¥ ¢á¥£®, E~ ¨¬¥¥â ⮫쪮 y-î ª®¬¯®­¥­âã. „ «¥¥, § ¢¨á¨¬®áâì E~ ®â x â ª ï ¦¥, ª ª ¨ ¤«ï ®á­®¢­®© ¬®¤ë, â ª ª ª £à ­¨ç­ë¥ ãá«®¢¨ï ­¥ § ¢¨áïâ ®â x. ’ ª ª ª Ex = 0, â® ¯à¨ ¢ëç¨á«¥­¨¨ ¨­â¥à¥á ¯à¥¤áâ ¢«ïîâ ⮫쪮 y-¥ á®áâ ¢«ïî騥 ᮡá⢥­­ëå ä㭪権, ¯à¨ç¥¬ ¢ª« ¤ ¤ îâ «¨èì ᮡá⢥­­ë¥ ä㭪樨 á n = 1, â.¥. § ¢¨áï騥 ®â x â ª ¦¥, ª ª ¨ ®á­®¢­ ï ¬®¤ . Šà®¬¥ ⮣®, ¨§ ᨬ¬¥âਨ á«¥¤ã¥â, çâ® ¢ª« ¤ ¤ îâ ⮫쪮 ⥠ᮡá⢥­­ë¥ ä㭪樨, ¤«ï ª®â®àëå m ç¥â­® (m = 2l). ®à¬¨à®¢ ­­ë¥ ᮡá⢥­­ë¥ ä㭪樨, ¯à¥¤áâ ¢«ïî騥 ¨­â¥à¥á, ¨¬¥îâ á«¥¤ãî騩 ¢¨¤ (á ãç¥â®¬ à á¯®«®¦¥­¨ï ®á¥©): Œ £­¨â­ë¥ ¬®¤ë (m = 2l)

161

‘«¥¢  ‘¯à ¢  2p sin x cos 2ly E 0 = 2p sin x cos 2ly E2ly = 2ly b b0 a
g2l ab a a
g20 l ab0 a à¨ l = 0 r r x 2 sin x 2 00y = E0y = ab E a ab0 sin a «¥ªâà¨ç¥áª¨¥ ¬®¤ë (m = 2l)

E2ly = ¯à¨ í⮬

4lp sin x cos 2ly E 0 = 4lp sin x cos 2ly 2ly b b0 b
g2l ab a b0
g20 l ab0 a

r

r

2 22 2 22 g2l = a2 + 4b2l ; g20 l = a2 + 40 2l : b ‚ ª ç¥á⢥ ä㭪樨 E~ , ª®â®à ï, ª ª 㪠§ ­® ¢ëè¥, ¨¬¥¥â ⮫쪮 y-î ª®¬¯®­¥­âã ¨ § ¢¨á¨â ®â x â ª ¦¥, ª ª ®á­®¢­ ï ¬®¤ , ¯à¨¬¥¬ äã­ªæ¨î á ª®¬¯®­¥­â ¬¨: Ex = 0; Ey = E (y) sin x a.

à¨ í⮬ ¢¥é¥á⢥­­ ï á®áâ ¢«ïîé ï ¯à®¢®¤¨¬®á⨠ࠢ­  R 2 E~
E~ 10  dS Z1 : G = A 2 0 Z R E~
E~ 1 dS 1 A

‡¤¥áì Z1 = Z10 (â ª ª ª § ¢¨á¨â «¨èì ®â a), E1y =

r

x 2 ab0 sin a .

r

2 x 0 ab sin a , E1y =

à¨ ¯®¤áâ ­®¢ª¥ E ¨­â¥£à «ë ᮪à é îâáï:  2 2

G =

p

0  2ab 2

p

ab

= bb0 :

Œ­¨¬ ï (ॠªâ¨¢­ ï) á®áâ ¢«ïîé ï ¯à®¢®¤¨¬®á⨠ࠢ­  162

B =

2 P 2 1 Z1 R 1 Z1 R P  0  E
E2l dS + 0 E
E2l dS l=1 jZ2l A l=1 jZ2l A : R 2 E
E1 dS A

‚ ª ç¥á⢥ ¯¥à¢®£® ¯à¨¡«¨¦¥­¨ï ¤«ï E~ ¯à¨¬¥¬ ¯®«¥ ®á­®¢­®© ¬®¤ë ¢ ¯¥à¢®¬ ¢®«­®¢®¤¥

r

2 sin x : ab a ‚ᥠAn ¤«ï ¬ £­¨â­ëå ¨ í«¥ªâà¨ç¥áª¨å ¬®¤ ᮤ¥à¦ â ¬­®¦¨â¥«ì Zb=2 2ly cos b dy = 0 (l 6= 0); ;b=2 ¯®í⮬㠢ᥠAn = 0. ‚ëç¨á«¨¬ ⥯¥àì Bn = B2l ¤«ï ¬ £­¨â­ëå ¨ í«¥ªâà¨ç¥áª¨å ¬®¤. Œ £­¨â­ë¥ ¬®¤ë r p 0 Za=2 x Zb=2 2ly 2  2 H B2l = ab 0 p 0 sin2 a dx cos b0 dy = 22p 20 2abl  ag2l ab ;a=2 a bb ;b=2

Ey =

p r

£¤¥ ' = b b0 . Š¢ ¤à â

 2 b sin l'  sin lb b0 = ag20 l b0 l' ;

2 0 2  (B2Hl )2 = bb
a22 g0 2 sinl'l' : 2l «¥ªâà¨ç¥áª¨¥ ¬®¤ë p

B2El = 2g0 2bl0 2l

r

b sin l' ; (B E )2 = b
82 l2  sin l' 2 : 2l b0 l' b0 b0 2 g20 l 2 l' 163

‚ëç¨á«¨¬ á㬬ã

P2l = jZZ01H (B2Hl )2 + jZZ01 E (B2El )2 : 2l 2l 0 0 H j! 0 E 2l ‡¤¥áì Z1 = ! , Z2l = 20 l , Z2l = j!" . ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï P2l , ¯®«ã稬

 P2l = 10 k2 (B2El )2 ; 20 l 2 (B2Hl )2 = 2l

 sin l' 2 b  82l2 2  2  1 2 0 2 = 0 l' b0 k b0 2 g20 l 2 ; 2l a2 g20 l 2 = 2l 2  4l2 0 2  2  = 10 bb0 g20 2 sinl'l' k2 b0 2 ; a22l : 2l 2l ’ ª ª ª 20 l 2 = g20 l 2 ; k2 , â® 2  4l2 1  2  P2l = 10 bb0 g20 2 sinl'l' k2 b0 2 ; a2 (g20 l 2 ; k2 ) : 2l 2l ‚ëà ¦¥­¨¥ ¢ ¯®á«¥¤­¨å ᪮¡ª å à ¢­®  4l2 1  g0 2 02 02 2 1 4 l 0 2 2 2 2 k b0 2 ; a2 (g2l ; k ) = k b0 2 + a2 ; a22l = k2 g2l2 ; ga22l = 2 02 02 = g2l2 (k2 ; a2 ) = g2l2 2 : ®¤áâ ¢«ïï, ¯®«ã稬  sin l' 2 2  sin l' 2 g20 l 2 2 b 2  2 b 1 P2l = 0 b0 g0 2 l' 2 = 20 l b0 l' : 2l 2l R R ®¤áâ ¢«ïï ¢ ¢ëà ¦¥­¨¥ ¤«ï B (á ãç¥â®¬ ⮣®, çâ® EE1 dS = 1, EE2l dS = 0), A A ¯®«ã稬 1 1 1  sin l' 2 X X 2 b B = P2l = b0 0 l' : l=1 2l l=1 Žâá ¯à®¢®¤¨¬®áâì à ¢­  164

1 1  sin l' 2 X b b Y = b0 + 2j b0 0 l' ; l=1 2l

£¤¥

0 ' = b b0 ; 2l =

q0

g2l 2 ; k2 :

’ ª ª ª B > 0, ¯à®¢®¤¨¬®áâì ¨¬¥¥â ¥¬ª®áâ­®© å à ªâ¥à ¨ íª¢¨¢ «¥­â­ ï á奬  ¨¬¥¥â ¢¨¤, ¯à¥¤áâ ¢«¥­­ë© ­  à¨á. n:1 Y0 =1

C

Y0 =1

r

‡¤¥áì n = bb0 . à¨ 㬥­ì襭¨¨ ᪠窠 ¥¬ª®áâì áâ६¨âáï ª ­ã«î ª ª ª¢ 
b à §¬¥à  2 ¤à â ᪠窠  b , ¤®¡ ¢ª  ª G { ª ª ¯¥à¢ ï á⥯¥­ì, ¯®í⮬㠯ਠ¬ «ëå áª çª å ¥¬ª®áâìî ¬®¦­® ¯à¥­¥¡à¥çì. ‚ í⮬ á«ãç ¥ ®âà ¦¥­¨¥ ®¯à¥¤¥«ï¥âáï ®â­®è¥­¨¥¬ ¢®«­®¢ëå ¯à®¢®¤¨¬®á⥩ ¢®«­®¢®¤®¢ b=b0. 4.6.10. ‘®£« á®¢ ­¨¥ ¢®«­®¢®¤®¢ ¬­®£®áâ㯥­ç â묨 ¯¥à¥å®¤ ¬¨

‡¤¥áì à áᬠâਢ ¥âáï ᮣ« á®¢ ­¨¥ ¢®«­®¢®¤®¢ ¯à¨ ¨§¬¥­¥­¨¨ à §¬¥à®¢ ¢ ¯«®áª®á⨠E , ª®£¤  ¬®¦­® ¯à¥­¥¡à¥çì ॠªâ¨¢­®áâìî ᪠窠 ¨ ®âà ¦¥­¨¥ ®¯à¥¤¥«ï¥âáï ⮫쪮 ᮮ⭮襭¨¥¬ ¢®«­®¢ëå ᮯà®â¨¢«¥­¨©, ¯®í⮬ã १ã«ìâ âë ¬®£ãâ ¡ëâì ¯à¨¬¥­¥­ë â ª¦¥ ª TEM «¨­¨ï¬. Šà®¬¥ ⮣® ¬ë ¡ã¤¥¬ ¯à¥¤¯®« £ âì ᪠窨 ¬ «ë¬¨. à¥­¥¡à¥£ ¥¬ â ª¦¥ ¬­®£®ªà â­ë¬¨ ®âà ¦¥­¨ï¬¨. ‘«¥¤ã¥â ®â¬¥â¨âì, çâ® ¢á¥ í⨠®£à ­¨ç¥­¨ï ¢¢®¤ïâáï ¤«ï ¯à®áâ®âë à áç¥â , ­®  ­ «®£¨ç­ë© १ã«ìâ â ¬®¦¥â ¡ëâì ¯®«ã祭 ¨ ¢ ¡®«¥¥ ®¡é¥¬ á«ãç ¥, ª®£¤  ᪠窨 ¢¥«¨ª¨ ¨ ॠªâ¨¢­®áâﬨ ¯à¥­¥¡à¥£ âì ­¥ á«¥¤ã¥â.  áᬮâਬ ¬­®£®áâ㯥­ç âë© ¯¥à¥å®¤ ¬¥¦¤ã ¤¢ã¬ï ¢®«­®¢®¤ ¬¨, á®áâ®ï騩 ¨§ ®â१ª®¢ à ¢­®© ¤«¨­ë, ­® à §«¨ç­®£® ¢®«­®¢®£® ᮯà®â¨¢«¥­¨ï (á¬. à¨á.). 165

Z0

Z1 Γ0

Z2

d

Zn

Γ1

Z Γn

‚¥«¨ç¨­ë ;0 ; ;1 ; : : : ; ;n { ¬¥áâ­ë¥ ª®íää¨æ¨¥­âë ®âà ¦¥­¨ï: Zi : ;i = ZZi+1 ; + Zi i+1 ãáâì ¤«ï ®¯à¥¤¥«¥­­®á⨠n { ç¥â­®¥ ç¨á«®. Žâà ¦¥­­ ï ¢®«­  ¢ ­ ç «¥ ¯¥à¥å®¤  à ¢­ , ®ç¥¢¨¤­®, á㬬¥ ¢®«­, ®âà ¦¥­­ëå ®â ®â¤¥«ì­ëå á®ç«¥­¥­¨© (¢ ¯à¥­¥¡à¥¦¥­¨¨ ¬­®£®ªà â­ë¬¨ ®âà ¦¥­¨ï¬¨):

X j(n;2m)' ;m e;2jm' = e;jn' ;m e = m=0 m=0 n o = e;jn' ;0 e jn' + ;1 e j(n;2)' + : : : + ;n=2 + : : : + ;n e;jn' ; £¤¥ ' = 2d : à¥¤¯®«®¦¨¬, çâ® ¯¥à¥å®¤ ¢ë¯®«­¥­ ᨬ¬¥âà¨ç­®, â ª çâ® ;m = ;n;m. ’®£¤    1 ; jn' b = 2e ;0 cos n' + ;1 cos(n ; 1)' + : : : + 2 ;n=2 ; â.¥. ¢ëà ¦ ¥âáï ç¥à¥§ ª®á¨­ãáë ªà â­ëå ¤ã£. ’ ª ª ª ª®á¨­ãáë ªà â­ëå ¤ã£ ¬®£ãâ ¡ëâì ¢ëà ¦¥­ë ç¥à¥§ á⥯¥­¨ cos ', â®  ¬¯«¨â㤠 ®âà ¦¥­­®© ¢®«­ë ¢ëà ¦ ¥âáï ç¥à¥§ ¯®«¨­®¬ ®â cos '. Ž¡®§­ ç¨¢ cos ' = x, ¬®¦¥¬ § ¯¨á âì b =

n X

n

jbj = M Pn (x); £¤¥ Pn (x) { ¯®«¨­®¬ á⥯¥­¨ n á ª®íää¨æ¨¥­â®¬ ¯à¨ x, à ¢­ë¬ ¥¤¨­¨æ¥. ¥à¥å®¤ë, ã ª®â®àëå ª®íää¨æ¨¥­â ®âà ¦¥­¨ï ¢ëà ¦ ¥âáï ç¥à¥§ ­¥ª®â®àë© ¯®«¨­®¬ á ¢¥é¥á⢥­­ë¬¨ ª®íää¨æ¨¥­â ¬¨ ®â cos ', ­ §ë¢ îâáï ¯®«¨­®¬¨ «ì­ë¬¨. “á«®¢¨¥¬ ¤«ï í⮣® á«ã¦ â à ¢¥­á⢮ ®â¤¥«ì­ëå ®â१ª®¢ ¯® ¤«¨­¥ ¨ ᨬ¬¥âà¨ï ª®íää¨æ¨¥­â®¢ ®âà ¦¥­¨ï. 166

®¤¡¨à ï ª®íää¨æ¨¥­âë ¯®«¨­®¬  ®¯à¥¤¥«¥­­ë¬ ®¡à §®¬, ¬®¦­® ¯®«ãç¨âì å®à®è¥¥ ᮣ« á®¢ ­¨¥ ¢ è¨à®ª®© ¯®«®á¥ ç áâ®â. …᫨ ¬¥áâ­ë¥ ª®íää¨æ¨¥­âë ¯®¤®¡à ­ë â ª, çâ® ¨á祧 î⠢ᥠá⥯¥­¨ x, ªà®¬¥ ¢ëá襩, â® â ª®© ¯¥à¥å®¤ ­ §ë¢ îâ ¡¨­®¬¨ «ì­ë¬ ¨«¨ ¯¥à¥å®¤®¬ á ¬ ªá¨¬ «ì­® ¯«®áª®© å à ªâ¥à¨á⨪®© (á¬. à¨á.). |b|

x= cos ϕ

„«ï í⮣® ¬¥áâ­ë¥ ª®íää¨æ¨¥­âë ®âà ¦¥­¨ï ¤®«¦­ë ¡ëâì ¯à®¯®à樮­ «ì­ë ª®íää¨æ¨¥­â ¬ ¡¨­®¬  ìîâ®­ : ;m =  Cnm : ’®£¤ 

b =

n X m=0

;
;m e j(n;2m)' =  e j' + e;j' n =  2n cosn ' = 2n  xn :

‚¥«¨ç¨­   ®¯à¥¤¥«ï¥âáï ¯à¨ x = 1 (' = 0):

b(' = 0) =

P

X m

;m = 2n;

®âªã¤   = ;2nT , £¤¥ ;T = ;m { ª®íää¨æ¨¥­â ®âà ¦¥­¨ï ¯à¨ ᮥ¤¨m ­¥­¨¨ ᮣ« á㥬ëå «¨­¨© ¡¥§ ¯¥à¥å®¤ . Žâá

b = ;T xn : …᫨ x ¢ § ¤ ­­®© ¯®«®á¥ ç áâ®â ¨§¬¥­ï¥âáï ¢ ¯à¥¤¥« å ; 1t  x  1t ¯à¨ t > 1, â® ¬ ªá¨¬ «ì­ë© ª®íää¨æ¨¥­â ®âà ¦¥­¨ï ¢ í⮩ ¯®«®á¥ ¡ã¤¥â à ¢¥­ 167

b¬ ªá = ;tnT :

‚¥«¨ç¨­ã K = tn ¢ í⮬ á«ãç ¥ ­ §ë¢ î⠢먣àë襬, â ª ª ª ®­  ¯®ª §ë¢ ¥â, ¢® ᪮«ìª® à § ¢ § ¤ ­­®© ¯®«®á¥ 㬥­ìè ¥âáï ª®íää¨æ¨¥­â ®âà ¦¥­¨ï. —¥¬ 㦥 ¯®«®á , ⥬ ¡®«ìè¥ ¢ë¨£àëè. ‡ ¬¥â¨¬, çâ® á¥à¥¤¨­  ¤¨ ¯ §®­  ®¯à¥¤¥«ï¥âáï ⥬, çâ® x = 0, â.¥. ' = 2 . ’®£¤  d = 4 , â.¥. ®â१ª¨ ¤®«¦­ë ¡ëâì ç¥â¢¥àâ좮«­®¢ë¬¨ ¤«ï á।­¥© ¤«¨­ë ¢®«­ë. „à㣮© ¢ à¨ ­â ¬®¦¥â ¡ëâì á«¥¤ãî騬. ‚롥६ ¬¥áâ­ë¥ ª®íää¨æ¨¥­âë ®âà ¦¥­¨ï â ª, ç⮡ë á㬬 à­ë© ª®íää¨æ¨¥­â ®âà ¦¥­¨ï ®¯¨á뢠«áï ¯®«¨­®¬®¬, ­ ¨¬¥­¥¥ 㪫®­ïî騬áï ®â ­ã«ï ¢ ¤ ­­®¬ ¤¨ ¯ §®­¥. „«ï í⮣® ¨á¯®«ì§ãîâáï ¯®«¨­®¬ë —¥¡ë襢  Tn(y): Tn(y) = 2n1;1 Tn (y); £¤¥ Tn (y) = cos(n arccos y). —¥¡ëè¥¢ë¬ ¡ë«® ¯®ª § ­®, çâ® ¯®«¨­®¬ Tn(y) ï¥âáï ¯®«¨­®¬®¬ á ª®íää¨æ¨¥­â®¬ ¯à¨ áâ à襬 ç«¥­¥, à ¢­ë¬ 1, ­ ¨¬¥­¥¥ 㪫®­ïî騬áï ®â ­ã«ï ¯à¨ ¨§¬¥­¥­¨¨ y ¢ ¯à¥¤¥« å ;1 < y < +1. ¨¦¥ ¯à¨¢¥¤¥­ë ¯¥à¢ë¥ ¯ïâì ¯®«¨­®¬®¢ Tn(y) (á¬. à¨á.):

T0 T1 T2 T3 T4

= = = = =

1;

y; 2y2 ; 1; 4y3 ; 3y; 8y4 ; 8y2 + 1:

®«¨­®¬ë 㤮¢«¥â¢®àïîâ ४ãà७⭮¬ã ᮮ⭮襭¨î

Tn+1 = 2Tny ; Tn;1:

168

Tn(x)

1

1 n= 2

n=

x 1

b = M Tn(tx):

n= 4

n=3

-1

’ ª ª ª x ¨§¬¥­ï¥âáï ¢ ¯à¥¤¥« å (; 1t ; 1t ), â® ­¥®¡å®¤¨¬® ¢¢¥á⨠¬ áèâ ¡­ë© ¬­®¦¨â¥«ì:

-1

‚¥«¨ç¨­  M ­ å®¤¨âáï â ª ¦¥, ª ª ¨ ¢ëè¥. à¨ x = 1 b = ;T . ®í⮬ã M = T;T(t) . Žâá ¯®«ãç ¥¬ n



jbj = ;T TTn((txt)) = ;T TTn ((txt)) : n n  ¨¡®«ì襥 §­ ç¥­¨¥ Tn (tx) ¢ à ¡®ç¥¬ ¤¨ ¯ §®­¥ à ¢­® 1, ¯®í⮬ã b¬ ªá = T;(Tt) : n Žâá ¢ë¨£àëè à ¢¥­ K = Tn(t). Œ®¦­® ¯®ª § âì, çâ® íâ®â ¢ë¨£àëè § ¬¥â­® ¯à¥¢ë蠥⠢먣àëè ¤«ï ¡¨­®¬¨ «ì­®£® ¯¥à¥å®¤  (¯à¨¬¥à­® ¢ 2n;1 à §). ¨¦¥ ¯à¨¢¥¤¥­ë ᮮ⭮襭¨ï ¬¥áâ­ëå ª®íää¨æ¨¥­â®¢ ®âà ¦¥­¨ï ¤«ï 祡ë襢᪮£® ¯¥à¥å®¤ : 1 1 1 1 2(1 ; 1=t2) 1 1 3(1 ; 1=t2) 3(1 ; 1=t2 ) 1; â.¥. ¡®«¥¥ ¯« ¢­ë© ¯¥à¥å®¤, 祬 ¡¨­®¬¨ «ì­ë©. Ž¯à¥¤¥«¨¬ ⥯¥àì ¤¨ ¯ §®­, â.¥. ¢¥«¨ç¨­ã t. „«¨­  áâ㯥­ìª¨ ®¯à¥¤¥«ï¥âáï 㣫®¬ ' = 2d :   ªà ïå ¤¨ ¯ §®­  (á¬. à¨á.) 169

'¬¨­ = 2d = arccos 1t ; ¬ ªá   2 '¬ ªá =  d = arccos ; 1t =  ; '¬¨­ : ¬¨­ ϕmin ϕmin

 §¤¥«¨¬ ®¤­® ᮮ⭮襭¨¥ ­  ¤à㣮¥: ¬ ªá =  ; '¬¨­ = q | ¬¨­ '¬¨­ § ¤ ­­®¥ ®â­®è¥­¨¥.

Žâá

'¬¨­ = 1 + q ; '¬ ªá = 1 q + q: Œ áèâ ¡­ë© ¬­®¦¨â¥«ì ¢ëà ¦ ¥âáï ç¥à¥§ q: 1 t = cos '1 =  : ¬¨­ cos 1 + q

Œ®¦­® ­ ©â¨ â ª¦¥ ¤«¨­ã áâ㯥­¨: '¬¨­ = 2d = 1 + q ; ¬ ªá ®âªã¤  ¬¨­q ¬¨­ ¬ ªá=¬¨­ d = 2(1¬ ªá + q) = 2(1 + q) = 2(1 + ¬ ªá=¬¨­) = 1  = 4áà ; =  2 1 +  1 ¬¨­ ¬ ªá £¤¥   1 = 1 1 + 1 : áà 2 ¬¨­ ¬ ªá 170

„«ï TEM-«¨­¨¨ 1 = fc , ¯®í⮬ã

d = 2(f c+ f ) = 4fc = 4áà ; ¬ ªá ¬¨­ áà

â.¥. ç¥â¢¥à⨠¢®«­ë ­  á।­¥© ç áâ®â¥. „«ï ¢®«­®¢®¤  ¤«¨­  d à ¢­  ç¥â¢¥à⨠¤«¨­ë ¢®«­ë ­  ­¥ª®â®à®© á।­¥© ¢®«­¥ ¤¨ ¯ §®­ . à¨¬¥à: ;T = 0:5; q = 2; n = 4. ’®£¤  t = 1  = 2: cos 3 ‚먣àëè (¤«ï 祡ë襢᪮£® ¯¥à¥å®¤ ) à ¢¥­

K4 = T4(2) = 97: Œ ªá¨¬ «ì­ë© ª®íää¨æ¨¥­â ®âà ¦¥­¨ï ¢ § ¤ ­­®¬ ¤¨ ¯ §®­¥ à ¢¥­ b = 097:5 = 0:005: ¨­®¬¨ «ì­ë© ¯¥à¥å®¤ ¯à¨ íâ¨å ãá«®¢¨ïå ¤ ¥â ¢ë¨£àëè K4 = 24 = 16. 5.

‚Ž‡“†„…ˆ… ‚Ž‹Ž‚Ž„Ž‚ ‡€„€›Œˆ ’ŽŠ€Œˆ

„® á¨å ¯®à à áᬠâਢ « áì ®¤­®à®¤­ ï § ¤ ç , ᮮ⢥âáâ¢ãîé ï à á¯à®áâà ­¥­¨î ᢮¡®¤­ëå ¢®«­. ® ¨­â¥à¥á ¯à¥¤áâ ¢«ï¥â â ª¦¥ § ¤ ç  ® ¢ë­ã¦¤¥­­ëå ª®«¥¡ ­¨ïå, ¨­ ç¥ £®¢®àï, § ¤ ç  ® ¢®§¡ã¦¤¥­¨¨ ¢®«­ ¢ ¢®«­®¢®¤¥.  §«¨ç îâ ¤¢  ᯮᮡ  ¢®§¡ã¦¤¥­¨ï: ¢®§¡ã¦¤¥­¨¥ § ¤ ­­ë¬¨ (áâ®à®­­¨¬¨) ⮪ ¬¨ ¨ ¢®§¡ã¦¤¥­¨¥ § ¤ ­­ë¬ ¯®«¥¬ ­  £à ­¨æ¥ (­¥®¤­®à®¤­ë¥ £à ­¨ç­ë¥ ãá«®¢¨ï). ‚®§¡ã¦¤¥­¨¥ ¢®«­®¢®¤®¢ ⮪ ¬¨ ¢áâà¥ç ¥âáï ¢ ‘‚—-í«¥ªâà®­¨ª¥, ª®£¤  ¢®«­®¢®¤ ¢®§¡ã¦¤ ¥âáï ¯®â®ª ¬¨ § à殮­­ëå ç áâ¨æ. ‚ ‘‚—-¬¥âਪ¥ ç é¥ ¢áâà¥ç ¥âáï ¢®§¡ã¦¤¥­¨¥ ¯®«¥¬ ­  £à ­¨æ¥. Š ª ¬ë 㢨¤¨¬ ­¨¦¥, ä®à¬ «ì­® ¢®§¡ã¦¤¥­¨¥ ç¥à¥§ £à ­¨æã ¬®¦¥â ¡ëâì ᢥ¤¥­® ª ¢®§¡ã¦¤¥­¨î ¯®¢¥àå­®áâ­ë¬¨ ⮪ ¬¨. 171

5.1.

‹¥¬¬  ‹®à¥­æ . «¥ªâà¨ç¥áª¨¥ ¨ ¬ £­¨â­ë¥ ⮪¨

ˆá室­ë¬¨ ¤«ï à¥è¥­¨ï § ¤ ç¨ ïîâáï ãà ¢­¥­¨ï Œ ªá¢¥«« 

~ rot H~ = j!"H~ + J: ~ rot E~ = ; j!H; ‚ â ª®¬ ¢¨¤¥ ãà ¢­¥­¨ï Œ ªá¢¥««  § ¯¨á ­ë ­¥á¨¬¬¥âà¨ç­®. Ž¤­ ª® ¯à¨ à¥è¥­¨¨ ªà ¥¢ëå § ¤ ç ç áâ® ®ª §ë¢ ¥âáï 㤮¡­®© ᨬ¬¥âà¨ç­ ï § ¯¨áì ãà ¢­¥­¨© Œ ªá¢¥«« . ‘¨¬¬¥âਧ æ¨ï ¤®á⨣ ¥âáï ¯ã⥬ ¢¢¥¤¥­¨ï ¬ £­¨â­ëå ⮪®¢. â¨ ⮪¨ ïîâáï 䨪⨢­ë¬¨; ®­¨ á«ã¦ â ¤«ï ⮣®, ç⮡ë, ®â¡à áë¢ ï ­¥ª®â®àãî ç áâì ¯®«ï ¢­¥ ¨­â¥à¥áãî饩 ­ á ®¡« áâ¨, § ¬¥­¨âì ¥£® ¬ £­¨â­ë¬¨ ⮪ ¬¨, ⥪ã騬¨ ­  £à ­¨æ¥. ‘¯®á®¡ â ª®© § ¬¥­ë ¬ë à áᬮâਬ ­¥áª®«ìª® ¯®§¤­¥¥. à¨ í⮬ ãá«®¢¨¨ ãà ¢­¥­¨ï Œ ªá¢¥««  ¬®¦­® § ¯¨á âì ¢ á«¥¤ãî饩 ä®à¬¥: rot E~ = ; j!H~ ; J~m ; rot H~ = j!"H~ + J~e ; £¤¥ J~m { ¯«®â­®áâì ¬ £­¨â­®£® ⮪ , J~e { ¯«®â­®áâì í«¥ªâà¨ç¥áª®£® ⮪ . ‘ãé¥á⢥­­ãî à®«ì ¢ § ¤ ç å ® ¢ë­ã¦¤¥­­ëå ª®«¥¡ ­¨ïå ¨£à ¥â «¥¬¬  ‹®à¥­æ .  ­¥¥ ¡ë«  ¯®«ã祭  «¥¬¬  ‹®à¥­æ  ¯à¨ ®âáãâá⢨¨ ⮪®¢.  áᬮâਬ ¢ëà ¦¥­¨¥ div (E~ (1)  H~ (2) ) ; div (E~ (2)  H~ (1) ) = H~ (2) rot E~ (1) ; E~ (1) rot H~ (2) ;

; H~ (1)rot E~ (2) + E~ (2) rot H~ (1) :

‡¤¥áì E~ (1) , H~ (1) ¨ E~ (2) , H~ (2) { ¤¢  à¥è¥­¨ï ãà ¢­¥­¨© Œ ªá¢¥«« , § ¯¨á ­­ëå ¢ëè¥ ¨ ᮮ⢥âáâ¢ãîé¨å ⮪ ¬ Je(1) ; Jm(1) ; Je(2) ; Jm(2) . “ç¨â뢠ï íâ® ¨ ¯®¤áâ ¢«ïï rot ¨§ ¤ ­­ëå ãà ¢­¥­¨©, ¯®«ã稬 h i div (E~ (1)  H~ (2) ) ; div (E~ (2)  H~ (1) ) = H~ (2) ;j!H~ (1) ; J~m(1) ;

h

h

i

h

i

; E~ (1) j!"E~ (2) + J~e(2) ; H~ (1) ;j!H~ (2) ; J~m(2) +

i

+ E~ (2) j!"E~ (1) + J~e(1) = E~ (2)
J~e(1) ; E~ (1)
J~e(2) + H~ (1)
J~m(2) ; H~ (2)
J~m(1) : „ ­­®¥ à ¢¥­á⢮ ¯à¥¤áâ ¢«ï¥â ᮡ®© «¥¬¬ã ‹®à¥­æ  ¢ ¤¨ää¥à¥­æ¨ «ì­®© ä®à¬¥. ˆ­â¥£à¨àãï ¯® ®¡ê¥¬ã V , ®£à ­¨ç¥­­®¬ã ¯®¢¥àå­®áâìî S , ¯®«ã稬 172

I =

Z V

S

(E~ (1)  H~ (2) ; E~ (2)  H~ (1) )
~n dS =

(E~ (2)
J~e(1) ; E~ (1)
J~e(2) + H~ (1)
J~m(2) ; H~ (2)
J~m(1) ) dV;

£¤¥ ¨­â¥£à « á«¥¢  ¢§ïâ ¯® ¯®¢¥àå­®á⨠S , ®£à ­¨ç¨¢ î饩 ®¡ê¥¬ V , ~n { ­ à㦭 ï ­®à¬ «ì ª ¯®¢¥àå­®áâ¨. â® à ¢¥­á⢮ ­ §ë¢ îâ «¥¬¬®©

‹®à¥­æ  ¢ ¨­â¥£à «ì­®© ä®à¬¥. à¥¤¯®«®¦¨¬, çâ® J~e(2) (~r) = ~a
(~r ; ~r0 ), J~m(2)  0, £¤¥ ~a { ­¥ª®â®àë© ¥¤¨­¨ç­ë© ¢¥ªâ®à, ~r { â®çª  ­ ¡«î¤¥­¨ï, ~r0 { â®çª  ¨á⮪  (¢­ãâਠ®¡ê¥¬ ). ’ ª¨¬ ®¡à §®¬, ¯®«ï E~ (2) , H~ (2) ¬ë ¯à¥¤¯®« £ ¥¬, ᮧ¤ ­­ë¬¨ í«¥ªâà¨ç¥áª¨¬ ¤¨¯®«¥¬, à á¯®«®¦¥­­ë¬ ¢ â®çª¥ ~r0 , í«¥ªâà¨ç¥áª¨© ¬®¬¥­â ª®â®à®£® à ¢¥­ ~a=j!. Ž¡®§­ ç¨¬ íâ® ¯®«¥ ¨­¤¥ªá®¬ \a": E~ (2) = E~ a (~r;~r0 ); H~ (2) = H~ a (~r;~r0 ): ˆ­¤¥ªá®¬ (1) ®¡®§­ ç¨¬ ¯®«ï ¨ ⮪¨ E~ (1) = E~ (~r); H~ (1) = H~ (~r); J~e(1) = J~e (~r); J~m(1) = J~m (~r):

‚ íâ¨å ¯à¥¤¯®«®¦¥­¨ïå ¨ á ¯à¨­ïâ묨 ®¡®§­ ç¥­¨ï¬¨ ¨§ ¢ëà ¦¥­¨ï ¤«ï «¥¬¬ë ‹®à¥­æ  (~n { ­ à㦭 ï ­®à¬ «ì) ¯®«ãç ¥¬

I S

(E~  H~ a + H~  E~ a ) ~n dS = ; E~ (~r0 )
~a +

®âªã¤  ­ å®¤¨¬

E~ (~r0 )
~a =

Zh V

Z

V

(J~e
E~ a ; J~m
H~ a ) dV;

i

J~e
E~ a (~r;~r0 ) ; J~m
H~ a (~r;~r0 ) dV +

I

+ (H~ a  E~ ; E~ a  H~ )~n dS = =

Z V

S

(J~e
E~ a ; J~m
H~ a ) dV +

Ih

i

S

(E~  ~n)
H~ a + (H~  ~n)
E~ a dS:

173

…᫨ áç¨â âì, çâ® ¯®«¥ ¢­¥ ¯®¢¥àå­®á⨠S à ¢­® ­ã«î, â® ­  á ¬®© ¯®¢¥àå­®á⨠᫥¤ã¥â ¢¢¥á⨠¯®¢¥àå­®áâ­ë¥ ⮪¨

K~ e = H~  ~n; K~ m = ; E~  ~n; £¤¥ ~n { ­ à㦭 ï ­®à¬ «ì. ’®£¤  ¯®«¥ ¢ â®çª¥ ~r0 ¢ëà ¦ ¥âáï ç¥à¥§

¯®¢¥àå­®áâ­ë¥ ⮪¨:

E~ (~r0 )
~a =

Z

V

(J~e
E~ a ; J~m
H~ a ) dV +

Z S

(K~ e
E~ a ; K~ m
H~ a ) dS:

„ ­­®¥ ᮮ⭮襭¨¥ ¯®ª §ë¢ ¥â, çâ® § ¤ ­­ë¥ ­  ¯®¢¥àå­®á⨠S â ­£¥­æ¨ «ì­ë¥ á®áâ ¢«ïî騥 E~ ¨ H~ ¬®£ãâ ¡ëâì § ¬¥­¥­ë ᮮ⢥âáâ¢ãî騬 ®¡à §®¬ à á¯à¥¤¥«¥­­ë¬¨ ⮪ ¬¨.  ©¤¥­­ë¥ ¢ëè¥ á®®â­®è¥­¨ï ¯®§¢®«ïîâ ­ ©â¨ ¯à®¥ªæ¨î ¢¥ªâ®à  E~ ­  ­ ¯à ¢«¥­¨¥ ¢¥ªâ®à  ~a. —â®¡ë ­ ©â¨ ¢¥ªâ®à­ãî äã­ªæ¨î E~ ¯®«­®áâìî, ­¥®¡å®¤¨¬® § ¤ âì âਠ®à⮣®­ «ì­ëå ¢¥ªâ®à  ~a, ­ ©¤ï ᮮ⢥âáâ¢ãî騥 à¥è¥­¨ï ¤«ï E~ a . ‘®¢®ªã¯­®áâì âà¥å ¢¥ªâ®à­ëå ä㭪権 E~ a (~r;~r0 ) ®¡à §ã¥â ⥭§®à­ãî äã­ªæ¨î, ª®â®àãî ­ §ë¢ îâ ⥭§®à®¬ ƒà¨­ . ‡­ ­¨¥ ⥭§®à  ƒà¨­  ¯®§¢®«ï¥â ­ ©â¨ à¥è¥­¨ï ãà ¢­¥­¨© Œ ªá¢¥««  ¯® § ¤ ­­ë¬ ®¡ê¥¬­ë¬ ¨ ¯®¢¥àå­®áâ­ë¬ ⮪ ¬. ‘«¥¤ã¥â ®â¬¥â¨âì, çâ® £à ­¨ç­ë¥ ãá«®¢¨ï ¤«ï ⥭§®à  ƒà¨­  ãáâ ­ ¢«¨¢ îâáï ¯à®¨§¢®«ì­®. à®é¥ ¢á¥£® ­ ©â¨ ¯®«¥ ¤¨¯®«ï ¢ ᢮¡®¤­®¬ ¯à®áâà ­á⢥, ­® ¢ í⮬ á«ãç ¥ ¤«ï ­ å®¦¤¥­¨ï à¥è¥­¨© ãà ¢­¥­¨© Œ ªá¢¥««  ¯® § ¤ ­­ë¬ ⮪ ¬ ¤®«¦­ë ¡ëâì § ¤ ­ë â ­£¥­æ¨ «ì­ë¥ á®áâ ¢«ïî騥 ­  ¯®¢¥àå­®á⨠S ª ª í«¥ªâà¨ç¥áª®£®, â ª ¨ ¬ £­¨â­®£® ¯®«¥© (¨«¨ ᮮ⢥âáâ¢ãî騥 ¯®¢¥àå­®áâ­ë¥ ⮪¨). …᫨ ¦¥ ¤«ï ⥭§®à  ƒà¨­  § ¤ âì ᯥ樠«ì­ë¥ £à ­¨ç­ë¥ ãá«®¢¨ï, â® § ¤ ç  ¬®¦¥â ¡ëâì ã¯à®é¥­ .  ¯à¨¬¥à, ­ ©¤¥¬ ¯®«¥ í«¥ªâà¨ç¥áª®£® ¤¨¯®«ï E~ a , H~ a á £à ­¨ç­ë¬ ãá«®¢¨¥¬

E~ at = 0 ­  S; â.¥. à ¢¥­á⢮ ­ã«î â ­£¥­æ¨ «ì­®© á®áâ ¢«ïî饩 í«¥ªâà¨ç¥áª®£® ¯®«ï ­  ¯®¢¥àå­®á⨠S (ª ª ­  ¨¤¥ «ì­® ¯à®¢®¤ï饩 ¯®¢¥àå­®áâ¨). ’®£¤  ®¤¨­ ¨§ ¯®¢¥àå­®áâ­ëå ¨­â¥£à «®¢,   ¨¬¥­­®, ᮤ¥à¦ é¨© í«¥ªâà¨ç¥áª¨© ¯®¢¥àå­®áâ­ë© ⮪, ®¡à é ¥âáï ¢ ­ã«ì. ˆáª®¬®¥ à¥è¥­¨¥ ãà ¢­¥­¨© Œ ªá¢¥««  ¢ í⮬ á«ãç ¥ ¢ëà ¦ ¥âáï ⮫쪮 ç¥à¥§ ¯®¢¥àå­®áâ­ë© ¬ £­¨â­ë© ⮪ ¨«¨ ç¥à¥§ â ­£¥­æ¨ «ì­ãî á®áâ ¢«ïîéãî í«¥ªâà¨ç¥áª®£® ¯®«ï ­  ¯®¢¥àå­®á⨠S : 174

Z

E~ (~r0 )
~a =

Z

V

= (J~e
E~ a ; J~m
H~ a ) dV ; V

5.2.

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11

I

(J~e
E~ a ; J~m
H~ a ) dV +

E

I S

S

(E~  ~n)
H~ a dS =

K~ m
H~ a dS:

„«ï ¨¤¥ «ì­® ¯à®¢®¤ïé¨å ¯®¢¥àå­®á⥩ â ­£¥­æ¨ «ì­ ï á®áâ ¢«ïîé ï í«¥ªâà¨ç¥áª®£® ¯®«ï à ¢­  ­ã«î ¨ ®â«¨ç ¥âáï ®â ­ã«ï «¨èì ­  ¯à®à¥§ ­­ëå ¢ á⥭ª å ®â¢¥àáâ¨ïå. ˆ§ ¯à¥¤ë¤ã饣® á«¥¤ã¥â, çâ® ®â¢¥àáâ¨ï ¬®£ãâ ¡ëâì § ¬¥­¥­ë ᮮ⢥âáâ¢ãî騬 à á¯à¥¤¥«¥­¨¥¬ ¬ £­¨â­ëå ⮪®¢ ­  á⥭ª å. …᫨ ®â¢¥àá⨥ ¯¥à¥à¥§ ¥â ¯®¢¥àå­®áâ­ë¥ ⮪¨, â® ¬¥¦¤ã ªà ï¬¨ ®â¢¥àáâ¨ï ¢®§­¨ª ¥â í«¥ªâà¨ç¥áª®¥ ¯®«¥ (á¬. à¨á.). ‡ ¬¥­ïï íâ® ¯®«¥ ¬ £­¨â­ë¬¨ ⮪ ¬¨, £à ­¨ç­ãî § ¤ çã ¬®¦­® § ¬¥­¨âì § ¤ ç¥© ® ­ å®¦¤¥­¨¨ ¯®«ï, ®¯à¥¤¥«ï¥¬®£® ⮪ ¬¨.

‚®§¡ã¦¤¥­¨¥ ¢®«­®¢®¤®¢ í«¥ªâà¨ç¥áª¨¬¨ ¨ ¬ £­¨â­ë¬¨ ⮪ ¬¨

à¥¤¯®«®¦¨¬, çâ® ¢á¥ ¨áâ®ç­¨ª¨ ¯®«ï ¢ ¢®«­®¢®¤¥ à á¯®«®¦¥­ë ¢ ®¡« á⨠z1 < z < z2 (á¬. à¨á.). ‚­¥ í⮩ ®¡« á⨠¯®«¥ 㤮¢«¥â¢®àï¥â ®¤­®à®¤­ë¬ ãà ¢­¥­¨ï¬ Œ ªá¢¥««  ¨ ®¤­®à®¤­ë¬ £à ­¨ç­ë¬ ãá«®¢¨ï¬ ­  á⥭ª å ¢®«­®¢®¤ . „«ï ⮣® çâ®¡ë § ¤ ç  à¥è « áì ®¤­®§­ ç­®, ¯®«ï á«¥¤ã¥â â ª¦¥ ¯®¤ç¨­¨âì ãá«®¢¨î ¨§«ã祭¨ï, ª®â®à®¥ âॡã¥â, ç⮡ë 00000000 11111111 00000000 11111111 á«¥¢  ®â z1 ¢®«­ë à á¯à®áâà ­ï00000000 11111111 00000000 11111111 «¨áì ⮫쪮 ¢ áâ®à®­ã ®âà¨æ â¥«ì00000000 11111111 ­ëå z ,   á¯à ¢  ®â z2 { ¢ áâ®à®­ã z =0 z z ¯®«®¦¨â¥«ì­ëå z . ¥è¥­¨¥ á¯à ¢  ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¢¨¤¥ 1

2

175

E~ ¯à =   á«¥¢ 

E~ «¥¢ =

X m

X m

Am E~ m e; m z ; H~ ¯à = Bm E~ ;m e m z ; H~ «¥¢ =

X m

Am H~ m e; m z ;

X m

Bm H~ ;m e m z :

‡¤¥áì E~ m , H~ m { ¢¥ªâ®à­ë¥ ä㭪樨 á¥ç¥­¨ï, ᮮ⢥âáâ¢ãî騥 à á¯à®áâà ­¥­¨î ¢ áâ®à®­ã +z ,   E~ ;m , H~ ;m { ¢ áâ®à®­ã ;z . ®¯¥à¥ç­ë¥ ¨ ¯à®¤®«ì­ë¥ ª®¬¯®­¥­âë íâ¨å ä㭪権 㤮¢«¥â¢®àïîâ ᮮ⭮襭¨ï¬ ~ m¯®¯ ; E~ ;¯®¯m = E~ m¯®¯ ; H~ ;¯®¯ m = ;H ¯à®¤ ~ ;¯à®¤ ~ m¯à®¤: E~ ;¯à®¤ m = ; E~ m ; H m = H Šà®¬¥ ⮣®, ®â¬¥â¨¬, çâ® ¯®¯¥à¥ç­ë¥ ¢¥ªâ®à­ë¥ ä㭪樨 ¬®¦­® áç¨â âì ¢¥é¥á⢥­­ë¬¨, ¯®áª®«ìªã ¨å ­ ç «ì­ ï ä §  ¬®¦¥â § ª«îç âìáï ¢ Am ¨ Bm . „«ï ®¯à¥¤¥«¥­¨ï ª®íää¨æ¨¥­â®¢ Am ¨ Bm ¬®¦¥â ¡ëâì ¨á¯®«ì§®¢ ­  «¥¬¬  ‹®à¥­æ . „«ï í⮣® ¢ «¥¬¬¥ ‹®à¥­æ  ¯®«®¦¨¬ E~ (1) ; H~ (1) { ¯®«ï, ¢®§¡ã¦¤ ¥¬ë¥ § ¤ ­­ë¬¨ ⮪ ¬¨ J~e ; J~m , E~ (2) = E~ ;n e n z ; H~ (2) = H~ ;n e nz ; J~e(2)  0; J~m(2)  0: ®¢¥àå­®áâì S ¢ë¡¥à¥¬, ª ª ¯®ª § ­® ­  à¨á. (S = S1 + S2 + S3 ). S1

S2

S3

11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111

n

z1

Z" X S1

;

m

ˆ­â¥£à « ¯® ¯®¢¥àå­®á⨠á⥭®ª (S3 ) ®¡à é ¥âáï ¢ ­ã«ì, â ª ª ª ­  ­¥© â ­£¥­æ¨ «ì­ë¥ á®áâ ¢«ïî騥 E~ (1) ¨ E~ (2) à ¢­ë ­ã«î. Žáâ îâáï ⮫쪮 ¨­â¥£à «ë ¯® S1 ¨ S2 :

n

z2

Bm (E~ ;m  H~ ;n )~z0 e m z e nz + +

X m

#

Bm (E~ ;n  H~ ;m )~z0 e m z e n z dS + 176

+

X S2

m

Am (E~ m  H~ ;n )~z0 e n z e; m z ;

=

Z V

;

X m

#

Am (E~ ;n  H~ m )~z0 e; m z e nz dS =

(E~ ;n J~e e n z ; H~ ;n J~m e nz ) dV:

“ç¨â뢠ï ãá«®¢¨ï ®à⮣®­ «ì­®á⨠¨ ᮮ⭮襭¨ï ¬¥¦¤ã ¯®¯¥à¥ç­ë¬¨ á®áâ ¢«ïî騬¨ ¯®«¥© ¯à®â¨¢®¯®«®¦­®£® ­ ¯à ¢«¥­¨ï à á¯à®áâà ­¥­¨ï, ­¥âà㤭® ¢¨¤¥âì, çâ® á« £ ¥¬ë¥, ᮤ¥à¦ é¨¥ ¨­â¥£à «ë ¯® S1 , ¨á祧 îâ (á« £ ¥¬ë¥ á m 6= n | ¢á«¥¤á⢨¥ ®à⮣®­ «ì­®áâ¨, á« £ ¥¬ë¥ á m = n | ¢§ ¨¬­® ã­¨ç⮦ îâáï). ‘« £ ¥¬ë¥ á ¨­â¥£à « ¬¨ ¯® S2 ᪫ ¤ë¢ îâáï, â ª ª ª (E~ n  H~ ;n )~z0 = ;(E~ ;n  H~ n )~z0 . ‚ १ã«ìâ â¥ ¯®«ãç ¥¬

Z

Z

S

V

;2 An (E~ n  H~ n )~z0 dS =

(E~ ;n J~e e nz ; H~ ;n J~m e nz ) dV:

ˆ­â¥£à « á«¥¢  ¡¥à¥âáï ¯® ¯à®¨§¢®«ì­®¬ã á¥ç¥­¨î ¢®«­®¢®¤ , â ª ª ª ¯®¤ë­â¥£à «ì­®¥ ¢ëà ¦¥­¨¥ ­¥ § ¢¨á¨â ®â z . ˆ­â¥£à «

Z S

(E~ n  H~ n )~z0 dS

®¯à¥¤¥«ï¥âáï ­®à¬¨à®¢ª®©. ’ ª ª ª H~ n¯®¯ = Z1 (~z0  E~ n ¯®¯), â® n R R 1 ¯®¯ 2 ~ ~ ~ (En  Hn )~z0 = Z (E ) , ®âªã¤  á«¥¤ã¥â, çâ® (E~ n H~ n )~z0 dS = Z1 (E~ n¯®¯ )2 dS . n nS S …᫨ ¯à¨­ïâì ­®à¬¨à®¢ªã ¤«ï ¯®¯¥à¥ç­ëå ä㭪権

Z

R â® (E~ n  H~ n )~z0 dS = Z1 . n S ’®£¤ 

S

(E~ n¯®¯ )2 dS = 1;

177

Z

An = ; 12 Zn
(E~ ;n J~e e n z ; H~ ;n J~m e n z ) dV: V

€­ «®£¨ç­® ¬®¦­® ­ ©â¨ ¢¥«¨ç¨­ë Bn : Z Bn = ; 12 Zn (E~n J~e e; n z ; H~ n J~m e; n z ) dV: V

®«ã祭­ë¥ ä®à¬ã«ë á¯à ¢¥¤«¨¢ë ¤«ï ¯®«¥© ¢­¥ ®¡« áâ¨, ¢ ª®â®à®© ¨¬¥îâáï ⮪¨. ‚ ®¡« áâ¨, § ­ï⮩ ⮪ ¬¨, ­¥®¡å®¤¨¬® ãç¨â뢠âì ¯®«ï, ᮧ¤ ¢ ¥¬ë¥ ­¥¯®á।á⢥­­® § à鸞¬¨ ¨ ⮪ ¬¨ (â ª ª ª ¢ í⮩ ®¡« á⨠div E~ 6= 0). …᫨ ¯®«ï ¢ ¢®«­®¢®¤¥ ¢®§¡ã¦¤ îâáï ⮫쪮 í«¥ªâà¨ç¥áª¨¬¨ ⮪ ¬¨, â®

Z 1 An = ; 2 Zn VZ 1 Bn = ; 2 Zn V

E~;n J~e e nz ) dV; E~n J~e e; n z dV:

ˆ§ ¯®«ã祭­ëå ä®à¬ã« ¢¨¤­®, çâ® ­ ¨¡®«¥¥ íä䥪⨢­® ¢®§¡ã¦¤ îâáï ⥠¢®«­ë, ¤«ï ª®â®àëå à á¯à¥¤¥«¥­¨¥ í«¥ªâà¨ç¥áª¨å ⮪®¢, ¨å ­ ¯à ¢«¥­¨¥ ¨ ä §  ᮢ¯ ¤ îâ á à á¯à¥¤¥«¥­¨¥¬ í«¥ªâà¨ç¥áª®£® ¯®«ï. €­ «®£¨ç­® í⮬㠭 ¯à ¢«¥­¨¥ 饫¨ ¢ á⥭ª¥ ¢®«­®¢®¤  ¤«ï íä䥪⨢­®£® ¢®§¡ã¦¤¥­¨ï ¤®«¦­® ᮢ¯ ¤ âì á ­ ¯à ¢«¥­¨¥¬ ¬ £­¨â­®£® ¯®«ï (â ª ª ª ¬ £­¨â­ë© ⮪ â¥ç¥â ¢¤®«ì 饫¨) (á¬. à¨á.). ’ ª ª ª ⮪¨ ¢ á⥭ª å ¤® ¯à®à¥§ ­¨ï 饫¨ ¯¥à¯¥­¤¨ªã«ïà­ë ¬ £­¨â­®¬ã ¯®«î, â® é¥«ì ¤®«¦­  ¯¥à¥à¥§ âì «¨­¨¨ ⮪®¢ ¢ á⥭ª å (á¬. à¨á.).

Et

Km

‚®§¡ã¦¤¥­¨¥ ¢®«­®¢®¤®¢ ®áãé¥á⢫ïîâ â ª¦¥ á ¯®¬®éìî  ­â¥­­ë ¨ ¯¥â«¨. 178

€­â¥­­  (èâëàì) ¯à¥¤áâ ¢«ï¥â ᮡ®© áâ¥à¦¥­ì, ¯® ª®â®à®¬ã ⥪ãâ ⮪¨, ¯®íâ®¬ã  ­â¥­­ã ¯®¬¥é îâ ¢ ¬ ªá¨¬ã¬ í«¥ªâà¨ç¥áª®£® ¯®«ï (á¬. à¨á.). ¥â«ï íª¢¨¢ «¥­â­  ¬ £­¨â­®¬ã ¤¨¯®«î. …¥ ­¥®¡å®¤¨¬® ¯®¬¥é âì ¢ ¬ ªá¨¬ã¬¥ ¬ £­¨â­®£® ¯®«ï, ¯à¨ç¥¬ ¯«®áª®áâì ¯¥â«¨ ¤®«¦­  ¡ëâì ¯¥à¯¥­¤¨ªã«ïà­  ¬ £­¨â­®¬ã ¯®«î (á¬. à¨á.). ™¥«ì ¯à®à¥§ îâ ¯¥à¯¥­¤¨ªã«ïà­® ⮪ ¬ ¢ á⥭ª¥, £¤¥ ⮪¨ ¬ ªá¨¬ «ì­ë. ¥à¥¤ª® ¢®«­®¢®¤ ¡ë¢ ¥â § ª®à®ç¥­ á ®¤­®© áâ®à®­ë. ‚ í⮬ á«ãç ¥ á«¥¢  ®â S1 ¨¬¥îâáï ¢®«­ë, à á¯à®áâà ­ïî騥áï ¢ ®¡¥ áâ®à®­ë, 㤮¢«¥â¢®àïî騥 £à ­¨ç­ë¬ ãá«®¢¨ï¬ ­  § ¬ëª î饩 ¯¥à¥¬ë窥:

E~ «¥¢ = H~ «¥¢ =

X

m X m

Bm (E~ ;m e m z ; E~ m e; m z ); Bm (H~ ;m e m z ; H~ m e; m z ):

„ ­­®¥ ¯®«¥ 㤮¢«¥â¢®àï¥â £à ­¨ç­®¬ã ãá«®¢¨î ­  ª®à®âª®§ ¬ëª î饩 ¯¥à¥¬ë窥 ¯à¨ z = 0. ‚ í⮬ á«ãç ¥ ¯à¨ ¢ëç¨á«¥­¨¨ An á«¥¤ã¥â ¯®«®¦¨âì

E~ (2) = E~ ;n e nz ; E~ n e; nz ; H~ (2) = H~ ;n e n z ; H~ n e; nz : ¥âà㤭® ¢¨¤¥âì, çâ® ¢ í⮬ á«ãç ¥ ¨­â¥£à « ¯® S1 ®¡à é ¥âáï ¢ ­ã«ì. ˆ­â¥£à « ¯® S2 ¨¬¥¥â â®â ¦¥ ¢¨¤, ¯®í⮬ã Zh An = ; 12 Zn (E~ ;n e n z ; E~ n e; n z )
J~e ; V

i

; (H~ ;n e n z ; H~ ;n e; n z )
J~m dV: ãáâì, ­ ¯à¨¬¥à, ¢®«­®¢®¤ ¢®§¡ã¦¤ ¥âáï  ­â¥­­®©, à á¯®«®¦¥­­®© ­  à ááâ®ï­¨¨ l ®â § ª®à ç¨¢ î饩 ¯¥à¥¬ë窨. ’®£¤  ¬®¦­® áç¨â âì, ç⮠⮪¨ J~e á®á।®â®ç¥­ë ¢ ¯«®áª®á⨠z = l ¨ ¯®í⮬ã ( n = j n ) 179

Z

An = ; 21 Zn (E~ ;n e j n l ; E~n e;j n l )
J~e dV: V

’ ª ª ª ⮪¨ ⥪ãâ ⮫쪮 ¢ ¯®¯¥à¥ç­®© ¯«®áª®áâ¨, â® ¨£à îâ ஫ì ⮫쪮 ¯®¯¥à¥ç­ë¥ ª®¬¯®­¥­âë En ; ¯®í⮬㠬®¦­® § ¬¥­¨âì En ­  E;n . ’®£¤  ¯®«ã稬 Z An = ; 12 Zn (e j n l ; e;j n l )
E~ ;n
J~e dV = V

Z

= ; jZn sin n l E~ ;n
J~e dV: V

Žç¥¢¨¤­®, çâ® ¢®§¡ã¦¤¥­¨¥ ¡ã¤¥â ­ ¨¡®«¥¥ íä䥪⨢­® ¯à¨ n l = 2 , â.¥. l = 4 . …᫨ ¯à®¨­â¥£à¨à®¢ âì ¯«®â­®áâì ⮪  ¯® á¥ç¥­¨î  ­â¥­­ë, â® ¯®«ã稬 ⮪ I~e . ’®£¤ 

Z

An = ; jZn sin n l E~ ;n I~e dl; l

£¤¥ ¨­â¥£à « ¡¥à¥âáï ¢¤®«ì  ­â¥­­ë. ‚ ª ç¥á⢥ ¯à¨¬¥à  à áᬮâਬ ¢®§¡ã¦¤¥­¨¥ ¯àאַ㣮«ì­®£® ¢®«­®¢®¤  èâë६, ¢ ª®â®à®¬ § ¤ ­® à á¯à¥¤¥«¥­¨¥ ⮪  (á¬. à¨á.). ®à¬¨à®¢ ­­ ï äã­ªæ¨ï E10 ¨¬¥¥â ¢¨¤ a b h

r

2 sin x : ab a ‚®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ à ¢­®

E10y =

Z10 = ! =

x0

£¤¥ 180

p1 ; (=2a)2 ;

 =

r

":

®¤áâ ¢«ïï E10y ¢ ¨­â¥£à «, ¯®«ã稬 ¢¥«¨ç¨­ã A10 :

r

2 sin x0 Z I~ d~l: A10 = ; j p  sin l 2 ab e a 1 ; (=2a) 0

Ž¡®§­ ç¨¬

Zh

h

I~e d~l = I0 h¤ ;

0 £¤¥ I0 { ⮪ 㠮᭮¢ ­¨ï èâëàï, h¤ { ¤¥©áâ¢ãîé ï ¢ëá®â  èâëàï. ’®£¤ 

A10 =

r  sin l 2 sin x0 I h : p ;j 2 ab a 0 ¤ 1 ; (=2a)

«¥ªâà¨ç¥áª®¥ ¯®«¥ à ¢­® 2 sin x0 I h sin x : E = A10 E10 = ; j p  sin l 2 ab a 0 ¤ a 1 ; (=2a) 5.3.

„à㣮© ¢ë¢®¤ ä®à¬ã«ë ¢®§¡ã¦¤¥­¨ï í«¥ªâà¨ç¥áª¨¬¨ ¨ ¬ £­¨â­ë¬¨ ⮪ ¬¨

‡ ¤ ­ë ⮪¨ ¢ ¢®«­®¢®¤¥ ¢ ®¡« á⨠z1 < z < z2 (á¬. à¨á.). S1

n

S2

S3

11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111

z1

z2

z

S

®«ï ¬®¦­® § ¯¨á âì á«¥¤ãî騬 ®¡à §®¬: 181

n

1. ‚ á¥ç¥­¨¨ S

Xn

E~ (x; y; z ) =

m

o

Am (z )
E~ m¯®¯(x; y) + A;m (z )
E~ ;¯®¯m (x; y) + E~ z :

2. ‚ á¥ç¥­¨¨ S1

E~ (x; y; z ) = 3. ‚ á¥ç¥­¨¨ S2

E~ (x; y; z ) =

X m

A;m (z )
E~ ;¯®¯m (x; y) + E~ z :

X m

Am (z )
E~ m¯®¯ (x; y) + E~ z :

‡¤¥áì E~ m¯®¯, E~ ;¯®¯ m { ¯®¯¥à¥ç­ë¥ ᮡá⢥­­ë¥ ä㭪樨 ¢®«­®¢®¤  ¯à¨ à á¯à®áâà ­¥­¨¨ ᮮ⢥âá⢥­­® ¢ ¯®«®¦¨â¥«ì­®¬ ¨ ®âà¨æ â¥«ì­®¬ ­ ¯à ¢«¥­¨¨ ®á¨ z . €­ «®£¨ç­® ¤«ï ¬ £­¨â­®£® ¯®«ï: 1. ‚ á¥ç¥­¨¨ S

H~ (x; y; z ) =

X m

~ z: Am (z )
H~ m¯®¯(x; y) + A;m (z )
H~ ;¯®¯ m (x; y ) + H

2. ‚ á¥ç¥­¨¨ S1

H~ (x; y; z ) = 3. ‚ á¥ç¥­¨¨ S2

H~ (x; y; z ) =

X m

~ z: A;m (z )
H~ ;¯®¯ m (x; y ) + H

X m

Am (z )
H~ m¯®¯(x; y) + H~ z :

Š®íää¨æ¨¥­âë Am ¨ A;m ¬®£ãâ ¡ëâì ­ ©¤¥­ë á ¯®¬®éìî «¥¬¬ë ‹®à¥­æ 

I S

(E~ (1)  H~ (2) ; E~ (2)  H~ (1) )
~n dS = 182

Z

V

(E~ (2)
J~(1) ; E~ (1)
J~(2) ) dV:

‡¤¥áì S { ¯à®¨§¢®«ì­ ï § ¬ª­ãâ ï ¯®¢¥àå­®áâì, ®£à ­¨ç¨¢ îé ï ®¡ê¥¬ V , ~n { ¢­¥è­ïï ®â­®á¨â¥«ì­® ¯®¢¥àå­®á⨠S ­®à¬ «ì, E~ (1) , H~ (1) , E~ (2) , H~ (2) { ¯®«ï, 㤮¢«¥â¢®àïî騥 ãà ¢­¥­¨ï¬ Œ ªá¢¥««  á ⮪ ¬¨ J~(1) , J~(2) . ãáâì E~ (1) , H~ (1) { ¨áª®¬ë¥ ¯®«ï. 1: E~ (2) = E~ ;n
e n z ; H~ (2) = H~ ;n
e n z ; J~(2) = 0: ‡¤¥áì E~ ;n
e n z ¨ H~ ;n
e n z { ᮡá⢥­­ë¥ ä㭪樨 n-© ¬®¤ë ¯ãá⮣® ¢®«­®¢®¤ , ᮮ⢥âáâ¢ãî騥 ®âà¨æ â¥«ì­®¬ã ­ ¯à ¢«¥­¨î à á¯à®áâà ­¥­¨ï. Ž¡« áâì ¨­â¥£à¨à®¢ ­¨ï { ãç á⮪ ¢®«­®¢®¤  ®â S ¤® S1 . ˆ­â¥£à « ¯® S3 ®¡à é ¥âáï ¢ ­ã«ì ª ª ¯à¨ ®âáãâá⢨¨, â ª ¨ ¯à¨ ­ «¨ç¨¨ ¯®â¥àì (á¬. [1], á.431). ®í⮬ã á«¥¢  ¢ «¥¬¬¥ ‹®à¥­æ  ®áâ îâáï «¨èì ¨­â¥£à «ë ¯® S1 ¨ S , ¯à¨ç¥¬ E~ z ¨ H~ z ¨á祧 îâ:

;

Z (X

S1

+

;

m

A;m (E~ ;¯®¯m  H~ ;n )~z0 e nz ;

Z Xn m

ZS X n S m

X m

)

z A;m (E~ ;n  H~ ;¯®¯ m )~z0 e n dS +

o

Am (E~ m¯®¯  H~ ;n )~z0 e n z + A;m (E~ ;¯®¯m  H~ ;n )~z0 e n z dS ;

o

z Am (E~ ;n  H~ m¯®¯)~z0 e nz + A;m (E~ ;n  H~ ;¯®¯ m )~z0 e n dS =

ˆ­â¥£à «ë, ᮤ¥à¦ é¨¥ ¯à®¨§¢¥¤¥­¨ï ᮡá⢥­­ëå ä㭪権 á à §«¨ç­ë¬¨ ¨­¤¥ªá ¬¨ ¨á祧 î⠢᫥¤á⢨¥ ®à⮣®­ «ì­®áâ¨. ‚ १ã«ìâ â¥, ¢ á㬬 å ¯®¤ ¨­â¥£à « ¬¨ ®áâ ¥âáï ¯® ®¤­®¬ã á« £ ¥¬®¬ã = ; +

;

Zn

S1

o

A;n (E~ ;n  H~ ;n )~z0 e nz ; A;n (E~ ;n  H~ ;n )~z0 e n z dS +

Zn

o

ZS n

o

S

An (E~ n  H~ ;n )~z0 e nz + A;n (E~ ;n  H~ ;n )~z0 e n z dS ;

An (E~ ;n  H~ n )~z0 e n z + A;n (E~ ;n  H~ ;n )~z0 e nz dS =

‚ ¨­â¥£à «¥ ¯® S1 á« £ ¥¬ë¥ ¢§ ¨¬­® ã­¨ç⮦ îâáï, ¢ ¨­â¥£à «¥ ¯® S ã­¨ç⮦ îâáï ¢â®à®¥ ¨ ç¥â¢¥à⮥ á« £ ¥¬ë¥. ¥à¢®¥ ¨ âà¥âì¥ á« £ ¥¬ë¥ 183

¢ ¨­â¥£à «¥ ¯® S , ­ ®¡®à®â, ᪫ ¤ë¢ îâáï ¯à¨ § ¬¥­¥ ;n ­  n á ®¤­®¢à¥¬¥­­®© ¯¥à¥¬¥­®© §­ ª®¢ ¢ ᮮ⢥âá⢨¨ á ¯à ¢¨«®¬ H;n = ;Hn , E;n = En

Z

Z

S

V (z1 ;z)

= ; 2An
e nz (E~ n  H~ n )~z0 dS = Žâá ­ å®¤¨¬

An (z ) =

J~
E~ ;n e n z dV:

R J~
E~ e nz dV ;n 1 V (z ;z) ;

z n ;2e R(E~  H~ )~z dS : 1

n

S

n 0

2. ãáâì E (2) = En e; nz , H (2) = Hn e; nz . Ž¡« áâì ¨­â¥£à¨à®¢ ­¨ï { ãç á⮪ ¢®«­®¢®¤  ®â S ¤® S2 . €­ «®£¨ç­® ¯®«ãç ¥¬

A;n (z ) =

R J~
E~ e; nz dV n 1 V (z;z )

z n ;2e R(E~  H~ )~z dS : 2

S

n

n 0

à®¤®«ì­ë¥ ª®¬¯®­¥­âë ­ å®¤ïâáï á ¯®¬®éìî ãà ¢­¥­¨© Œ ªá¢¥«« 

~ rot H~ = j!"E~ + J: ~ rot E~ = ; j!H; ˆ§ ¢â®à®£® ãà ¢­¥­¨ï 1 rot H~ ; 1 J~ = 1 rot H~ ¯®¯ ; 1 J~ = Ez = j!" z j!" z j!" z j!" z 1 X nA (z )rot H~ ¯®¯ + A (z )rot H~ ¯®¯o ; 1 J~ : = j!" m z m ;m z ;m j!" z m

’ ª®¥ ¢ëà ¦¥­¨¥ ¯®«ãç ¥âáï ¯®â®¬ã, çâ® rot z H~ m¯®¯ ­¥ ᮤ¥à¦¨â ¤¨ää¥à¥­æ¨à®¢ ­¨ï ¯® z , ¯®í⮬ã Am (z ) ¨ A;m (z ) ¢ë室ïâ ¨§-¯®¤ §­ ª  rot z . ‘®¡á⢥­­ë¥ ä㭪樨 㤮¢«¥â¢®àïîâ ãà ¢­¥­¨î rot (H~ m
e; m z ) = j!"
E~ m e; m z ; ®âªã¤  184

rot z H~ m = j!"
E~ mz : ’ ª¨¬ ®¡à §®¬,

X

1 J: [Am (z )Emz + A;m (z )Emz ] ; j!" z m ‘ª« ¤ë¢ ï ¯®¯¥à¥ç­®¥ ¨ ¯à®¤®«ì­®¥ ¯®«ï, ­ å®¤¨¬ ¯®«­®¥ ¯®«¥:

Ez =

E~ (x; y; z ) =

Xh

i

1 J~ : Am (z )
E~ m + A;m (z )
E~ ;m ; j!" z m

€­ «®£¨ç­® ¤«ï ¬ £­¨â­®£® ¯®«ï ¯®«ãç ¥¬

H~ (x; y; z ) = 6.

6.1.

Xh m

i

Am (z )
H~ m + A;m (z )
H~ ;m :

…‡Ž€’Ž›

‘¢®¡®¤­ë¥ ª®«¥¡ ­¨ï ¢ १®­ â®à å

à¨ ®ç¥­ì ª®à®âª¨å ¢®«­ å ª®«¥¡ â¥«ì­ë¥ á¨á⥬ë, á®áâ®ï騥 ¨§ ¨­¤ãªâ¨¢­®á⥩ ¨ ª®­¤¥­á â®à®¢, ®áãé¥á⢨¬ë á âà㤮¬ ¢á«¥¤á⢨¥ ¬ «ëå à §¬¥à®¢. Šà®¬¥ ⮣®, ¢ ­¨å áâ ­®¢ïâáï ᫨誮¬ ¡®«ì訬¨ ¯®â¥à¨, ¯®í⮬㠢 í⮩ ®¡« á⨠ç áâ®â ®á­®¢­ãî à®«ì ¨£à îâ ®¡ê¥¬­ë¥ १®­ â®àë, ¯à¥¦¤¥ ¢á¥£® § ªàëâë¥. à®á⥩訩 ®¡ê¥¬­ë© १®­ â®à ¯à¥¤áâ ¢«ï¥â ᮡ®© ®¡ê¥¬, ®£à ­¨ç¥­­ë© á® ¢á¥å áâ®à®­ ¬¥â ««¨ç¥áª®© ®¡®«®çª®©. ‚ ¯¥à¢®¬ ¯à¨¡«¨¦¥­¨¨ ¬®¦­® ¯à¥¤¯®«®¦¨âì, çâ® á⥭ª¨ ïîâáï ¨¤¥ «ì­® ¯à®¢®¤ï騬¨. …᫨ ¢ â ª®¬ ®¡ê¥¬¥ ¢®§¡ã¦¤¥­ë í«¥ªâ஬ £­¨â­ë¥ ª®«¥¡ ­¨ï, â® ®­¨ ¡ã¤ãâ ¯®áâ®ï­­® â ¬ áãé¥á⢮¢ âì. 6.2.

‘®¡á⢥­­ë¥ ä㭪樨 ¨ ᮡá⢥­­ë¥ §­ ç¥­¨ï. Žà⮣®­ «ì­®áâì ᮡá⢥­­ëå ä㭪権

‘¢®¡®¤­ë¥ ª®«¥¡ ­¨ï ¢ १®­ â®à¥ ®¯¨á뢠îâáï ®¤­®à®¤­ë¬¨ ãà ¢­¥­¨ï¬¨ Œ ªá¢¥««  185

~

~

rot E~ = ;  @@tH ; rot H~ = " @@tE : à¨ í⮬ ¯®«ï E~ ¨ H~ ¤®«¦­ë 㤮¢«¥â¢®àïâì £à ­¨ç­ë¬ ãá«®¢¨ï¬ ­  ¯®¢¥àå­®á⨠¬¥â ««  (¨¤¥ «ì­® ¯à®¢®¤ï饣®)

~n  E~ = 0; â® ¥áâì E~ t = 0; ~n
H~ = 0; â® ¥áâì Hn = 0; £¤¥ n { ¢¥ªâ®à ­®à¬ «¨. ã¤¥¬ ¨áª âì à¥è¥­¨¥ ãà ¢­¥­¨© ¢ ¢¨¤¥

E~ = E~ (x; y; z ) e j!t ; H~ = H~ (x; y; z ) e j!t ; £¤¥ ! { ¯®ª  ­¥¨§¢¥áâ­ ï ¢¥«¨ç¨­ . ®¤áâ ¢«ïï í⨠¯®«ï ¢ ãà ¢­¥­¨¥ Œ ªá¢¥«« , ¯®«ã稬

~ rot H~ = j!"E: ~ rot E~ = ; j!H;

ˆáª«îç ï H~ ¨ ãç¨â뢠ï, çâ® div E~ = 0, ¯®«ã稬 ãà ¢­¥­¨¥ ¤«ï E~
E~ + k2 E~ = 0; £¤¥ k2 = !2 ",
{ « ¯« á¨ ­, ¯à¨¬¥­ï¥¬ë© ª ¤¥ª àâ®¢ë¬ á®áâ ¢«ïî騬 E~ . ‚ ®¡é¥¬ á«ãç ¥ ¯à®¨§¢®«ì­ëå ª®®à¤¨­ â ãà ¢­¥­¨¥ ¨¬¥¥â ¢¨¤ rot rot E~ ; k2 E~ = 0: „ ­­®¥ ãà ¢­¥­¨¥ ¤®«¦­® ¡ëâì à¥è¥­® ¯à¨ 㪠§ ­­ëå à ­¥¥ £à ­¨ç­ëå ãá«®¢¨ïå. ¥âਢ¨ «ì­®¥ à¥è¥­¨¥ áãé¥áâ¢ã¥â «¨èì ¯à¨ ­¥ª®â®àëå ®¯à¥¤¥«¥­­ëå §­ ç¥­¨ïå km2 ¯®áâ®ï­­®© k2 , ¢å®¤ï饩 ¢ ãà ¢­¥­¨¥. â¨ §­ ç¥­¨ï ­ §ë¢ îâ ᮡá⢥­­ë¬¨. ‘®®â¢¥âáâ¢ãî騥 ¨¬ à¥è¥­¨ï E~ m (x; y; z ) ¨ H~ m (x; y; z ) ­ §ë¢ îâ ᮡá⢥­­ë¬¨ äã­ªæ¨ï¬¨. €­ «®£¨ç­®¬ã ãà ¢­¥­¨î 㤮¢«¥â¢®àï¥â ¬ £­¨â­®¥ ¯®«¥: rot rot H~ ; k2 H~ = 0: 186

‘®¡á⢥­­ë¥ §­ ç¥­¨ï, ª ª ¬®¦­® ¯®ª § âì, á®áâ ¢«ïîâ ¡¥áª®­¥ç­ãî áç¥â­ãî ¯®á«¥¤®¢ â¥«ì­®áâì, ­¥ ¨¬¥îéãî ª®­¥ç­ëå â®ç¥ª á£ã饭¨ï. ®ª ¦¥¬, ç⮠ᮡá⢥­­ë¥ §­ ç¥­¨ï ¢¥é¥á⢥­­ë ¨ ¯®«®¦¨â¥«ì­ë. „«ï í⮣® ¢ëç¨á«¨¬ á«¥¤ãî饥 ¢ëà ¦¥­¨¥: div (H~ m  rot H~ m ) = rot H~ m
rot H~ m ; H~ m
rot rot H~ m = = jrot H~ m j2 ; km2
jH~m j2 :

ˆ­â¥£à¨à®¢ ­¨¥ ¯® ®¡ê¥¬ã १®­ â®à  ¤ ¥â

k2

Z

Z

V

V

~ m j2 dV = m jH

jrot H~ m j2 dV ;

I S

(H~ m  rot H~ m )~n dS:

‚â®à®© ¨­â¥£à « á¯à ¢  ®¡à é ¥âáï ¢ ­ã«ì ¢ ᨫ㠣࠭¨ç­ëå ãá«®¢¨© ¨ ⮣¤  R jrot H~ j2 dV m : km2 = V R ~ 2 V

jHm j dV

®«ã祭­®¥ ᮮ⭮襭¨¥ ¤®ª §ë¢ ¥â, ç⮠ᮡá⢥­­ë¥ §­ ç¥­¨ï ¢¥é¥á⢥­­ë ¨ ¯®«®¦¨â¥«ì­ë. ’ ª ª ª km2 = !2 ", â® !2 > 0 ¨ ! { ¢¥é¥á⢥­­®¥ ç¨á«®. ’ ª¨¬ ®¡à §®¬, à¥è¥­¨ï ãà ¢­¥­¨ï Œ ªá¢¥««  ¤«ï १®­ â®à  á ¨¤¥ «ì­® ¯à®¢®¤ï騬¨ á⥭ª ¬¨ ¯à¥¤áâ ¢«ïîâ ᮡ®© ­¥§ âãå î騥 í«¥ªâ஬ £­¨â­ë¥ ª®«¥¡ ­¨ï. — áâ®âë ¨ ¤«¨­ë ¢®«­ íâ¨å ª®«¥¡ ­¨© ®¯à¥¤¥«ïîâáï á«¥¤ãî騬 ®¡à §®¬: ! = ck = km ;  = 2 : m

p"

m

m

km

Ž¤­®¬ã ¨ ⮬㠦¥ ᮡá⢥­­®¬ã §­ ç¥­¨î km2 ¬®¦¥â ᮮ⢥âá⢮¢ âì ­¥áª®«ìª® ᮡá⢥­­ëå ä㭪権. ’ ª¨¥ à¥è¥­¨ï ­ §ë¢ îâ ¢ë஦¤¥­­ë¬¨. ‘®¡á⢥­­ë¥ ä㭪樨 ®¡« ¤ îâ ᢮©á⢮¬ ®à⮣®­ «ì­®á⨠¢ ⮬ á¬ëá«¥, çâ® ¯à¨ km2 6= kn2 ¨¬¥îâ ¬¥áâ® à ¢¥­á⢠

Z

V

E~ m
E~ n dV = 0;

Z

V

H~ m
H~ n dV = 0;

¯à¨ç¥¬ ¨­â¥£à «ë ¡¥àãâáï ¯® ¢á¥¬ã ®¡ê¥¬ã १®­ â®à . 187

—⮡ë íâ® ¯®ª § âì, § ¯¨è¥¬ ¤¢  ãà ¢­¥­¨ï, ª®â®àë¬ ã¤®¢«¥â¢®àïîâ E~ m ¨ E~ n : rot rot E~ m ; km2 E~ m = 0; rot rot E~ n ; kn2 E~ n = 0: ‚ëç¨á«¨¬ ¢ëà ¦¥­¨¥ div (E~ n  rot E~ m ) ; div (E~ m  rot E~ n ) = rot E~ m
rot E~ n ; E~ n
rot rot E~ m ; ; rot E~ n
rot E~ m + E~ m
rot rot E~ n = E~ m
rot rot E~ n ; E~ n
rot rot E~ m : “ç¨â뢠ï ãà ¢­¥­¨ï, ª®â®àë¬ ã¤®¢«¥â¢®àïîâ E~ m ¨ E~ n , ¯®«ã稬 div (E~ n  rot E~ m ) ; div (E~ m  rot E~ n ) = (kn2 ; km2 )E~ m
E~ n : ˆ­â¥£à¨àãï ¯® ®¡ê¥¬ã १®­ â®à , ¯®«ã稬

Z

I

V

S

(kn2 ; km2 ) E~ m
E~ n dV =

(E~ n  rot E~ m ; E~ m  rot E~ n )~n dS:

‘¯à ¢  ¨­â¥£à « à ¢¥­ ­ã«î ¢ ᨫ㠣࠭¨ç­ëå ãá«®¢¨©:

Z

(kn2 ; km2 ) E~ m
E~ n dV = 0; V

’ ª ª ª ¯® ãá«®¢¨î km2 6= km2 , â®

Z

V

E~ m
E~ n dV = 0 :

€­ «®£¨ç­® ¤®ª §ë¢ ¥âáï ¢â®à®¥ à ¢¥­á⢮. …᫨ E~ m ¨ E~ n ᮮ⢥âáâ¢ãîâ ®¤­®¬ã ¨ ⮬㠦¥ ᮡá⢥­­®¬ã §­ ç¥­¨î, â® í⨠ä㭪樨, ¢®®¡é¥ £®¢®àï, ­¥ ®à⮣®­ «ì­ë, ­® ¬®¦­® ¯®«ãç¨âì ®à⮣®­ «ì­ë¥ ä㭪樨 á ¯®¬®éìî ¯à®æ¥áá  ®à⮣®­ «¨§ æ¨¨. à¥¤¯®«®¦¨¬, çâ® E~ m ¨ H~ m { à¥è¥­¨ï, ᮮ⢥âáâ¢ãî騥 ­¥ª®â®à®¬ã ᮡá⢥­­®¬ã ç¨á«ã, ¯à¨ç¥¬ ¢ë஦¤¥­¨¥ ®âáãâáâ¢ã¥â. …᫨ E~ m ¨ H~ m { ª®¬¯«¥ªá­ë¥ ä㭪樨, â® ¨å ¬®¦­® ¯à¥¤áâ ¢¨âì ¢ ¢¨¤¥ áã¬¬ë ¢¥é¥á⢥­­®© ¨ ¬­¨¬®© ç á⥩ 188

E~ m = E~ m(r) + E~ m(i) ; H~ m = H~ m(r) + H~ m(i) : ¥âà㤭® ¢¨¤¥âì, çâ® ¢ १®­ â®à¥ ¡¥§ ¯®â¥àì £à ­¨ç­ë¬ ãá«®¢¨ï¬ ¤®«¦­ë 㤮¢«¥â¢®àïâì ®â¤¥«ì­® ¢¥é¥á⢥­­ ï ¨ ¬­¨¬ ï ç á⨠E~ m ¨ H~ m , â.¥. (r) = 0; E~ (i) = 0; H (r) = 0; H (i) = 0 ­  S: E~ mt mt mn mn

®¤áâ ¢«ïï E~ m ¨ H~ m ¢ ãà ¢­¥­¨ï Œ ªá¢¥««  ¨ à §¤¥«ïï ¢¥é¥á⢥­­ë¥ ¨ ¬­¨¬ë¥ ç áâ¨, ¯®«ã稬 rot E~ m(r) = ; j!m 
H~ m(i) ; rot E~ m(i) = ; j!m 
H~ m(r); rot H~ m(i) = j!m "
E~ m(r); rot H~ m(r) = j!m"
E~ m(i) :

’ ª¨¬ ®¡à §®¬, E~ m(r) ¨ H~ m(i) â ª¦¥ 㤮¢«¥â¢®àïîâ ãà ¢­¥­¨ï¬ Œ ªá¢¥««  ¨ £à ­¨ç­ë¬ ãá«®¢¨ï¬ ¨, á«¥¤®¢ â¥«ì­®, ¯à¨ ®âáãâá⢨¨ ¢ë஦¤¥­¨ï ¬®£ã⠮⫨ç âìáï ®â E~ m ¨ H~ m ⮫쪮 ¯®áâ®ï­­ë¬ ¬­®¦¨â¥«¥¬. ’® ¦¥ ¬®¦­® ᪠§ âì ®â­®á¨â¥«ì­® E~ m(i) ¨ H~ m(r). ’ ª ª ª ¬­®¦¨â¥«ì ­¥áãé¥á⢥­, â® ¢ ª ç¥á⢥ ᮡá⢥­­ëå ä㭪権 ¬®¦­® ¯à¨­ïâì, ­ ¯à¨¬¥à, E~ m(r) ¨ H~ m(i) . â® ®§­ ç ¥â, çâ® í«¥ªâà¨ç¥áª®¥ ¯®«¥ ¢® ¢á¥å â®çª å १®­ â®à  ª®«¥¡«¥âáï ¢ ®¤­®© ¨ ⮩ ¦¥ ä §¥. Œ £­¨â­®¥ ¯®«¥ â ª¦¥ ª®«¥¡«¥âáï ¢® ¢á¥å â®çª å ¢ ®¤­®© ä §¥, ­® ᮠᤢ¨£®¬ ­  90o ®â­®á¨â¥«ì­® í«¥ªâà¨ç¥áª®£® ¯®«ï. ‘ª § ­­®¥ á¯à ¢¥¤«¨¢® «¨èì ¯à¨ ®âáãâá⢨¨ ¢ë஦¤¥­¨ï. …᫨ ¨¬¥¥â ¬¥áâ® ¢ë஦¤¥­¨¥, â® íâ®, ¢®®¡é¥ £®¢®àï, ­¥á¯à ¢¥¤«¨¢®. à¨¬¥à { १®­ â®àë ¡¥£ã饩 ¢®«­ë. ‚¥à­¥¬áï ⥯¥àì ª ¢ëà ¦¥­¨î ¤«ï km2 :

Z

Z

V

V

km2 jH~ m j2 dV =

jrot H~ m j2 dV:

®¤áâ ¢«ïï á rot H~ m = j!m "E~ m ¨ k2 = !m2
", ¯®«ã稬 Z jHmj2 Z "jEm j2 dV = 2 2 dV; V

V

â.¥. á।­¨¥ § ¯ áë í«¥ªâà¨ç¥áª®© ¨ ¬ £­¨â­®© í­¥à£¨© ¢ १®­ â®à¥ ¡¥§ ¯®â¥àì à ¢­ë. ’®â ¦¥ ¢ë¢®¤ ¬®¦­® ᤥ« âì ¨§ ä §®¢ëå ᮮ⭮襭¨© 189

¤«ï í«¥ªâà¨ç¥áª®£® ¨ ¬ £­¨â­®£® ¯®«¥© ¢ á«ãç ¥ ®âáãâáâ¢¨ï ¢ë஦¤¥­¨ï. ‡ ¬¥â¨¬, çâ® ¯à¨¢¥¤¥­­ë© ¢ëè¥ ¢ë¢®¤ ¯à¨£®¤¥­ ¨ ¯à¨ ­ «¨ç¨¨ ¢ë஦¤¥­¨ï. 6.3.

¥§®­ â®àë, ®¡à §®¢ ­­ë¥ ¨§ ®â१ª®¢ «¨­¨¨ ¯¥à¥¤ ç¨

à®á⥩訥 १®­ â®àë ¯®«ãç îâáï, ¥á«¨ ¯¥à¥£®à®¤¨âì «¨­¨î ¯¥à¥¤ ç¨ ¬¥â ««¨ç¥áª¨¬¨ ¯¥à¥£®à®¤ª ¬¨ ¢ ¤¢ãå á¥ç¥­¨ïå. ‹¥£ª® ¯®ª § âì, çâ® ¢ â ª®© á¨á⥬¥ ¢®§¬®¦­ë ᢮¡®¤­ë¥ ª®«¥¡ ­¨ï. „¥©á⢨⥫쭮, ¯ãáâì ­  á⥭ªã, § ª®à ç¨¢ îéãî ¢®«­®¢®¤, ¯ ¤ ¥â ¢®«­ . à¨ í⮬ ¢á«¥¤á⢨¥ ¯®«­®£® ®âà ¦¥­¨ï ¢®§­¨ª ¥â áâ®ïç ï ¢®«­ , ¢ ª®â®à®© ¨¬¥îâáï á¥ç¥­¨ï, £¤¥ Ex = 0 ¨ Ey = 0. …᫨ ¢ â ª®¥ á¥ç¥­¨¥ ¯®¬¥áâ¨âì ¬¥â ««¨ç¥áªãî ¯¥à¥£®à®¤ªã, â® ­¨ç¥£® ­¥ ¨§¬¥­¨âáï. ‚®«­  ¡ã¤¥â ¬­®£®ªà â­® ®âà ¦ âìáï, ­¥ § âãå ï (¯à¨ ¨¤¥ «ì­®© ¯à®¢®¤¨¬®á⨠á⥭®ª). Žç¥¢¨¤­®, çâ® íâ® ¢®§¬®¦­®, ª®£¤  ¤«¨­  १®­ â®à  ªà â­  ¤«¨­¥ ¯®«ã¢®«­ë (=2) ¢ ¢®«­®¢®¤¥ (á¬. à¨á.). ‚ â ª¨å ¢®«­®¢®¤­ëå १®­ â®à å ª®«¥¡ â¥«ì­ë¥ ¬®¤ë ¤¥«ïâáï, ª ª ¨ ¢ ¢®«­®¢®¤ å, ­  í«¥ªâà¨ç¥áª¨¥ ¨ ¬ £­¨â­ë¥, ¨ ¤«ï ¨å ®¯¨á ­¨ï ¬®¦­® ¯®«ì§®¢ âìáï ⥬ ¦¥  ¯¯ à â®¬ ¯®â¥­æ¨ «ì­ëå ä㭪権  ¨ . à¨ í⮬, ®¤­ ª®, § ¢¨á¨¬®áâì ®â z ®¯¨á뢠¥âáï 㦥 ­¥ ¬­®¦¨â¥«ï¬¨ ej z ,   ¬­®¦¨â¥«ï¬¨ sin z ¨ cos z . â® á¢ï§ ­® á ⥬, çâ® ¯®«ï ¯®«ãç îâáï ¯ã⥬ á㬬¨à®¢ ­¨ï ¯àאַ© ¨ ®¡à â­®© ¢®«­ à ¢­®©  ¬¯«¨âã¤ë.  ©¤¥¬, ­ ¯à¨¬¥à, ¯®«ï ¬ £­¨â­ëå ¬®¤. „«ï ¢®«­®¢®¤  (¯àï¬ ï ¢®«­ ) L Hz = (k2 ; 2 ) e;j z : z ®«¥ ¢ १®­ â®à¥ ¯®«ãç ¥âáï ¯ã-:

⥬ ¢ëç¨â ­¨ï ¯ ¤ î饩 ¨ ®âà ¦¥­­®© ¢®«­ (ç⮡ë 㤮¢«¥â¢®à¨âì £à ­¨ç­®¬ã ãá«®¢¨î ¯à¨ z = 0)

z=0

Hz = 21 (k2 ; 2 ) (e;j z ; ej z ) = ; j (k2 ; ) sin z:

„ «¥¥, ¤«ï ¢®«­®¢®¤ 

Hx = ; j @@x e;j z ; Hy = ; j @@y e;j z : 190

„«ï १®­ â®à  ᮮ⢥âá⢥­­®

Hx = ; 12 j @@x (e;j z + e j z ) = ; j @@x cos z; Hy = ; 12 j @@y (e;j z + e j z ) = ; j @@y cos z: „«ï ¢®«­®¢®¤ 

Ex = ; j!
@@y e;j z ; Ey = j!
@@x e;j z : „«ï १®­ â®à  ᮮ⢥âá⢥­­®

Ex = ; 12 j! @@y (e;j z ; e j z ) = ; ! @@y sin z; Ey = 12 j! @@x (e;j z ; e j z ) = ; ! @@x sin z: —⮡ë 㤮¢«¥â¢®à¨âì £à ­¨ç­®¬ã ãá«®¢¨î ­  ¢â®à®© ¯¥à¥£®à®¤ª¥, ¤®«¦­® ¡ëâì sin L = 0; ®âªã¤  L = l; â.¥.

2 = l; L = l  ; l = 1; 2; : : : :  2 Œ®¦­® ¯®«ãç¨âì â ª¦¥ ᮡá⢥­­ë¥ §­ ç¥­¨ï, ãç¨â뢠ï, çâ®  l 2 2 2 0 2 2 2 2 2 2 = k ; gmn ; L = l  ; = L : ®¤áâ ¢«ïï 2 ¢ ¯à¥¤ë¤ã饥 ¢ëà ¦¥­¨¥, ¯®«ã稬  2 2 = gmn 0 2 + l : kmnl L

¥âà㤭® ¢¨¤¥âì, çâ® ¯à¨ l = 0 ¯®«¥ ¨á祧 ¥â, â ª ª ª ¤«ï í⮣® ­¥®¡å®¤¨¬®, ç⮡ë = 0 (â ª ª ª L 6= 0). 191

«¥ªâà¨ç¥áª¨¥ ¬®¤ë ¢ १®­ â®à¥ ¬®£ãâ ¡ëâì § ¯¨á ­ë ¢ á«¥¤ãî饬 ¢¨¤¥:

Ez = (k2 ; 2 )
 cos z; Ex = ;
@ @x sin z; Ey = ;
@ @y sin z; Hx = j!"
@ @y cos z; Hy = ; j!" @ @x cos z: ƒà ­¨ç­ë¥ ãá«®¢¨ï ¯à¨ z = L ¢ë¯®«­ïîâáï â ª¦¥ ¯à¨ sin L = 0, â.¥. ¯à¨ L = l, ®âªã¤  k2

2 mnl = gmn +

 l 2 L

:

‚ í⮬ á«ãç ¥ ¯à¨ l = 0 (â.¥. ¯à¨ = 0) ¯®«¥ ­¥ ¨á祧 ¥â. Ž¡à é îâáï ¢ ­ã«ì «¨èì Ex ¨ Ey . â® §­ ç¨â, çâ® ¯®«¥ ®ª §ë¢ ¥âáï ç¨áâ® ¯à®¤®«ì­ë¬. ‘®¡á⢥­­ë¥ §­ ç¥­¨ï ¯à¨ í⮬ à ¢­ë 2 0 = gmn 2 ; kmn â.¥. १®­ ­á­ë¥ ç áâ®âë ᮢ¯ ¤ îâ á ªà¨â¨ç¥áª¨¬¨ ç áâ®â ¬¨ ¢®«­®¢®¤  ¨ ­¥ § ¢¨áï⠮⠤«¨­ë १®­ â®à . 6.4.

à¨¬¥àë १®­ â®à®¢, ®¡à §®¢ ­­ëå ¨§ ®â१ª®¢ «¨­¨¨ ¯¥à¥¤ ç¨

àאַ㣮«ì­ë© १®­ â®à „«ï ¯àאַ㣮«ì­®£® ¢®«­®¢®¤  2 = gnm

®âªã¤ 

 n 2 a

192

+

 m 2 b

;

k2

nml =

 n 2 a

+

 m 2 b

+

 l 2 L

:

Š ª ¢¨¤­®, ¢á¥ à §¬¥àë १®­ â®à  ¢å®¤ïâ ᨬ¬¥âà¨ç­®, çâ® ¢¯®«­¥ ¥áâ¥á⢥­­®, â ª ª ª ¢ â ª®¬ १®­ â®à¥ «î¡ ï ¨§ ®á¥© ¬®¦¥â ¡ëâì ¯à¨­ïâ  §  ®áì z . (H ) = gnm (E) ¤«ï ¯àאַ㣮«ì­®£® ¢®«­®¢®¤ , â® ª ¦¤ ï ¨§ ᮡ’ ª ª ª gnm á⢥­­ëå ¬®¤ १®­ â®à  ï¥âáï ¤¢ ¦¤ë ¢ë஦¤¥­­®© (¬ £­¨â­ë¥ ¨ í«¥ªâà¨ç¥áª¨¥ ¬®¤ë). Šà®¬¥ ⮣®, ¢ â ª®¬ १®­ â®à¥ ­¥â à §«¨ç¨ï ¬¥¦¤ã ¬ £­¨â­ë¬¨ ¨ í«¥ªâà¨ç¥áª¨¬¨ ¬®¤ ¬¨, â ª ª ª íâ® § ¢¨á¨â ®â ¢ë¡®à  ®á¥©. ˆáª«î祭¨¥ ¢ á¬ëá«¥ ¢ë஦¤¥­¨ï á®áâ ¢«ïîâ ¬®¤ë á ®¤­¨¬ ¨§ ¨­¤¥ªá®¢, à ¢­ë¬ ­ã«î, â ª ª ª í«¥ªâà¨ç¥áª¨¥ ¬®¤ë á ­ã«¥¢ë¬ ¨­¤¥ªá®¬ ®âáãâáâ¢ãîâ. „«ï ¢ë஦¤¥­­ëå ¬®¤ ᢮¡®¤­ë¥ ª®«¥¡ ­¨ï ¬®£ãâ ¡ëâì ¯à¥¤áâ ¢«¥­ë ¢ ¢¨¤¥ á㯥௮§¨æ¨¨ ¤¢ãå ᮡá⢥­­ëå ä㭪権. Šà㣮¢®© 樫¨­¤à¨ç¥áª¨© १®­ â®à ‚ ¤ ­­®¬ á«ãç ¥ à §«¨ç îâ ¬ £­¨â­ë¥ ¨ í«¥ªâà¨ç¥áª¨¥ ¬®¤ë ª®«¥¡ ­¨©:  t0 2  l 2 ( H ) 2 kmnl = mn + L ; a  tmn 2  l 2 (E ) 2 = kmnl + L : a à¨ í⮬, ª ª ¨ ¢ ªà㣫®¬ ¢®«­®¢®¤¥, ¯à¨ m 6= 0 ¬®¤ë ¤¢ ¦¤ë ¢ë஦¤¥­ë (¯®«ïਧ æ¨®­­®¥ ¢ë஦¤¥­¨¥). ‚ë஦¤¥­¨¥ ®âáãâáâ¢ã¥â ¤«ï ᨬ¬¥âà¨ç­ëå ¬®¤, â.¥. ¤«ï ¬®¤ á m = 0.  ¨¡®«ì訩 ¨­â¥à¥á ¯à¥¤áâ ¢«ïîâ १®­ â®àë á ¬®¤ ¬¨ E010 ¨ H01n . „«ï ¯¥à¢®£® १®­ ­á­ ï ç áâ®â  § ¢¨á¨â ­¥ ®â ¤«¨­ë १®­ â®à ,   (E)  2:4 , ®âªã¤  ⮫쪮 ®â ¥£® à ¤¨ãá . ‚¥«¨ç¨­  t01  2:4. ®í⮬ã k010 a 2  2 : 4 á«¥¤ã¥â, çâ®  = a ¨ १  2:6 a = 1:3 d, ¨«¨ d  0:75 १ = 34 १ १

193

(â®ç­¥¥ 0.764१), £¤¥ d { ¤¨ ¬¥âà १®­ â®à . ’ ª®© १®­ â®à (á¬. à¨á.) ®ç¥­ì 㤮¡¥­ ¢ ã᪮à¨â¥«ì­®© â¥å­¨ª¥, â ª ª ª ­  ®á¨ ¯à®¤®«ì­ ï ª®¬¯®­¥­â  í«¥ªâà¨ç¥áª®£® ¯®«ï ¬ ªá¨¬ «ì­ ,   ¤«¨­  १®­ â®à  ¬®¦¥â ¡ëâì ¢ë¡à ­  ®¯â¨¬ «ì­®© ¤«ï ã᪮७¨ï ç áâ¨æ ¢ § ¢¨á¨¬®á⨠®â ¨å ᪮à®áâ¨. „«ï ¬®¤ë E010 ¨¬¥¥¬ á«¥¤ãî騥 ᮮ⭮襭¨ï ¤«ï á®áâ ¢«ïîé¨å ¯®«ï:

Ez = J0 (kr); r" 1 dE k z 0 H' = j! dr = j! J0 (kr) = j  J1 (kr): „«ï १®­ â®à  á ¬®¤®© H01 ¨¬¥¥¬  2 (H ) 2 = t001 +   2 = k011 a L

 3:83 2 a

+

  2

L :

Š ª ¢¨¤¨¬, १®­ ­á­ ï ç áâ®â  ¢ í⮬ á«ãç ¥ § ¢¨á¨â ®â ¤«¨­ë १®­ â®à . « £®¤ àï ¬ «ë¬ ¯®â¥àï¬ â ª®© १®­ â®à ¬®¦¥â ¨¬¥âì ¡®«ìèãî ¤®¡à®â­®áâì. Š® ªá¨ «ì­ë¥ १®­ â®àë Žâ«¨ç¨¥ ª® ªá¨ «ì­ëå १®­ â®à®¢ ®â ¢®«­®¢®¤­ëå á®á⮨⠢ ⮬, çâ® ¢ ª® ªá¨ «ì­®© «¨­¨¨ «¥£ç¥ ®áãé¥á⢨âì ­¥ ⮫쪮 ª®à®âª®¥ § ¬ëª ­¨¥, ­® ¨ 宫®á⮩ 室. ®í⮬㠧¤¥áì ¬ë ¨¬¥¥¬ âன­®© ­ ¡®à १®­ ­á­ëå ¬®¤. ‚®§¬®¦­ë âਠá®ç¥â ­¨ï £à ­¨ç­ëå ãá«®¢¨©, ¯®ª § ­­ë¥

194

­  à¨á. “á«®¢¨¥ १®­ ­á :

१ L = n
१ 2 = 4 2n n = 1; 2; 3; : : :

L

L = १ 4 (2n + 1) n = 0; 1; 2; : : : L

L = १ 4 2n n = 1; 2; 3; : : : L

‚ १®­ â®à å á 宫®áâë¬ å®¤®¬ ­  ®¤­®¬ ¨«¨ ¤¢ãå ª®­æ å ¨­®£¤  ­¥®¡å®¤¨¬® ãç¨â뢠âì ª®­æ¥¢ãî ¥¬ª®áâì. “ç¥â ª®­æ¥¢®© ¥¬ª®á⨠¯à¨¢®¤¨â ª 㤫¨­¥­¨î १®­ ­á­®© ¢®«­ë ¨«¨ 㪮à®ç¥­¨î á ¬®£® १®­ â®à  ¯à¨ § ¤ ­­®© १®­ ­á­®© ¤«¨­¥ ¢®«­ë, ¯®í⮬㠥¬ª®áâì ­ §ë¢ îâ 㪮à ç¨¢ î饩. Š®­æ¥¢ãî ¥¬ª®áâì ¬®¦­® ¨áªãáá⢥­­® 㢥«¨ç¨âì,ç⮡ë 㬥­ìè¨âì ¤«¨­ã १®­ â®à . à¨ ®ç¥­ì ᨫ쭮¬ 㪮à®ç¥­¨¨ ®á­®¢­ ï ç áâì § ¯ á¥­­®© í«¥ªâà¨ç¥áª®© í­¥à£¨¨ ­ å®¤¨âáï ¢

195

§ §®à¥,   ¬ £­¨â­®© { ¢ ª® ªá¨ «ì­®© ç áâ¨, â® ¥áâì í«¥ªâà¨ç¥áª ï ¨ ¬ £­¨â­ ï í­¥à£¨¨ ¯à®áâà ­á⢥­­® à §¤¥«¥­ë. ’ ª®© १®­ â®à ­ §ë¢ îâ ª¢ §¨áâ â¨ç¥áª¨¬. …£® à §¬¥àë ¬­®£® ¬¥­ìè¥ ¤«¨­ë ¢®«­ë,   ¯®«ï ¬®¦­® à ááç¨â뢠âì ¯® ãà ¢­¥­¨ï¬ áâ â¨ç¥áª¨å ¯®«¥©. ’ ª¨¥ १®­ â®àë ¯à¨¬¥­ïîâ ¢ ã᪮à¨â¥«ì­®© â¥å­¨ª¥ ¨ ‘‚—í«¥ªâà®­¨ª¥. 6.5.

à¨¡«¨¦¥­­ë¥ ¬¥â®¤ë à áç¥â  ç áâ®âë ᢮¡®¤­ëå ª®«¥¡ ­¨© १®­ â®à®¢

’®ç­ë© à áç¥â १®­ ­á­ëå ç áâ®â १®­ â®à®¢ ¢ ¡®«ì設á⢥ á«ãç ¥¢ ­¥¢®§¬®¦¥­, ç é¥ ¢®§¬®¦¥­ ¯à¨¡«¨¦¥­­ë© à áç¥â á ¯®¬®éìî ç¨á«¥­­ëå ¬¥â®¤®¢. ‘ãé¥áâ¢ãîâ â ª¦¥ ¯à¨¡«¨¦¥­­ë¥  ­ «¨â¨ç¥áª¨¥ ¬¥â®¤ë, ª®â®àë¥ ¡ã¤ãâ à áᬮâà¥­ë ¢ ¤ ­­®¬ ¯®¤à §¤¥«¥,   ¨¬¥­­®: ª¢ §¨áâ â¨ç¥áª¨© ¬¥â®¤, ¬¥â®¤ á訢 ­¨ï, ¨«¨ ¬¥â®¤ ç áâ¨ç­ëå ®¡« á⥩, ¢ à¨ æ¨®­­ë¥ ¬¥â®¤ë ¨ ¬¥â®¤ ¢®§¬ã饭¨©. 6.5.1. Š¢ §¨áâ â¨ç¥áª¨© ¬¥â®¤

®«ì讥 ç¨á«® ®¡ê¥¬­ëå १®­ â®à®¢ ¯à¨­ ¤«¥¦¨â ª« ááã ª¢ §¨áâ â¨ç¥áª¨å १®­ â®à®¢. ’ ª¨¥ १®­ â®àë ᮤ¥à¦ â ¤¢¥ ç¥âª® à §¤¥«¥­­ë¥ ®¡« áâ¨, ¢ ®¤­®© ¨§ ª®â®àëå áãé¥á⢥­­® ¯à¥®¡« ¤ îâ § ¯ áë ¬ £­¨â­®© í­¥à£¨¨,   ¢ ¤à㣮© { § ¯ áë í«¥ªâà¨ç¥áª®© í­¥à£¨¨. â® ¢®§¬®¦­® «¨èì ¢ ⮬ á«ãç ¥, ª®£¤  à §¬¥àë १®­ â®à  ¬ «ë ¯® áà ¢­¥­¨î á ¤«¨­®© ¢®«­ë. ‘ª § ­­®¥ ¢ëè¥ ¯®§¢®«ï¥â ¯à¨ ®¯à¥¤¥«¥­¨¨ í«¥ªâà¨ç¥áª®£® ¯®«ï ¯à¥­¥¡à¥çì ¢ ¯¥à¢®¬ ãà ¢­¥­¨¨ Œ ªá¢¥««  ç«¥­®¬  @H @t ¢ í«¥ªâà¨ç¥áª®© ®¡« á⨠¨ ⥬ á ¬ë¬ á¢¥á⨠§ ¤ çã ª í«¥ªâà®áâ â¨ç¥áª®©. à¨ ®¯à¥¤¥«¥­¨¨ ¦¥ ª®­ä¨£ãà æ¨¨ ¬ £­¨â­®£® ¯®«ï ¢ ¬ £­¨â­®© ®¡« á⨠¬®¦­® ®â¡à®á¨âì ç«¥­ " @E @t , ¨ ⮣¤  ¬ £­¨â­®¥ ¯®«¥ ®¯¨á뢠¥âáï áâ â¨ç¥áª¨¬ ãà ¢­¥­¨¥¬

~ rot H~ = J: 196

ˆâ ª, ª¢ §¨áâ â¨ç¥áª¨© ¬¥â®¤ á®á⮨⠢ ⮬, çâ® ¯à¨ ®¯à¥¤¥«¥­¨¨ ª®­ä¨£ãà æ¨¨ í«¥ªâà¨ç¥áª®£® ¨ ¬ £­¨â­®£® ¯®«¥© í⨠¯®«ï áç¨â îâáï ­¥ § ¢¨áï騬¨ ¤à㣠®â ¤à㣠 ¨ ¢á«¥¤á⢨¥ í⮣® áâ â¨ç¥áª¨¬¨. “á«®¢¨¥ १®­ ­á  á®á⮨⠢ ⮬, çâ® § ¯ áë í­¥à£¨¨ í«¥ªâà¨ç¥áª®£® ¨ ¬ £­¨â­®£® ¯®«¥© à ¢­ë ¤à㣠¤àã£ã. ’ ª®© ¬¥â®¤ ¬®¦¥â ¡ëâì ¯à¨¬¥­¥­ ¯à¨ ãá«®¢¨¨ ¬ «®á⨠ࠧ¬¥à®¢ १®­ â®à  ¯® áà ¢­¥­¨î á ¤«¨­®© ¢®«­ë. 6.5.2. à¨¬¥àë: â®à®¨¤ «ì­ë© १®­ â®à, १®­ â®à \饫ì { ®â¢¥àá⨥"

‚ ª ç¥á⢥ ¯à¨¬¥à  à áᬮâਬ ­¥ª®â®àë¥ ª¢ §¨áâ â¨ç¥áª¨¥ १®­ â®àë. ’®à®¨¤ «ì­ë© १®­ â®à ®¡à §ãr ¥âáï ¢à é¥­¨¥¬ ¯«®áª®© ®¡« á⨠S ¢®ªà㣠®á¨, «¥¦ é¥© ¢ í⮩ 1010 d ¦¥ ¯«®áª®áâ¨, ­® ­¥ ¯¥à¥á¥ª î饩 S1 ®¡« áâì S (á¬. à¨á.). «¥ªâà¨ç¥010 ᪮¥ ¯®«¥ á®á।®â®ç¥­® ¢ § §®à¥ 10 è¨à¨­®© d, ¬ £­¨â­®¥ ¯®«¥ { ¢ ¯®2a «®á⨠S . ­¥à£¨ï í«¥ªâà¨ç¥áª®£® dS ¯®«ï ¢ § §®à¥ à ¢­  2 2 2 2 2 WE = V "02E = a2 d "02E = ad "0 U2 = CU2 ; 2 £¤¥ C = ad "0 , U = d E .

Œ £­¨â­ ï í­¥à£¨ï à ¢­ 

WH =

Z 0H 2 dV;

VH

2

I , dV = 2r dS . ˆ­â¥£à¨àãï, ¯®«ãç ¥¬ ¯à¨ç¥¬ H = 2r W =

Z 0H 2 2r
dS

SH

£¤¥ L = 40

2

R dS .

SH

=

Z 0 I 2 2
42 r2 2r
dS S H

r

197

2 Z LI 2 ; = I40 dS = r 2 SH

¥§®­ ­á­ ï ¤«¨­  ¢®«­ë १®­ â®à  ¬®¦¥â ¡ëâì ¢ëç¨á«¥­  ¯® ¥¬ª®á⨠¨ ¨­¤ãªâ¨¢­®á⨠C ¨ L:

p

१ = 2c LC = ˆ­â¥£à «

v u a2 "0 0 Z dS 2cu t d 2
r SH

=

v u 1 Z dS 2au t 2d r : SH

Z dS S r = ráà :

SH

®¤áâ ¢«ïï á L ¨ C , ­ å®¤¨¬

s

१ = 2a 2dSr : áà “á«®¢¨¥ ¯à¨¬¥­¨¬®á⨠ª¢ §¨áâ æ¨®­ à­®£® ¬¥â®¤  ¢ ¤ ­­®¬ á«ãç ¥

S 2d ráà  1; â.¥. § §®à d ¤®«¦¥­ ¡ëâì ¬ « ¯® áà ¢­¥­¨î á ¯®¯¥à¥ç­ë¬¨ à §¬¥à ¬¨ ®¡« á⨠S (¯«®é ¤ì d ráà  S ). „à㣮© ¯à¨¬¥à { १®­ â®à \饫ì { ®â¢¥àá⨥", ¯à¨¬¥­ï¥¬ë© ¢ ¬ £­¥âà®­ å (á¬. à¨á.).

11111111 00000000 00000000 11111111 a 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 000 111 00000000 11111111 000 l 111

â®â १®­ â®à â ª¦¥ ï¥âáï ª¢ §¨áâ â¨ç¥áª¨¬. …᫨ ¢ëá®â  १®­ â®à  h §­ ç¨â¥«ì­® ¡®«ìè¥ ¤¨ ¬¥âà  ®â¢¥àáâ¨ï (ª ª íâ® ®¡ëç­® ¡ë¢ ¥â), â® ¬ £­¨â­®¥ ¯®«¥ ¢ ®â¢¥àá⨨ ¬®¦­® ¯à¨¡«¨¦¥­­® áç¨â âì ®¤­®à®¤­ë¬. Œ £­¨â­®¥ ¯®«¥ à ¢­®

H = hI :

d

Œ £­¨â­ë© ¯®â®ª

2  = L I = 0 H S = 0 hI a2 = 0 ah I:

198

Žâá ¨­¤ãªâ¨¢­®áâì ¯®«ãç ¥âáï à ¢­®© 2 L = 0 ah : …¬ª®áâì ª®­¤¥­á â®à  (饫¨) à ¢­ 

C = "0 lhd :

¥§®­ ­á­ ï ¤«¨­  ¢®«­ë ¢ëà ¦ ¥âáï ç¥à¥§ ¨­¤ãªâ¨¢­®áâì ¨ ¥¬ª®áâì: r 2 r a lh १ = 2c 0 h "0 d = 2a ld : 6.5.3. Œ¥â®¤ ç áâ¨ç­ëå ®¡« á⥩

„ ­­ë© ¬¥â®¤ á®á⮨⠢ ⮬, çâ® ¢¥áì १®­ â®à à §¡¨¢ îâ ­  ®â¤¥«ì­ë¥ ®¡« áâ¨, ¯®«ï ¢ ª®â®àëå ¨§¢¥áâ­ë ¨«¨ «¥£ª® à ááç¨â뢠îâáï. ‡ â¥¬ à¥è¥­¨ï ¤«ï ®â¤¥«ì­ëå ®¡« á⥩ \á訢 îâáï" ­  ¯®¢¥àå­®áâïå à §¤¥«  ¬¥¦¤ã ®¡« áâﬨ. ‘訢 ­¨¥ á®á⮨⠢ ¯à¨¡«¨¦¥­­®¬ 㤮¢«¥â¢®à¥­¨¨ £à ­¨ç­ëå ãá«®¢¨© ­  ¯®¢¥àå­®áâïå à §¤¥« . à¨ í⮬ ¯®«ãç îâáï ãà ¢­¥­¨ï, ¨§ ª®â®àëå ¬®£ãâ ¡ëâì ®¯à¥¤¥«¥­ë १®­ ­á­ë¥ ç áâ®âë. ˆ§¢¥áâ­ë à §«¨ç­ë¥ ¢ à¨ ­âë ãá«®¢¨© ­  £à ­¨æ¥. à®á⥩襥 ¨§ ­¨å { ãá«®¢¨¥ ¤«ï ᮯà®â¨¢«¥­¨ï ­  £à ­¨æ¥, ª®â®à®¥ ¬®¦¥â ¡ëâì ¯®«ã祭® á«¥¤ãî騬 ®¡à §®¬. ‚áï ®¡« áâì à §¤¥«¥­  ­  ®¡« á⨠1 ¨ 2 (á¬. à¨á.). Š ª ¨§¢¥áâ­®, ¯à¨ १®­ ­á¥ á।­¨¥ § ¯ áë í«¥ªâà¨ç¥áª®© ¨ ¬ £­¨â­®© í­¥à£¨¨ à ¢­ë ¤à㣠¤àã£ã, i2 2 â.¥. i1 1

Z1 i1

Z2

WE1 + WE2 = WH 1 + WH 2 :

i2

â® à ¢¥­á⢮ ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¢¨¤¥

(WH 1 ; WE1 ) + (WH 2 ; WE2 ) = 0:  §­®á⨠¢ ᪮¡ª å ¬®£ãâ ¡ëâì ¢ëà ¦¥­ë ç¥à¥§ ¢å®¤­ë¥ ᮯà®â¨¢«¥­¨ï: 199

1 Z 1 jI j2 ; WH 1 ; WE1 = 2j! 12 1 1 Z 1 jI j2 : WH 2 ; WE2 = 2j! 22 2 à¨à ¢­¨¢ ï á㬬㠭ã«î, ¯®«ã稬 ãà ¢­¥­¨¥ á訢 ­¨ï Z1
jI1 j2 + Z2
jI2 j2 = 0; ¯à¨ç¥¬ Z1 ¨ Z2 { ¯®«­ë¥ ᮯà®â¨¢«¥­¨ï ç áâ¨ç­ëå ®¡« á⥩ ­  £à ­¨ç­®© ¯®¢¥àå­®áâ¨. „®¯®«­¨â¥«ì­®¥ ãá«®¢¨¥ ­  £à ­¨æ¥ á®á⮨⠢ ⮬, çâ® jI1 j = jI2 j; â ª ª ª «¨­¨¨ ⮪  ­¥¯à¥à뢭ë. ®í⮬ã ãá«®¢¨¥ ­  £à ­¨æ¥ ᢮¤¨âáï ª á«¥¤ãî饬ã (¤«ï ¤¢ãå ®¡« á⥩): Z1 + Z2 = 0: „à㣮© ¢¨¤ â ª®£® £à ­¨ç­®£® ãá«®¢¨ï Y1 + Y2 = 0: ‚ëç¨á«¥­¨¥ Z1 ¨ Z2 ç áâ® ¯à®¨§¢®¤¨âáï ¯à¨¡«¨¦¥­­®, ¡¥§ ãç¥â  ¢ëáè¨å ¬®¤ ­  £à ­¨æ¥. à¨ í⮬ १ã«ìâ â â ª¦¥ ¯®«ãç ¥âáï ¯à¨¡«¨¦¥­­ë¬. ‚ á«ãç ¥ ­¥®¡å®¤¨¬®á⨠¬®¦­® ãç¥áâì ¨ ¢ëá訥 ¬®¤ë, çâ® ¯®§¢®«ï¥â ãâ®ç­¨âì १ã«ìâ â. 6.5.4. à¨¬¥àë

Š® ªá¨ «ì­ ï «¨­¨ï, ­ £à㦥­­ ï ¥¬ª®áâìî

11111111111 00000000000 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 l 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111

Š® ªá¨ «ì­ ï «¨­¨ï å à ªâ¥à¨§ã¥âáï ¢®«­®¢ë¬ ᮯà®â¨¢«¥­¨¥¬ ãáâì ª®à®âª®§ ¬ª­ãâë© 0. C Z ®â१®ª «¨­¨¨ ­ £à㦥­ ­  ¥¬ª®áâì.  ©¤¥¬ १®­ ­á­ë¥ ç áâ®âë â ª®£® १®­ â®à  (á¬. à¨á.). ‚室­®¥ ᮯà®â¨¢«¥­¨¥ ª®à®âª®§ ¬ª­ã⮣® ®â१ª  «¨­¨¨ à ¢­® 200

Z¢å = jZ0 tg kl; â ª ª ª k = !c , â®

Z¢å = jZ0 tg !c l:

“á«®¢¨¥ १®­ ­á 

1 ! j!C + jZ0 tg c l = 0: â® ãá«®¢¨¥ ¬®¦­® ¨­ ç¥ § ¯¨á âì ¢ ¢¨¤¥

ctg !c l = Z0 !C:

…᫨ ®¡®§­ ç¨âì !c l = x, â® ãà ¢­¥­¨¥ ¯à¨¬¥â ¢¨¤ ctg x = Ax: ‡¤¥áì A = Z0lC c = CC l , £¤¥ C1 { ¯®£®­­ ï ¥¬ª®áâì «¨­¨¨. 1 ®«ã祭­®¥ âà ­á業¤¥­â­®¥ ãà ¢­¥­¨¥ à¥è ¥âáï £à ä¨ç¥áª¨ (á¬. à¨á.).  ©¤ï ª®à­¨ x1 ; x2 ; : : :, ¬®¦­® ®¯à¥¤¥«¨âì १®­ ­á­ë¥ ¤«¨­ë ¢®«­, ¨áå®¤ï ¨§ ᮮ⭮襭¨ï  = 2 l: n

xn

Žâ¬¥â¨¬, çâ® íâ® à¥è¥­¨¥ â¥àï¥â ᨫã â ¬, £¤¥ ¥¬ª®áâì C à áᬠâਢ ¥âáï ­¥ ª ª ç¨áâ ï ¥¬ª®áâì,   ª ª à ¤¨ «ì­ ï «¨­¨ï.

x x1

x2

x3

„¢¥ ª® ªá¨ «ì­ë¥ «¨­¨¨ á à §«¨ç­ë¬¨ ¢®«­®¢ë¬¨ ᮯà®â¨¢«¥­¨ï¬¨ 201

“áâனá⢮ १®­ â®à  ¨§ ¤¢ãå ª® ªá¨ «ì­ëå «¨­¨© á à §«¨ç­ë¬ ¢®«­®¢ë¬ ᮯà®â¨¢«¥­¨¥¬ ¯®ª § ­® ­  à¨á. Z 01

Z 02

l1

l2

l1

l2

‡¤¥áì ãá«®¢¨¥ १®­ ­á  ¨¬¥¥â ¢¨¤

jZ01 tg !lc 1 + jZ02 tg !lc 2 = 0

¨«¨

Z02 !l2 !l1 Z01 tg c = ; tg c :

Ž¡®§­ ç¨¢ ¯à¥®¡à §ã¥¬ ãà ¢­¥­¨¥ ª ¢¨¤ã

!l1 = x; c

Z02 tg l2
x = ; tg x: Z01 l1 „ ­­®¥ ãà ¢­¥­¨¥ â ª¦¥ ¬®¦¥â ¡ëâì à¥è¥­® £à ä¨ç¥áª¨ (á¬. à¨á.).  ©¤ï ª®à­¨ ãà ¢­¥­¨ï x, ­ ©¤¥¬ १®­ ­á­ë¥ ¤«¨­ë ¢®«­. à¨ â ª®¬ à¥è¥­¨¨ ­¥ ãç¨â뢠¥âáï ¯®«¥, ¢®§­¨ª î饥 ¢¡«¨§¨ ᪠窮®¡à §­®£® ¨§¬¥­¥­¨ï á¥ç¥­¨ï. â® ¯®«¥ ¬®¦­® áç¨â âì í«¥ªâà®áâ â¨ç¥áª¨¬. Ž­® x x íª¢¨¢ «¥­â­® ­¥ª®â®à®© ¥¬ª®x áâ¨, ¢ª«î祭­®© ¯ à ««¥«ì­® «¨­¨¨ ¢ ¬¥á⥠᪠窠 (á¬. à¨á.). ‘ ãç¥â®¬ ¥¬ª®á⨠ãá«®¢¨¥ १®­ ­á  ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¢¨¤¥ (á㬬  ¯à®¢®¤¨¬®á⥩) 2

1

202

1 1 + + j!C = 0: !l 1 jZ01
tg c jZ02
tg !lc2 Z 01

C Z 02

–¨«¨­¤à¨ç¥áª¨© १®­ â®à, ç áâ¨ç­® § ¯®«­¥­­ë© ä¥àà¨â®¬ 2

h

1111 0000 0000 1111 0000 1111 0000 1111 0000 1111

1

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 2a 2b

 áᬮâਬ 樫¨­¤à¨ç¥áª¨© १®­ â®à ­  ¬®¤¥ E01 , ç áâ¨ç­® § ¯®«­¥­­ë© ä¥àà¨â®¬, â.¥. ­¥¯à®¢®¤ï騬 ¬ £­¥â¨ª®¬ (á¬. à¨á.). ‚ ®¡« á⨠1 ᨬ¬¥âà¨ç­ ï ¬®¤  ¨¬¥¥â í«¥ªâà¨ç¥áª®¥ ¯®«¥ Ez = J0 (k1 r). Œ £­¨â­®¥ ¯®«¥ à ¢­®

1 @Ez = k1 J 0 (k r) = ; k1 J (k r): H' = j! j!1 0 1 j!1 1 1 1 @r   à ¤¨ãᥠr = a ⮪ à ¢¥­ k1 J (k a): I = 2a H' = ; 2a j! 1 1 1 ‚室­®¥ ᮯà®â¨¢«¥­¨¥ ¤«ï ®¡« á⨠1 ¡ã¤¥â j!1 J0 (k1 a) : Z1 = EIz h = ; h2a k1 J1 (k1 a) „«ï ®¡« á⨠2

Ez = A J0 (k2 r) + B N0 (k2 r): Žâ­®è¥­¨¥ ¯®áâ®ï­­ëå A=B ®¯à¥¤¥«ï¥âáï ¨§ ãá«®¢¨ï ­  ¢­¥è­¥¬

à ¤¨ãá¥

203

A J0 (k2 b) + B N0 (k2 b) = 0; ®âªã¤ 

B = ; J0 (k2 b) : A N0 (k2 b) ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï Ez , ¯®«ã稬 Ez = A J0 (k2 r) N0 (k2 bN) (;k bJ)0 (k2 b) N0 (k2 r) : 0 2 Œ £­¨â­®¥ ¯®«¥ à ¢­® 1 @Ez = A k2 J00 (k2 r) N0 (k2 b) ; J0 (k2 b) N00 (k2 r) : H' = j! j!2 N0 (k2 b) 2 @r ‚室­®¥ ᮯà®â¨¢«¥­¨¥ ®¡« á⨠2 à ¢­® h E j!2 J0 (k2 a) N0 (k2 b) ; J0 (k2 b) N0 (k2 a) : z Z2 = 2a H = h2a k2 J00 (k2 a) N0 (k2 b) ; J0 (k2 b) N00 (k2 a) ' r=a “á«®¢¨¥ १®­ ­á  ¨¬¥¥â ¢¨¤

Z1 + Z2 = 0; ¯®¤áâ ¢«ïï á Z1 ¨ Z2 , ¯®«ã稬 १®­ ­á­®¥ ãà ¢­¥­¨¥ r 2 J0(k2 r) N0 (k2 b) ; J0 (k2b) N0(k2 r) r 1 J0(k1 a) "2 J00 (k2 r) N0 (k2 b) ; J0 (k2 b) N00 (k2 r) ; "1 J1 (k1 a) = 0: ¥è ï íâ® ãà ¢­¥­¨¥, ¬®¦­® ­ ©â¨ १®­ ­á­ë¥ ç áâ®âë १®­ â®à  ¨ í«¥ªâ஬ £­¨â­®¥ ¯®«¥. ‘®ç¥â ­¨ï ä㭪権, ¢å®¤ï騥 ¢ í⨠ãà ¢­¥­¨ï, ⠡㫨஢ ­ë, çâ® ¯®§¢®«ï¥â ­ ©â¨ à¥è¥­¨ï. 6.5.5. ‚ à¨ æ¨®­­ë© ¬¥â®¤. ”®à¬ã«  ¢®§¬ã饭¨©

‚ à¨ æ¨®­­ë© ¬¥â®¤ ®á­®¢ ­ ­  ¢ à¨ æ¨®­­ëå ᢮©á⢠å ä®à¬ã«ë R jrot H j2 dV 2 k = R jH j2 dV ; 204

£¤¥ ¨­â¥£à «ë ¢§ïâë ¯® ¢á¥¬ã ®¡ê¥¬ã १®­ â®à . …᫨ à áᬠâਢ âì ¤ ­­®¥ ᮮ⭮襭¨¥ ª ª ä㭪樮­ « ®â­®á¨â¥«ì­® ä㭪樨 ¬ £­¨â­®£® ¯®«ï H~ , â® ¬®¦­® ¯®ª § âì, çâ® íâ®â ä㭪樮­ « áâ æ¨®­ à¥­ ®â­®á¨â¥«ì­® ¬ «ëå ¨§¬¥­¥­¨© H~ , ¥á«¨ ¤ ­­ ï äã­ªæ¨ï 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î rot rot H~ ; k2 H~ = 0 ¯à¨ ãá«®¢¨¨ ­  £à ­¨æ¥ ®¡« á⨠~n  rot H~ = 0. ˆ­ ç¥ £®¢®àï, ä㭪樮­ « áâ æ¨®­ à¥­, ¥á«¨ H~ ï¥âáï à¥è¥­¨¥¬ ãà ¢­¥­¨© Œ ªá¢¥««  ¨ 㤮¢«¥â¢®àï¥â ®¡ëç­ë¬ í«¥ªâத¨­ ¬¨ç¥áª¨¬ £à ­¨ç­ë¬ ãá«®¢¨ï¬. „®ª § â¥«ìá⢮. ‡ ¯¨è¥¬ ᮮ⭮襭¨¥ ¢ ¢¨¤¥

Z

Z

V

V

k2 jH j2 dV =

jrot H~ j2 dV:

ã¤¥¬ ¢ àì¨à®¢ âì äã­ªæ¨î H~ , ¯à¥¤¯®« £ ï ¤«ï ã¯à®é¥­¨ï H~ ¢¥é¥á⢥­­®© ä㭪樥© ª®®à¤¨­ â. ‡ ¬¥â¨¬, çâ®  ­ «®£¨ç­® ¬®¦­® ¤®ª § âì áâ æ¨®­ à­®áâì (­¥áª®«ìª® ¡®«¥¥ £à®¬®§¤ª®) ¨ ­¥ ¤¥« ï í⮣® ¯à¥¤¯®«®¦¥­¨ï. ®«ãç ¥¬

Z

Z

Z

V

V

V

k2 jH j2 dV + 2k2 H~ H~ dV = 2 rot H~ rot H~ dV: ’¥¯¥àì ¢®á¯®«ì§ã¥¬áï ⮦¤¥á⢮¬ ¢¥ªâ®à­®£®  ­ «¨§ 

~ div (H~  rot H~ ) = rot H~
rot H~ ; H~
rot rot H; ®âªã¤  rot H~
rot H~ = H~
rot rot H~ + div (H~  rot H~ ): ®¤áâ ¢«ïï íâ® ¢ à ­¥¥ ­ ©¤¥­­®¥ ãà ¢­¥­¨¥, ¯®«ã稬

Z

Z

Z

V

V

V

k2 jH~ j2 dV = 2 (rot rot H~ ; k2 H~ )
H~ dV +

div (H~  rot H~ ) dV:

‡ ¬¥­ïï ¨­â¥£à « ®â ¤¨¢¥à£¥­æ¨¨ ¯®¢¥àå­®áâ­ë¬ ¨­â¥£à «®¬, ¨¬¥¥¬ 205

Z

Z

I

V

V

S

k2 jH~ j2 dV = 2 (rot rot H~ ; k2 H~ )
H~ dV ;

(~n  rot H~ )
H~ dS:

ˆ§ ¯®«ã祭­®£® ᮮ⭮襭¨ï á«¥¤ã¥â, çâ® k2 = 0 ¢á«¥¤á⢨¥ ¯à®¨§¢®«ì­®á⨠H~ ⮫쪮 ¯à¨ ãá«®¢¨¨: 1)rot rot H~ ; k2 H~ = 0 ¢® ¢á¥¬ ®¡ê¥¬¥ V ¨ 2)~n  rot H~ = 0 ­  ¯®¢¥àå­®á⨠S; ®£à ­¨ç¨¢ î饩 íâ®â ®¡ê¥¬: ’ ª¨¬ ®¡à §®¬, í«¥ªâத¨­ ¬¨ç¥áª®¥ £à ­¨ç­®¥ ãá«®¢¨¥ ï¥âáï ¢ ¤ ­­®¬ á«ãç ¥ ¥áâ¥á⢥­­ë¬ £à ­¨ç­ë¬ ãá«®¢¨¥¬ ¢ à¨ æ¨®­­®© § ¤ ç¨. ®í⮬ã ä㭪樨 áà ¢­¥­¨ï ­¥ ®¡ï§ â¥«ì­® ¤®«¦­ë 㤮¢«¥â¢®àïâì £à ­¨ç­ë¬ ãá«®¢¨ï¬. Œë ¤®ª § «¨, çâ® k2 áâ æ¨®­ à­® ¯à¨ H~ , ïî饩áï ®¤­®© ¨§ ᮡá⢥­­ëå ä㭪権 १®­ â®à . ‘¢®©á⢮ áâ æ¨®­ à­®á⨠¯®§¢®«ï¥â ¯à¨¬¥­¨âì ¬¥â®¤ ¨âæ , ­ ¯à¨¬¥à, ¤«ï ­ å®¦¤¥­¨ï ᮡá⢥­­ëå ä㭪権 ¨ ᮡá⢥­­ëå §­ ç¥­¨©. „à㣮© ¢ à¨ ­â { ¯®¤áâ ­®¢ª  ¢ ä㭪樮­ « ¯à¨¡«¨¦¥­­®© ä㭪樨 ¯®«ï. ‚ í⮬ á«ãç ¥ áâ æ¨®­ à­®áâì ¤ ¥â ¢®§¬®¦­®áâì 㬥­ìè¨âì ¯®£à¥è­®áâì. ‚ ç áâ­®áâ¨, â ª¨¬ ®¡à §®¬ ¬®¦­® ¯®«ãç¨âì â ª ­ §ë¢ ¥¬ãî ä®à¬ã«ã ¢®§¬ã饭¨©. ãáâì, ­ ¯à¨¬¥à, १®­ â®à ¤¥ä®à¬¨à®¢ ­ â ª, çâ® ¥£® ®¡ê¥¬ V ¯à¥¢à â¨«áï ¢ ®¡ê¥¬ V1 §  áç¥â ¨áª«î祭¨ï ¬ «®£® ®¡ê¥¬  V2 (á¬. à¨á.). ‚ëïá­¨¬, ª ª ¨§¬¥­¨âáï ᮡV 1111 0000 á⢥­­®¥ §­ ç¥­¨¥ k12 : V1

1111 0000 V 0000 1111 0000 1111 2

k2 =

R jrot H~ (2)j2 dV V R jH~ (2)j2 dV ; 1

V1

£¤¥ H~ (2) { ᮡá⢥­­ ï äã­ªæ¨ï ¤¥ä®à¬¨à®¢ ­­®£® १®­ â®à . ®«ì§ãïáì áâ æ¨®­ à­®áâìî k2 , ¢¬¥áâ® H~ (2) ¯®¤áâ ¢¨¬ H~ (1) { ᮡá⢥­­ãî äã­ªæ¨î ­¥¤¥ä®à¬¨à®¢ ­­®£® १®­ â®à . ’®£¤  ¯®«ã稬: 206

R jrot H~ (1)j2 dV k2  V R ~ (1) 2 jH j dV 1

V1

=

8 > < 2 k1 >1 :

+

R jrot H~ (1)j2 dV ; R jrot H~ (1)j2 dV V = V R ~ (1) 2 R jH j dV ; jH~ (1) j2 dV V V R jH~ (1)j2 dV R jrot H~ (1)j2 dV 9 > = V V R jH~ (1)j2 dV ; R jrot H~ (1)j2 dV > = ; V V 
W ;
W  2

2

2

= k12 1 +

=

2

E

W0

H

;

£¤¥ Z
WH = ; 2 jH~ (1) j2 dV ; ¯à¨à é¥­¨¥ § ¯ á®¢ ¬ £­¨â­®© í­¥à£¨¨;

"Z

V2


WE = ; 2 jE~ (1) j2 dV ; ¯à¨à é¥­¨¥ § ¯ á®¢ í«¥ªâà¨ç¥áª®© í­¥à£¨¨; V2

(â ª ª ª rot H~ = j!"E~ ); W0 { ¯®«­ë© § ¯ á í­¥à£¨¨ ¢ १®­ â®à¥. ®«ã祭­®¥ ᮮ⭮襭¨¥ ¯®§¢®«ï¥â ª ç¥á⢥­­®,   ¢ ­¥ª®â®àëå á«ãç ïå ¨ ª®«¨ç¥á⢥­­® ®æ¥­¨¢ âì ¨§¬¥­¥­¨ï १®­ ­á­®© ç áâ®âë ¯à¨ ¬ «ëå ¤¥ä®à¬ æ¨ïå १®­ â®à . Ž­® ¤ ¥â ¢®§¬®¦­®áâì ¯®«ãç¨âì ¥é¥ ®¤­® ¢ ¦­®¥ ᮮ⭮襭¨¥. à¥¦¤¥ ¢á¥£®, ¯®«ã祭­ãî ä®à¬ã«ã ¬®¦­® § ¯¨á âì ¢ ¢¨¤¥ k2 ; k12 =
WE ;
WH k12 W0 ¨«¨, ãç¨â뢠ï, çâ® k2 = !2", k2 ; k12  2
! =
WE ;
WH : k12 !1 W0 ˆ§¬¥­¥­¨ï í­¥à£¨¨ ¬®£ãâ ¡ëâì ¢ëà ¦¥­ë ç¥à¥§ ¯«®â­®áâ¨ í«¥ªâà¨ç¥áª®© ¨ ¬ £­¨â­®© í­¥à£¨© ¢¡«¨§¨ á⥭ª¨ wE ¨ wH :
WE ;
WH = (wE ; wH )

V; 207

£¤¥
V { ¨§¬¥­¥­¨¥ ®¡ê¥¬  १®­ â®à . Ž¤­ ª® ¢¥«¨ç¨­  12 (wH ; wE ) = p (p { á।­¥¥ ¤ ¢«¥­¨¥ ­  á⥭ªã १®­ â®à  á® áâ®à®­ë í«¥ªâ஬ £­¨â­®£® ¯®«ï). ’ ª¨¬ ®¡à §®¬,
WE ;
WH = ; 2p
V; ­® p

V = ;
W0 , £¤¥
W0 ¥áâì ¨§¬¥­¥­¨¥ § ¯ á¥­­®© í­¥à£¨¨ í«¥ªâ஬ £­¨â­®£® ¯®«ï ¯à¨  ¤¨ ¡ â¨ç¥áª®© ¤¥ä®à¬ æ¨¨ १®­ â®à . ®í⮬ã
WE ;
WH = 2
W0 : ®¤áâ ¢«ïï, ¯®«ã稬
! =
W0 ; !0 W0 ®âªã¤ , ¨­â¥£à¨àãï, ¯®«ãç ¥¬ ᮮ⭮襭¨¥

W0 = const: !0

0 ‚¥«¨ç¨­  W !0 , ª®â®à ï ®áâ ¥âáï ¯®áâ®ï­­®© ¯à¨ ¬¥¤«¥­­®© ( ¤¨ ¡ â¨ç¥áª®©) ¤¥ä®à¬ æ¨¨ á⥭®ª, ï¥âáï  ¤¨ ¡ â¨ç¥áª¨¬ ¨­¢ à¨ ­â®¬.  «¨ç¨¥  ¤¨ ¡ â¨ç¥áª®£® ¨­¢ à¨ ­â  ¯®§¢®«ï¥â ¢ à拉 á«ãç ¥¢ ®¯à¥¤¥«ïâì ã室ë ç áâ®âë ¯à¨ ¤¥ä®à¬ æ¨¨ á⥭®ª (á¬. à¨á.).

000 111 111 000 6.6.

11 00 00 11 0 0 1 001 11 0 1 0 01 1 0 1 0 1 0 01 1 0 1

®â¥à¨ ¢ १®­ â®à å

…᫨ á⥭ª¨ १®­ â®à  ­¥ ïîâáï ¨¤¥ «ì­® ¯à®¢®¤ï騬¨, ⮠᢮¡®¤­ë¥ ª®«¥¡ ­¨ï ¡ã¤ãâ § âãå î騬¨. Œ â¥¬ â¨ç¥áª¨ íâ® ¯à®ï¢«ï¥âáï ¢ ⮬, ç⮠ᮡá⢥­­ë¥ §­ ç¥­¨ï ¯à¨®¡à¥â îâ ¬­¨¬ãî ç áâì (®¡ëç­® 208

¬ «ãî). …¥ ¢¥«¨ç¨­ã ¬®¦­® ¢ëç¨á«¨âì á ¯®¬®éìî ᮮ⭮襭¨ï, ¯®«ã祭­®£® à ­¥¥:

Z

Z

V

V

km2 jH~ m j2 dV =

I

jrot H~ m j2 dV ; (H~ m  rot H~ m )~n dS: S

’ ª ª ª rot H~ m = j!m "E~ m , â® í⮠ᮮ⭮襭¨¥ ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¢¨¤¥

Z

Z

V

V

km2 jH~ m j2 dV =

I

jrot H~ m j2 dV + j!m " (E~ m  H~ m )~n dS: S

à¨ ­¥¨¤¥ «ì­®© ¯à®¢®¤¨¬®á⨠¯®¢¥àå­®áâ­ë© ¨­â¥£à « ¢ ­ã«ì ­¥ ®¡à é ¥âáï. ˆ§ í⮣® ᮮ⭮襭¨ï ¬®¦­® ­ ©â¨ km2 : R jrot H~ j2 dV H(E~  H~  )~n dS m m m km2 = V R ~ 2 + j!" S R ~ 2 ; V

jHm j dV

V

jHm j dV

 §¤¥«ïï ¢¥é¥á⢥­­ãî ¨ ¬­¨¬ãî ç á⨠¢â®à®£® á« £ ¥¬®£®, ¯®«ã稬

H

Re (E~ m  H~ m )~n dS 2 0 2 ; km = km + j!m " S R ~ 2 jHm j dV V

£¤¥ km0 2 { ᮡá⢥­­®¥ §­ ç¥­¨¥ á ¬ «®© ¯®¯à ¢ª®© §  áç¥â ¢¥é¥á⢥­­®© ç á⨠¤®¡ ¢®ç­®£® ç«¥­ . ‚â®à®¥ á« £ ¥¬®¥ ¤ ¥â ¬­¨¬ãî ¤®¡ ¢ªã, ®¡ãá«®¢«¨¢ îéãî § âãå ­¨¥ ᢮¡®¤­ëå ª®«¥¡ ­¨© ¢ १®­ â®à¥. ‚ ¯¥à¢®¬ ¯à¨¡«¨¦¥­¨¨ km0 2 = !m0 2 ", !m0  !m. “ç¨âë¢ ï ¬ «®áâì ¬­¨¬®© ¯®¯à ¢ª¨, ¬®¦­® § ¯¨á âì

8 9 H > Re (E~ m  H~ m )~n dS > < = km  km0 >1 + j 2!km0 "2 S R ~ 2 > jHm j dV m : ; V 8 1 Re H(E~  H~  )~n dS 9 > > m m < = 2 S = km0 >1 + j : R 2!m 2 jH~ m j2 dV > : ; V

209

=

‚ ᪮¡ª å

1 Re I (E~  H~  )~n dS = P | m m 2 S

¯®â®ª í­¥à£¨¨, ­ ¯à ¢«¥­­ë© ¢ á⥭ª¨, â.¥. ¬®é­®áâì ¯®â¥àì ¢ á⥭ª å,

Z ~ 2 2 jHm j dV

V

= W|

§ ¯ á í­¥à£¨¨ ¢ १®­ â®à¥. Žâ­®è¥­¨¥

!m W = Q P

­ §ë¢ îâ ¤®¡à®â­®áâìî १®­ â®à , ¢¥«¨ç¨­ã Pr = !m W { ॠªâ¨¢­®© ¬®é­®áâìî ¢ १®­ â®à¥. „®¡à®â­®áâì, â ª¨¬ ®¡à §®¬, à ¢­  ®â­®è¥­¨î ॠªâ¨¢­®© ¬®é­®á⨠ª ¬®é­®á⨠¯®â¥àì. ‚ íâ¨å ®¡®§­ ç¥­¨ïå ¯®«ãç ¥¬ km = km0 (1 + j 21Q ): Žâá ­ å®¤¨¬, çâ® § âãå ­¨¥ ¯à®¨á室¨â ¯® íªá¯®­¥­æ¨ «ì­®¬ã § ª®­ã e j!m t e;!m t=2Q : à¨¢¥¤¥¬ ¢ë¢®¤ í⮣® १ã«ìâ â  ¤à㣨¬, ¡®«¥¥ ¯à®§à ç­ë¬ ᯮᮡ®¬. ˆ§¬¥­¥­¨¥ § ¯ á  í­¥à£¨¨ §  ¢à¥¬ï dt à ¢­®

dW = ; P dt: Œ®é­®áâì ¯®â¥àì, ¢ á¢®î ®ç¥à¥¤ì, ¯à®¯®à樮­ «ì­  § ¯ áã í­¥à£¨¨ ¢ १®­ â®à¥, ¯à¨ç¥¬ ª®íää¨æ¨¥­â ¯à®¯®à樮­ «ì­®á⨠¨¬¥¥â à §¬¥à­®áâì ®¡à â­®£® ¢à¥¬¥­¨. …£® ¬®¦­® § ¯¨á âì ¢ ¢¨¤¥ !m =Q, ¯à¨ç¥¬ !m { ç áâ®â  ᢮¡®¤­ëå ª®«¥¡ ­¨©. ®¤áâ ¢«ïï, ¯®«ã稬 ®âªã¤ 

dW = ; !Qm W dt; 210

W = W0 e;!m t=Q ; ¯à¨ç¥¬ Q = !mPW . ‚ëç¨á«¥­¨¥ ¯®â¥àì ¢ á⥭ª å âॡã¥â §­ ­¨ï â ­£¥­æ¨ «ì­®© á®áâ ¢«ïî饩 í«¥ªâà¨ç¥áª®£® ¯®«ï ­  ¯®¢¥àå­®á⨠á⥭ª¨ E~ mt , ®­  ¬®¦¥â ¡ëâì ­ ©¤¥­  á ¯®¬®éìî £à ­¨ç­®£® ãá«®¢¨ï ‹¥®­â®¢¨ç , á¯à ¢¥¤«¨¢®£® ¯à¨ ᨫ쭮¬ ᪨­-íä䥪â¥. „¥©á⢨⥫쭮, I I P = 21 Re (E~ m  H~ m )~n dS = 21 Re (~n  E~ m )H~ m dS: S

S

‘®£« á­® £à ­¨ç­®¬ã ãá«®¢¨î ‹¥®­â®¢¨ç 

~n  E~ m = 
H~ m ; + j { ¯®¢¥àå­®áâ­®¥ ᮯà®â¨¢«¥­¨¥. ®¤áâ ¢«ïï íâ® ¯®¤ ¨­â¥£¤¥  = 1  £à «, ­ å®¤¨¬ I 1 I jH~ j2 dS: P = 12 Re  jH~ m j2 dS = 2 m S

S

®¤áâ ¢«ïï ¬®é­®áâì ¢ ¢ëà ¦¥­¨¥ ¤«ï Q, ¯®«ã稬

Q =

R !m 2 jH~ m j2 dV V

=

H~ 2 2 jHm j dS 1

R jH~ j2 dV m V !m H ~ 2 : jH j dS S

S

m

“ç¨â뢠ï, çâ® !m  = 2 2 , £¤¥ Œ { ¬ £­¨â­ ï ¯à®­¨æ ¥¬®áâì Œ ¬¥â ««  á⥭ª¨, ¯®«ã稬 R jH~ j2 dV m Q = 2 VH ~ 2 : Œ jHm j dS S

211

…᫨ ¨§¢¥áâ­® à á¯à¥¤¥«¥­¨¥ ¯®«ï ¢ १®­ â®à¥, â® ¤®¡à®â­®áâì ¬®¦¥â ¡ëâì ¢ëç¨á«¥­  ¯® ¯®«ã祭­®© ä®à¬ã«¥. Ž¤­ ª® ®¯à¥¤¥«¥­­ë© ¨­â¥à¥á ¯à¥¤áâ ¢«ï¥â ¢®§¬®¦­®áâì áà ¢­¨â¥«ì­® £àã¡® ¯à®¨§¢®¤¨âì ®æ¥­ªã ¤®¡à®â­®áâ¨. ’ ª ï ®æ¥­ª  ¬®¦¥â ¡ëâì ¯®«ã祭 , ¥á«¨ ¨­â¥£à «ë ¢ëà §¨âì ç¥à¥§ á।­¨¥ §­ ç¥­¨ï: Hm j2V ; Q = 2 VS jjH Œ m j2S £¤¥ V { ®¡ê¥¬ १®­ â®à , S { ¯®¢¥àå­®áâì १®­ â®à , jHm j2V { á।­¨© ª¢ ¤à â ¬ £­¨â­®£® ¯®«ï ¯® ®¡ê¥¬ã १®­ â®à , jHm j2S { á।­¨© ª¢ ¤à â ¬ £­¨â­®£® ¯®«ï ¯® ¯®¢¥àå­®á⨠१®­ â®à . ‚¢¥¤¥¬ ª®íää¨æ¨¥­â ä®à¬ë १®­ â®à  A ᮣ« á­® á«¥¤ãî饬ã ᮮ⭮襭¨î: 2 jHm j2V = A jHm j2S : à¨ â ª®¬ ®¯à¥¤¥«¥­¨¨ ¤«ï १®­ â®à®¢, ¨¬¥îé¨å ¡«¨§ª¨¥ à §¬¥àë ¢® ¢á¥å ­ ¯à ¢«¥­¨ïå, ¢¥«¨ç¨­  A ¡«¨§ª  ª ¥¤¨­¨æ¥. ‚ íâ¨å ®¡®§­ ç¥­¨ïå

Q = VS  A = VV  A; Œ  Œ

£¤¥ V { ®¡ê¥¬ ᪨­-á«®ï. Žá®¡¥­­® ¯à®áâ® ¢ë£«ï¤¨â íâ  ä®à¬ã« , ¥á«¨  = Œ ¨ A = 1. ’®£¤  ¤®¡à®â­®áâì à ¢­  Q = VV . ‡ ¬¥â¨¬, çâ® íâ  ã¯à®é¥­­ ï ä®à¬ã«  ­¥ ¤ ¥â ¯à¨«¨ç­®© â®ç­®áâ¨, ¥á«¨ A ᨫ쭮 ®â«¨ç ¥âáï ®â ¥¤¨­¨æë.  ¯à¨¬¥à, ¤«ï ¬®¤ ⨯  H0mn ¢ ªà㣫®¬ ¢®«­®¢®¤¥ íâ  ä®à¬ã«  ­¥¯à¨¬¥­¨¬ , â ª ª ª ¯à¨ ¡®«ì讬 ¤¨ ¬¥âॠ(  ªà ) ¯®«¥ ­  á⥭ª å ¬ «® ¯® áà ¢­¥­¨î á ¯®«¥¬ ¢ ®¡ê¥¬¥, ¡« £®¤ àï 祬ã A  1. „«ï १®­ â®à®¢, ¨¬¥îé¨å ¯à¨¬¥à­® ®¤¨­ ª®¢ë¥ à §¬¥àë ¢® ¢á¥å ­ ¯à ¢«¥­¨ïå, íâã ä®à¬ã«ã ¬®¦­® ¥é¥ ã¯à®áâ¨âì.  ¯à¨¬¥à, ¤«ï ªã¡  V = a3 ; S = 6a2; VS = 61 a; ¤«ï è à  3 V = d6 ; S = d2 ; VS = 16 d: à¨¡«¨¦¥­­® ¬®¦­® ¯®«®¦¨âì 212

V = 1a : S 6 áà

®¤áâ ¢«ïï íâ® ®â­®è¥­¨¥ ¢ ¢ëà ¦¥­¨¥ ¤«ï ¤®¡à®â­®áâ¨, ¯®«ã稬 ®æ¥­ªã Q  16 aáà  A: Œ p “ç¨â뢠ï, çâ® aáà  १ ¨   १, ­ å®¤¨¬, çâ®

Q  1=2 ; â.¥. á 㬥­ì襭¨¥¬ ¤«¨­ë ¢®«­ë ¤®¡à®â­®áâì ¯ ¤ ¥â. 6.6.1. à¨¬¥àë à áç¥â  ¤®¡à®â­®áâ¨

 áç¥â ¤®¡à®â­®á⨠१®­ â®à  á ¬®¤®© E010 . Œ £­¨â­®¥ ¯®«¥ ¢ â ª®¬ १®­ â®à¥ ­¥ § ¢¨á¨â ®â ª®®à¤¨­ âë z . ‡ ¢¨á¨¬®áâì ®â r ¨¬¥¥â ¢¨¤

H' = J1 (kr): ˆ­â¥£à « ¯® ®¡ê¥¬ã à ¢¥­

Z V

jH~ m j2 dV =

Za 0

J12 (kr) 2r h dr = 2 h

Za 0

2 J12 (kr)r dr = 2h a2 J12 (ka) =

= ha2 J12 (ka): ®¢¥àå­®áâ­ë© ¨­¥â¥£à « ᪫ ¤ë¢ ¥âáï ¨§ 㤢®¥­­®£® ¨­â¥£à «  ¯® â®à楢®© ¯®¢¥àå­®á⨠¨ ¨­â¥£à «  ¯® ¡®ª®¢®© 樫¨­¤à¨ç¥áª®© ¯®¢¥àå­®áâ¨: I Za 2 2 jH~ m j dS = 2 J12 (kr) 2r dr + 2ah J12 (ka) = 2
2 a2 J12 (ka) + 0 S + 2ah J12(ka) = 2a(a + h) J12 (ka): 213

®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï ¤®¡à®â­®á⨠(¯à¨ Œ = ) ¯®«ã稬 2 J12 (ka) 1 ah Q = 2 2a(ha a + h) J12 (ka) =  a + h : —¨á«¥­­ë© ¯à¨¬¥à. ãáâì, ­ ¯à¨¬¥à, १®­ â®à ¨§ ¬¥¤¨ ¨¬¥¥â á«¥¤ãî騥 ¯ à ¬¥âàë: a=30 á¬, h=30 á¬, =80 á¬, =2:9
10;4 á¬. ’®£¤  Q = 1 aa+
hh = 5
104. „®¡à®â­®áâì १®­ â®à , ®¡à §®¢ ­­®£® ¨§ ®â१ª  «¨­¨¨, § ª®à®ç¥­­®£® ­  ª®­æ å. …᫨ ¤«¨­  â ª®£® १®­ â®à  ¢¥«¨ª  ¯® áà ¢­¥­¨î á ¯®¯¥à¥ç­ë¬ à §¬¥à®¬, â® ¢ ¯¥à¢®¬ ¯à¨¡«¨¦¥­¨¨ ¬®¦­® ¯à¥­¥¡à¥çì ¯®â¥àﬨ ¢ § ¬ëª îé¨å ¯¥à¥¬ë窠å. ’®£¤  ãç¨â뢠îâáï ⮫쪮 ¯®â¥à¨ ¢ ¡®ª®¢ëå á⥭ª å.  ­¥¥ ¤«ï § âãå ­¨ï ¢ ¢®«­®¢®¤¥ ¬ë ¨¬¥«¨ á«¥¤ãî饥 ¢ëà ¦¥­¨¥: H jH~ j2 dC m    1 CR  =
= 2  ¬ s   2 jH~ mj2 dS : S 1 ;  ªà Žâ­®è¥­¨¥ ¨­â¥£à «®¢, ­¥®¡å®¤¨¬®¥ ¤«ï ¢ëç¨á«¥­¨ï ¤®¡à®â­®áâ¨, ¬®¦­® ¢ëç¨á«¨âì á«¥¤ãî騬 ®¡à §®¬: R jH~ j2 dV R jH~ j2 dS m m 2 2   V S Q =  R ~ 2  H ~ 2 : Œ jHm j dS Œ jHm j dC S

C

‚ â® ¦¥ ¢à¥¬ï ¨§ ᮮ⭮襭¨ï ¤«ï § âãå ­¨ï ­ å®¤¨¬ R jH~ j2 dS m 1 SH = 2  Œ s   2 : 2 ~ jHm j dC C 1;  ªà ®¤áâ ¢«ïï íâ® ®â­®è¥­¨¥ ¢ ¢ëà ¦¥­¨¥ ¤«ï ¤®¡à®â­®áâ¨, ¯®«ã稬 214

 s Q = 

1

1

  2 : ; ªà

®«ã祭­®¥ ᮮ⭮襭¨¥ ¯®§¢®«ï¥â ­¥¯®á।á⢥­­® ®æ¥­¨âì ¤®¡à®â­®áâì, ¥á«¨ ¨§¢¥áâ­® § âãå ­¨¥ ¢®«­®¢®¤ , ¨§ ª®â®à®£® ®¡à §®¢ ­ १®­ â®à. „«ï ¨««îáâà æ¨¨ ¢ëç¨á«¨¬ ¤®¡à®â­®áâì १®­ â®à , ®¡à §®¢ ­­®£® ¨§ ªà㣫®£® ¢®«­®¢®¤  á ¬®¤®© H~ 01 , ¤«ï ª®â®à®©   2  s ªà  =    2 : 1 ;  ªà ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï ¤®¡à®â­®áâ¨, ¯®«ã稬  ªà 2 a Q =   ; â.¥. ¯à¨ ¯®áâ®ï­­®© ¢¥«¨ç¨­¥ ªà (¯à¨ ¯®áâ®ï­­®¬ à §¬¥à¥) ¤®¡à®â­®áâì à áâ¥â á 㬥­ì襭¨¥¬ ¤«¨­ë ¢®«­ë ª ª ;5=2 . ®â¥à¨ ¢ â®à楢ëå á⥭ª å á à®á⮬ ç áâ®âë à áâãâ, ¯®í⮬㠨¬¥¥âáï ®¯â¨¬ã¬, ª®£¤  ¤®¡à®â­®áâì ®¡ãá«®¢«¥­  ¢ ®á­®¢­®¬ ¯®â¥àﬨ ¢ â®àæ å. —¨á«¥­­ë© ¯à¨¬¥à. ãáâì  = 10 á¬, a = 20 á¬, ªà = 1:64a = 33 á¬,   10;4 ᬠ(¤«ï ¬¥¤¨). ®«ì§ãïáì ¯à¨¢¥¤¥­­®© ¢ëè¥ ä®à¬ã«®©, ¯®«ã稬 Q = 2
106: à¨¬¥à. ‚ ª ç¥á⢥ ¤à㣮£® ¯à¨¬¥à  ¢ëç¨á«¨¬ ¤®¡à®â­®áâì ®â१ª  ª® ªá¨ «ì­®© «¨­¨¨, ¤«ï ª®â®à®©  = 2RZ1 ; 0 215

£¤¥ R1 { ᮯà®â¨¢«¥­¨¥ ¥¤¨­¨æë ¤«¨­ë «¨­¨¨. ®¤áâ ¢«ïï ¢ ä®à¬ã«ã, ¯®«ã稬 Q = 2RZ0 : 1 • à ªâ¥à­®, çâ® ¢ëç¨á«¥­­ ï â ª¨¬ ®¡à §®¬ (¡¥§ ãç¥â  ¯®â¥àì ¢ â®àæ å) ¤®¡à®â­®áâì ­¥ § ¢¨á¨â ®â ç¨á«  ¯®«ã¢®«­ ¢ १®­ â®à¥. à¨ç¨­  ®ç¥¢¨¤­ . 6.6.2. ®â¥à¨ ¢ á।¥

…᫨ १®­ â®à § ¯®«­¥­ ¯à®¢®¤ï饩 á।®©, â® ¤®¡à®â­®áâì ®¯à¥¤¥«ï¥âáï â ª¦¥ ¯®â¥àﬨ ¢ í⮩ á।¥. ®â¥à¨ í­¥à£¨¨ à ¢­ë Z E 2 P = 2 dV; V   ¯®«­ ï í­¥à£¨ï ¢ १®­ â®à¥ ¢ëà ¦ ¥âáï ¨­â¥£à «®¬ Z 2 W = "E2 dV; ¯®í⮬㠤®¡à®â­®áâì à ¢­  ‚¥«¨ç¨­ 

V

Q = !PW = !" :

=! = "00 = tg  | " "0

â ­£¥­á 㣫  ¯®â¥àì ¢ á।¥, ¯®í⮬ã

Q = tg1 :

…᫨ ¯®â¥à¨ ¥áâì ¨ ¢ á⥭ª å ¨ ¢ á।¥, â® ®¡é ï ¤®¡à®â­®áâì ¡ã¤¥â ¨«¨

Q = P !+W P áà áâ

Páà Páâ 1 1 1 Q = ! W + ! W = Qáà + Qáâ :

’ ª¨¬ ®¡à §®¬, á㬬¨àãîâáï ®¡à â­ë¥ ¢¥«¨ç¨­ë ¤®¡à®â­®á⥩. 216

6.6.3. ®â¥à¨ ­  ¨§«ã祭¨¥

…᫨ ¢ á⥭ª å १®­ â®à  ¨¬¥îâáï ®â¢¥àáâ¨ï, â® ç¥à¥§ ­¨å ¬®¦¥â ¨§«ãç âìáï í«¥ªâ஬ £­¨â­ ï í­¥à£¨ï. ˆ§«ã祭¨¥ ¬®¦¥â ¯à®¨á室¨âì ¢ ᢮¡®¤­®¥ ¯à®áâà ­á⢮ ¨«¨ ¢®«­®¢®¤.  ¨¡®«ì訩 ¨­â¥à¥á ¯à¥¤áâ ¢«ï¥â ¨§«ã祭¨¥ ¢ ᮣ« á®¢ ­­ãî «¨­¨î ¯¥à¥¤ ç¨. ’ ª ª ª ¬®é­®áâì ¨§«ã祭¨ï ¯à®¯®à樮­ «ì­  § ¯ á¥­­®© í­¥à£¨¨, â® ¨§«ã祭¨¥ ¬®¦¥â ¡ëâì ®å à ªâ¥à¨§®¢ ­® ­¥ª®â®à®© ¤®¡à®â­®áâìî, ª®â®àãî ­ §ë¢ îâ ¢­¥è­¥© ¤®¡à®â­®áâìî

Q¢­ = P! W : ¨§«

„®¡à®â­®áâì १®­ â®à , ­ £à㦥­­®£® ᮣ« á®¢ ­­®© «¨­¨¥©, ­ §ë¢ îâ ­ £à㦥­­®© ¤®¡à®â­®áâìî Q­,   ¤®¡à®â­®áâì ᮡá⢥­­® १®­ â®à  (®¯à¥¤¥«ï¥¬ãî ¯®â¥àﬨ ¢ १®­ â®à¥) ­ §ë¢ îâ ᮡá⢥­­®© ¤®¡à®â­®áâìî Q0 . Žç¥¢¨¤­®, çâ® 1 = 1 + 1 : Q­ Q0 Q¢­ ˆ­®£¤  ¤«ï ¢ëç¨á«¥­¨ï ¤®¡à®â­®á⨠¨á¯®«ì§ãîâ ¯®­ï⨥ \£¥®¬¥âà¨ç¥áª¨© ä ªâ®à". ® ®¯à¥¤¥«¥­¨î £¥®¬¥âà¨ç¥áª¨¬ ä ªâ®à®¬ ­ §ë¢ îâ ¢¥«¨ç¨­ã, ¢ëç¨á«ï¥¬ãî ¯® ä®à¬ã«¥ R jH j2 dV G = ! 0 VR jH j2 dS : S

â  ¢¥«¨ç¨­  ¨¬¥¥â à §¬¥à­®áâì ᮯà®â¨¢«¥­¨ï. ®áª®«ìªã  ­ «®£¨ç­®¥ á®ç¥â ­¨¥ ¨­â¥£à «®¢ ¨¬¥¥â ¬¥áâ® â ª¦¥ ¢ ¢ëà ¦¥­¨¨ ¤«ï ᮡá⢥­­®© ¤®¡à®â­®áâ¨, â® ¤®¡à®â­®áâì ¬®¦¥â ¡ëâì ¢ëà ¦¥­  á«¥¤ãî騬 ®¡à §®¬:

Q = G ;

£¤¥

 = ¯®¢¥àå­®áâ­®¥ ᮯà®â¨¢«¥­¨¥.

r !

2 |

217

6.7.

‚ë­ã¦¤¥­­ë¥ ª®«¥¡ ­¨ï ¢ १®­ â®à å

 ­¥¥ à áᬠâਢ «¨áì ᢮¡®¤­ë¥ ª®«¥¡ ­¨ï, ª®â®àë¥ ¢ ॠ«ì­®¬ १®­ â®à¥ § âãå îâ. ¥§ âãå î騥 ª®«¥¡ ­¨ï ¬®£ãâ áãé¥á⢮¢ âì ¢ १®­ â®à¥, ¥á«¨ ¬®é­®áâì ¯®â¥àì ª®¬¯¥­á¨àã¥âáï ­¥§ ¢¨á¨¬ë¬ ¨áâ®ç­¨ª®¬. ‚®§¡ã¦¤¥­¨¥ १®­ â®à  ¬®¦¥â ®áãé¥á⢫ïâìáï § ¤ ­­ë¬¨ ⮪ ¬¨ ¨«¨ ç¥à¥§ £à ­¨æ㠮⠢­¥è­¥£® £¥­¥à â®à . ®á«¥¤­¨© ᯮᮡ ¬®¦¥â ¡ëâì ᢥ¤¥­ ª ¢®§¡ã¦¤¥­¨î ¯®¢¥àå­®áâ­ë¬¨ ⮪ ¬¨. à¨ à¥è¥­¨¨ § ¤ ç¨ ® ¢ë­ã¦¤¥­­ëå ª®«¥¡ ­¨ïå ¬ë ¡ã¤¥¬ ¯à¥¤¯®« £ âì, çâ® ¨§¢¥áâ­® à¥è¥­¨¥ § ¤ ç¨ ® ᢮¡®¤­ëå ª®«¥¡ ­¨ïå ¤«ï ¤ ­­®£® १®­ â®à  á ®¡ëç­ë¬¨ £à ­¨ç­ë¬¨ ãá«®¢¨ï¬¨ ­  ¢á¥© ¯®¢¥àå­®áâ¨, ®£à ­¨ç¨¢ î饩 ®¡ê¥¬ १®­ â®à . 6.7.1. ‚®§¡ã¦¤¥­¨¥ १®­ â®à  § ¤ ­­ë¬¨ ⮪ ¬¨

à¥¤¯®«®¦¨¬, çâ® ¢­ãâਠ®¡ê¥¬  १®­ â®à  § ¤ ­ë ¯«®â­®áâ¨ í«¥ªâà¨ç¥áª®£® (J~e ) ¨ ¬ £­¨â­®£® (J~m ) ⮪®¢. …᫨ ¢®§¡ã¦¤¥­¨¥ ®áãé¥á⢫ï¥âáï ç¥à¥§ £à ­¨æã, ­  ª®â®à®© § ¤ ­  â ­£¥­æ¨ «ì­ ï á®áâ ¢«ïîé ï í«¥ªâà¨ç¥áª®£® ¯®«ï, â® § ¤ ç  ᢮¤¨âáï ª ¢®§¡ã¦¤¥­¨î १®­ â®à  ¯®¢¥àå­®áâ­ë¬ ¬ £­¨â­ë¬ ⮪®¬ á ¯«®â­®áâìî K~ m = ; E~  ~n; £¤¥ ~n { ¢­¥è­ïï ­®à¬ «ì. ¥è¥­¨¥ § ¤ ç¨ ­ ©¤¥¬ ¢ ¢¨¤¥ à §«®¦¥­¨ï ¯® ᮡá⢥­­ë¬ ¢¥ªâ®à­ë¬ äã­ªæ¨ï¬ १®­ â®à . Ž¤­ ª® ®¤­¨å ᮡá⢥­­ëå ä㭪権, ¢®®¡é¥ £®¢®àï, ­¥¤®áâ â®ç­®. „¥©á⢨⥫쭮, ᮡá⢥­­ë¥ ä㭪樨 㤮¢«¥â¢®àïîâ ãá«®¢¨î div E~ s = 0 ¨ div H~ s = 0, ª®â®à®¬ã ¯®«­®¥ ¯®«¥ ¬®¦¥â ­¥ 㤮¢«¥â¢®àïâì (â ª ª ª ¯à¨ ­ «¨ç¨¨ ⮪®¢ ¨¬¥îâáï ¨ § àï¤ë). ®í⮬ã à¥è¥­¨ï á«¥¤ã¥â § ¯¨á âì ¢ ¢¨¤¥ X ~ X ~ E~ = As Es ; grad 'e ; H~ = Bs Hs ; grad 'm : s

s

ˆ§ £à ­¨ç­ëå ãá«®¢¨© ¤«ï E~ ¨ H~ á«¥¤ã¥â, çâ® ­  ¯®¢¥àå­®áâ¨, ®£à ­¨ç¨¢ î饩 ®¡ê¥¬ १®­ â®à ,



m = 0: 'e jS = 0; @' @n S

Šà®¬¥ ⮣®, ¯®â¥­æ¨ «ì­ë¥ á« £ ¥¬ë¥ ¯®«ï ®¡« ¤ îâ ᢮©á⢮¬ ®à⮣®­ «ì­®á⨠¯® ®â­®è¥­¨î ª ᮡá⢥­­ë¬ äã­ªæ¨ï¬. —⮡ë íâ® ¯®ª § âì, ¢ëç¨á«¨¬ div (H~ s  grad 'e ): 218

div (H~ s  grad 'e ) = grad 'e
rot H~ s = j!s "E~ s
grad 'e : ˆ­â¥£à¨àãï ¯® ®¡ê¥¬ã १®­ â®à , ¯®«ã稬

I S

Z

(H~ s  grad 'e )
~n dS = j!" E~ s
grad 'e
dV: V

Žâá (¢ ᨫ㠣࠭¨ç­ëå ãá«®¢¨© «¥¢ ï ç áâì à ¢­  ­ã«î)

Z

V

€­ «®£¨ç­® í⮬ã

Z V

E~ s
grad 'e
dV = 0:

H~ s
grad 'm
dV = 0:

 ©¤¥¬ ãà ¢­¥­¨ï ¤«ï ä㭪権 'e ¨ 'm . „«ï í⮣® § ¯¨è¥¬ ãà ¢­¥­¨ï Œ ªá¢¥««  rot E~ = ; j!H~ ; J~m ; rot H~ = j!"E~ + J~e ¨ ¢®§ì¬¥¬ ¤¨¢¥à£¥­æ¨î ®â ®¡®¨å ãà ¢­¥­¨©. ®«ã稬 0 = ;j!
div H~ ; div J~e ; 0 = j!"
div E~ + div J~m : „¨¢¥à£¥­æ¨ï ®â«¨ç­  ®â ­ã«ï ⮫쪮 ¤«ï ¯®â¥­æ¨ «ì­®© ç á⨠E~ ¨ H~ :

j!
div grad 'm = div J~m ; j!"
div grad 'e = div J~e ; ¨«¨ 219

J~m ;
' = div J~e :
'm = div e j! j!" ‚ â® ¦¥ ¢à¥¬ï ¨¬¥îâ ¬¥áâ® ãà ¢­¥­¨ï ­¥¯à¥à뢭®á⨠®âªã¤ 

@e + div J~ = 0; e @t div J~e = ; j!e ;

¨ ®âªã¤ 

@m + div J~ = 0; m @t

div J~m = ; j!m : ®¤áâ ¢«ïï íâ® ¢ ãà ¢­¥­¨¥, ã¡¥¦¤ ¥¬áï, çâ® 'e ¨ 'm 㤮¢«¥â¢®àïîâ áâ â¨ç¥áª¨¬ ãà ¢­¥­¨ï¬ ã áá®­ 
'e = ; "e ;
'm = ; m : —â®¡ë ­ ©â¨ As ¨ Bs , ¢ëç¨á«¨¬ á«¥¤ãî騥 ¢ëà ¦¥­¨ï:

~ div (E~ s  H~ ) = H~
rot E~ s ; E~ s
rot H; div (E~  H~ s ) = H~ s
rot E~ ; E~
rot H~ s : à¨ í⮬ á«¥¤ã¥â ãç¥áâì, çâ® E~ s ¨ H~ s 㤮¢«¥â¢®àïîâ ãà ¢­¥­¨ï¬

rot E~ s = ; j!s H~ s ; rot H~ s = j!s "E~ s : “ç¨â뢠ï í⨠ãà ¢­¥­¨ï ¤«ï E~ s ¨ H~ s , ¯®«ã稬

div (E~ s  H~ ) = H~
(j!s H~ s ) ; E~ s
(j!"E~ + J~e ) = = j!s H~
H~ s ; j!"E~
E~ s ; J~e
E~ s ; div (E~  H~ s ) = H~ s
(;j!H~ ; J~m ) ; E~
(;j!s "E~ s ) = = ; j!H~
H~ s + j!"E~
E~ s ; J~m H~ s : 220

ˆ­â¥£à¨àãï ¯® ®¡ê¥¬ã १®­ â®à  ¨ ãç¨â뢠ï, çâ® â ­£¥­æ¨ «ì­ë¥ á®áâ ¢«ïî騥 E~ ¨ E~ s à ¢­ë ­ã«î ­  ¢á¥© ¯®¢¥àå­®á⨠S (á ãç¥â®¬ § ¬¥­ë E~ t ¬ £­¨â­ë¬ ⮪®¬ ­  ¢®§¡ã¦¤ î饩 £à ­¨æ¥), § ¬¥­ïï E~ ¨ H~ ¨å à §«®¦¥­¨¥¬ ¢ àï¤ ¨ ãç¨â뢠ï ᮮ⭮襭¨ï ®à⮣®­ «ì­®áâ¨, ¯®«ã稬

Z

Z

Z

V

Z

V

Z

V

V

V

V

j!s Bs H~ s
H~ s dV ; j!"As E~ s
E~ s dV =

;j!Bs H~ s
H~ s dV + j!s "A E~ s
E~ s dV =  ­¥¥ ¬ë ¯®«ã稫¨

Z

Z

V

V

Z

J~e E~ s dV; J~m H~ s dV:

" E~ s
E~ s dV =  H~ s
H~ s dV: ®¤áâ ¢«ïï, ¯®«ã稬 á¨á⥬㠨§ ¤¢ãå ãà ¢­¥­¨©

j!
As ; j!s Bs = as ; j!s As ; j!
Bs = bs ; £¤¥

as =

R J~ E~  dV e s ; RV ~ ~  ; bs  H
H dV V

s

s

=

R J~ H~  dV m s VR :  H~
H~  dV V

s

s

¥è ï íâã á¨á⥬㠮⭮á¨â¥«ì­® As ¨ Bs , ¯®«ã稬

!s as ; !bs s ; !s bs As = !a j (!2 ; !s2 ) ; Bs = j (!2 ; !s2 ) : …᫨ १®­ â®à ­¥ ¨¬¥¥â ¯®â¥àì, â® !s { ¢¥é¥á⢥­­ ï ¢¥«¨ç¨­ . ®í⮬ã As ¨ Bs ®¡à é îâáï ¢ ¡¥áª®­¥ç­®áâì ¯à¨ ! = !s , â.¥. ¯à¨ १®-

­ ­á¥. ¥ «ì­® à®áâ  ¬¯«¨âã¤ë ®£à ­¨ç¨¢ ¥âáï ¯®â¥àﬨ ¢ १®­ â®à¥. à¨ ­ «¨ç¨¨ ¯®â¥àì

!s2 = !s0 2 (1 + Qj ); s 221

¯®í⮬ã 02 !2 ; !s2 = !2 ; !s0 2 ; j !Qs : s

’ ª ª ª ! { ¢¥é¥á⢥­­ ï ¢¥«¨ç¨­ , â® §­ ¬¥­ â¥«ì ¢ ¢ëà ¦¥­¨ïå ¤«ï As ¨ Bs ­¥ ¬®¦¥â ®¡à â¨âìáï ¢ ­ã«ì. à¨ ! = !s ¯®«ã稬

As = Qs as !;0 bs ; Bs = Qs as !;0 bs ; s

s

â.¥. As = Bs . ˆ§ ¯®«ã祭­ëå ᮮ⭮襭¨© á«¥¤ã¥â, çâ® ¥á«¨ ç áâ®â  ¡«¨§ª  ª ®¤­®© ¨§ ᮡá⢥­­ëå ç áâ®â, â®  ¬¯«¨â㤠 ᮮ⢥âáâ¢ãîé¨å ¯®«¥© ®á®¡¥­­® ¢¥«¨ª . ’ ª ª ª ¤®¡à®â­®áâì Qs ®¡ëç­® ¢¥á쬠 ¡®«ìè ï ¢¥«¨ç¨­ , â® ¢¡«¨§¨ १®­ ­á  ¬®¦­® ¯à¥­¥¡à¥çì ¢á¥¬¨ ­¥à¥§®­ ­á­ë¬¨ ¬®¤ ¬¨ ¢ à §«®¦¥­¨¨ E~ ¨ H~ . ‡ ¯¨è¥¬ § ¢¨á¨¬®áâì  ¬¯«¨â㤠¢¡«¨§¨ १®­ ­á­®© ç áâ®âë. à¨ í⮬ ¬®¦­® ¯®« £ âì !  !s0 ¢¥§¤¥, ªà®¬¥ à §­®áâ­®£® ç«¥­  ¢ §­ ¬¥­ â¥«¥. ’®£¤  !s0 (as ; bs ) Qs (as ; bs )=!s0 = As  Bs  = 2 0 1 + j !Q0s2 (!2 ; !s0 2 ) j (!2 ; !s0 2 ) + !s

Qs

£¤¥

s

0 = Qs (1as+;jQbs )x=!s ; s 0 !: x = !!0 ; !!s  2
0 !s s

â® १®­ ­á­ ï § ¢¨á¨¬®áâì á 㧪®© ¯®«®á®©, ®¯à¥¤¥«ï¥¬®© ¢¥«¨ç¨­®© Qs . ‘«¥¤ã¥â ®â¬¥â¨âì, çâ® As ¨ Bs â®ç­® à ¢­ë ¤à㣠¤àã£ã ¯à¨ १®­ ­á­®© ç áâ®â¥.   ç áâ®â¥, ­¥ à ¢­®© १®­ ­á­®©, í⨠ª®íää¨æ¨¥­âë ­¥ à ¢­ë, çâ® ®¡ãá«®¢«¨¢ ¥â ­¥à ¢¥­á⢮ § ¯ á¥­­®© í«¥ªâà¨ç¥áª®© ¨ ¬ £­¨â­®© í­¥à£¨©.

222

6.7.2. ‚®§¡ã¦¤¥­¨¥ १®­ â®à  ¯®â®ª®¬ § à殮­­ëå ç áâ¨æ

‚ í«¥ªâà®­­ëå ¯à¨¡®à å ‘‚— ¨ ã᪮à¨â¥«ïå ᪢®§ì १®­ â®à ¯à®å®¤¨â ¯®â®ª § à殮­­ëå ç áâ¨æ, á£à㯯¨à®¢ ­­ëå ¢ á£ãá⪨, ¡« £®¤ àï 祬㠢 १®­ â®à¥ ¬®£ãâ ¢®§¡ã¦¤ âìáï í«¥ªâ஬ £­¨â­ë¥ ¯®«ï. ®â®ª § à殮­­ëå ç áâ¨æ ¯à¥¤áâ ¢«ï¥â ᮡ®© í«¥ªâà¨ç¥áª¨© ¯¥à¥¬¥­­ë© ⮪. à¥¤¯®«®¦¨¬, ­ ¯à¨¬¥à, çâ® ç áâ®â  á£ãá⪮¢, ¯à®«¥â îé¨å ç¥à¥§ १®­ â®à, à ¢­  १®­ ­á­®© ç áâ®â¥. ’®£¤ 

As = Q! s as = s

R J~ E~  dV e s ; Q! s RV ~ ~  : s  H
H dV V

s

s

à¥¤¯®«®¦¨¬, çâ® ¯ã箪 ¤®áâ â®ç­® â®­ª¨©, â ª çâ® ¢ ¥£® á¥ç¥­¨¨

E~ s = const. ’®£¤ 

Z V

J~e E~ s dV =

Z L

E~ s
I~e dl;

R £¤¥ I~e = J~e dS { ⮪ ¯ã窠 (¨­â¥£à « ¯® á¥ç¥­¨î ¯ã窠). à¥¤¯®«®¦¨¬

â ª¦¥, çâ® § §®à, ç¥à¥§ ª®â®àë© ¯à®«¥â ¥â ¯ã箪, ¬­®£® ¬¥­ìè¥, 祬 à ááâ®ï­¨¥ ¬¥¦¤ã á£ãá⪠¬¨. ’®£¤  ¬®¦­® ¯à¨­ïâì, çâ® I~e = const ¢® ¢á¥¬ § §®à¥ १®­ â®à  ¨

Z L

Z

E~ s
I~e dl = Ie E~ s d~l = ; Ie Us ; L

£¤¥ Us { ­ ¯à殮­¨¥ ­  § §®à¥ ¯à¨ ¥¤¨­¨ç­®©  ¬¯«¨â㤥 (As = 1). ®¤áâ ¢«ïï, ¯®«ã稬  As = Q! s R jHUsj2 dV Ie : s s V

‚ â® ¦¥ ¢à¥¬ï ¬ë ¨¬¥«¨

E~ = As
E~ s :

ˆ­â¥£à¨àãï E~ ¢¤®«ì ¯ã窠, ¯®«ã稬

U = As Us : 223

®¤áâ ¢«ïï á ­ ©¤¥­­®¥ ¢ëè¥ ¢ëà ¦¥­¨¥ ¤«ï As , ¤«ï ­ ¯à殮­¨ï, ¢®§¡ã¦¤ ¥¬®£® ¯ã窮¬ ­  १®­ â®à¥, ¯®«ã稬 2 U = Q! s R jjHUsjj2 dV Ie : s s V

‚ â® ¦¥ ¢à¥¬ï

Z V

jH j2 dV = 2Ws ;

£¤¥ Ws { § ¯ á í­¥à£¨¨ ¢ १®­ â®à¥ ¯à¨ ¥¤¨­¨ç­®©  ¬¯«¨â㤥, ¯®í⮬ã U { â®à¬®§ï饥 ­ ¯à殮­¨¥ ­  १®­ â®à¥ à ¢­® 2 2 U = Qs 2!U0 sW Ie = Qs 2UPs Ie ; rs s s £¤¥ Prs = !s0 Ws { ॠªâ¨¢­ ï ¬®é­®áâì ¢ १®­ â®à¥ ¯à¨ ¥¤¨­¨ç­®©

 ¬¯«¨â㤥. ‚¥«¨ç¨­ 

2 2 2 s = 2UPs = 2!U0 sW = 2!U0 W rs s s s

¨¬¥¥â à §¬¥à­®áâì ᮯà®â¨¢«¥­¨ï ¨ ­®á¨â ­ §¢ ­¨¥ å à ªâ¥à¨áâ¨ç¥áª®£® ᮯà®â¨¢«¥­¨ï १®­ â®à  (®â­¥á¥­­®£® ª ®¯à¥¤¥«¥­­®¬ã ¯ã⨠¤¢¨¦¥­¨ï ¯ã窠). —¥à¥§ å à ªâ¥à¨áâ¨ç¥áª®¥ ᮯà®â¨¢«¥­¨¥ ¬®¦¥â ¡ëâì ¢ëà ¦¥­  ॠªâ¨¢­ ï ¬®é­®áâì: 2 Pr = 2U : s ‚¥«¨ç¨­ 

Rè = Qs s ­®á¨â ­ §¢ ­¨¥ èã­â®¢®£® ᮯà®â¨¢«¥­¨ï १®­ â®à . ˆ­ ç¥ èã­â®¢®¥ ᮯà®â¨¢«¥­¨¥ (ãç¨â뢠ï, çâ® Qs = PPr ) ¬®¦¥â ¡ëâì ®¯à¥¤¥«¥­® á«¥¤ãa î騬 ®¡à §®¬: 2 2 Rè = PPr 2UP = 2UP : a r a 224

—¥à¥§ èã­â®¢®¥ ᮯà®â¨¢«¥­¨¥ ¢ëà ¦ ¥âáï  ªâ¨¢­ ï ¬®é­®áâì (¬®é­®áâì ¯®â¥àì) 2 Pa = 2UR : è …á«¨ ¤«¨­  § §®à  á®áâ ¢«ï¥â § ¬¥â­ãî ¤®«î à ááâ®ï­¨ï ¬¥¦¤ã á®á¥¤­¨¬¨ á£ãá⪠¬¨, â® ­ ¯à殮­¨¥ ¢ëà ¦ ¥âáï ç¥à¥§ ⮪ ­¥áª®«ìª® á«®¦­¥¥. ‚ í⮬ á«ãç ¥ ⮪ à ¢¥­

Ie = Iem e;jke l ; £¤¥ ke = v! , ve { ᪮à®áâì ¤¢¨¦¥­¨ï í«¥ªâà®­®¢. ’®£¤  e

Z L

Z

E~ s
I~e dl = Iem E~ s
e;jke l dl: L

ãáâì, ­ ¯à¨¬¥à, Es = const. ’®£¤  Z Zd=2 E~ s
I~e dl = Iem Es e;jke l dl = Iem Es jk1 (e jke d=2 ; e;jke d=2 ) = e L ;d=2 2; = Iem Es 2 sinkke d=2 = Iem Us sin== 2 e

£¤¥ Us = Es d,  = ke d { 㣮« ¯à®«¥â . ®í⮬㠭 ¯à殮­¨¥ ­  § §®à¥ U = Iem Rè sin==2 2 : Ž¡®§­ ç¨¬  = sin==2 2 | ¯à®«¥â­ë© ä ªâ®à ¨ § ¯¨è¥¬ U = Iem Rè  ¨ U = Iem Rè 2 : Ž¡®§­ ç¨¬ ⥯¥àì 225

Uíää = U
 ; íä䥪⨢­®¥ ­ ¯à殮­¨¥ ¨ Rè íää = Rè
2 : ’®£¤  ¬®é­®áâì ¬®¦¥â ¡ëâì ¢ëà ¦¥­  ç¥à¥§ íä䥪⨢­ë¥ §­ ç¥­¨ï: 2 2 2 U2 P = 2UR = 2UR 2 = 2R íää : è è è íää • à ªâ¥à¨áâ¨ç¥áª®¥ ᮯà®â¨¢«¥­¨¥ â ª¦¥ 㬭®¦ ¥âáï ­  ª¢ ¤à â ¯à®«¥â­®£® ä ªâ®à  íää =  2 : …᫨ ç áâ®â  á«¥¤®¢ ­¨ï á£ãá⪮¢ ¢ ¯ã窥 ­¥ à ¢­  १®­ ­á­®© ç áâ®â¥, ­® ¡«¨§ª  ª ­¥©, â® ¯®ï¢«ï¥âáï ç áâ®â­ë© ¬­®¦¨â¥«ì U = 1 +U१ jQs x : à¨¬¥à. —¥â¢¥àâ좮«­®¢ë© ®â१®ª TEM-«¨­¨¨ (á¬. à¨á.).

U λ

4

‡¤¥áì à á¯à¥¤¥«¥­¨¥ ­ ¯à殮­¨ï ¢¤®«ì «¨­¨¨ ¨¬¥¥â ¢¨¤

U = Um sin kz: ‡ ¯ á í­¥à£¨¨ dW ¢ í«¥¬¥­â¥ dz à ¢¥­ 2 2 kz C1 dz: dW = Um sin 2

®«­ ï § ¯ á¥­­ ï í­¥à£¨ï ¢ëà ¦ ¥âáï ç¥à¥§ ¨­â¥£à « Z=4U 2 sin2 kz C1 Um2 : m C dz = W = 1 2 16 0 226

’¥¯¥àì ¬®¦­® ¢ëç¨á«¨âì å à ªâ¥à¨áâ¨ç¥áª®¥ ᮯà®â¨¢«¥­¨¥ 2 2  = 2!UmW = 2! C UUm2 =16 = ! C8  : 0 0 1 m 0 1 ’ ª ª ª  = c=f , â®  = 2 8C c = 4 Z0 ; 1 £¤¥ Z0 { ¢®«­®¢®¥ ᮯà®â¨¢«¥­¨¥ «¨­¨¨. à¥¤¯®«®¦¨¬, ç⮠१®­ â®à ¨¬¥¥â ¤«¨­ã 2n + 1 ç¥â¢¥àâ좮«­®¢ëå ®â१ª®¢ (á¬. à¨á.). ’®£¤  ¯à¨ ⮬ ¦¥ ­ ¯à殮­¨¨ § ¯ á í­¥à£¨¨ ¢®§à á⠥⠢ 2n + 1 à §. â® ¯à¨U ¢®¤¨â ª 㬥­ì襭¨î å à ªâ¥à¨áâ¨ç¥áª®£® ᮯà®â¨¢«¥­¨ï â ª¦¥ ¢ 2n + 1 à §. ’ ª ª ª ¤®¡à®â­®áâì ¯à¨ í⮬ ¯®ç⨠­¥ ¨§¬¥­ï¥âáï, â® èã­â®¢®¥ ᮯà®â¨¢«¥­¨¥ â ª¦¥ 㬥­ìè ¥âáï ¢ 2n + 1 à §. à¨¬¥à. ¥§®­ â®à E010 ‚ëç¨á«¨¬ å à ªâ¥à¨áâ¨ç¥áª®¥ ᮯà®â¨¢«¥­¨¥ 2  = 2!UsW : s s ‡¤¥áì Ez = J0 (g01 r), ¯à¨ç¥¬ g01 = ta01  2a:4 . ‚ 業âॠ(r = 0) Ez = 1, ¯®í⮬ã Us = h. ‚ëç¨á«¨¬ í­¥à£¨î, § ¯ á¥­­ãî ¢ १®­ â®à¥:

Z Z 2 2 Ws = "2h J02 (g01 r) 2r dr = "h J02 (g01 r) r dr = "ha 2 J1 (g01 a): a

Šà®¬¥ ⮣®,

a

0

0

t01 !01 = pg01 " = ap" :

®¤áâ ¢«ïï, ¯®«ã稬 227

h2 = 2 2 t "ha 01 p 2 a " 2 J1 (t01 ) „ «¥¥, t01  2:4, J1 (t01 )  0:52, ®âªã¤  010 =

r h

010  0:49 " a =

r h

1

" a t01 J12 (t01 ) :

r h

" 2a =

r h

" d:

à¨¬¥à. „«ï ¨««îáâà æ¨¨ ¯à¨¬¥¬: a = 7:5 á¬, h = 2:5 á¬,  = 20 á¬,  = 1:41
10;4 á¬, 010 = 0:49
377
13  62 Ž¬. „®¡à®â­®áâì 1 7:5
2:5 3 Q = 1 a ah + h = 1:4
10;4 10  13
10 : Žâá ­ å®¤¨¬ èã­â®¢®¥ ᮯà®â¨¢«¥­¨¥ Rè = 62
13
103  0:8
106 Ž¬ = 0:8 ŒŽ¬: 6.7.3. „à㣨¥ á¯®á®¡ë ¢®§¡ã¦¤¥­¨ï १®­ â®à®¢

¥§®­ â®àë, â ª ¦¥ ª ª ¨ ¢®«­®¢®¤ë, ¬®¦­® ¢®§¡ã¦¤ âì èâë६, ¯¥â«¥© ¨ ç¥à¥§ ®â¢¥àá⨥. …᫨ ãáâனá⢮ ¤«ï ¢®§¡ã¦¤¥­¨ï ¯à¥¤áâ ¢«ï¥â ᮡ®© â®­ª¨© ¯à®¢®¤­¨ª, â®  ¬¯«¨â㤠 ¯®«ï ¢ëà ¦ ¥âáï ç¥à¥§ ¨­â¥£à « ®â ⮪  ¯® ¯à®¢®¤­¨ªã:

Z

â.¥.

L

As =

E~ s
I~e dl;

R E~
I~ dl s e ; RLjH j2 dV : V

228

s

Ie

Z L

‚ ª ç¥á⢥ ¯à¨¬¥à  à áᬮâਬ ¢®§¡ã¦¤¥­¨¥ १®­ â®à  ¯¥â«¥© (á¬. à¨á.). …᫨ ¯¥â«ï ª®à®âª , â® ¢ ¯¥à¢®¬ ¯à¨¡«¨¦¥­¨¨ ⮪ ¢¤®«ì ¯¥â«¨ ¬®¦­® áç¨â âì ¯®áâ®ï­­ë¬. ’®£¤  ¨¬¥¥¬

I

Z

Z

L

S

S

E~ s
I~e dl = Ie E~ s d~l = Ie rot E~ s
~n
dS = ; j!Ie H~ s
~n
dS;

£¤¥ S { ¯®¢¥àå­®áâì, ­ âï­ãâ ï ­  ¯¥â«î, ~n { ¥¤¨­¨ç­ë© ¢¥ªâ®à ­®à¬ «¨ ª í⮩ ¯®¢¥àå­®áâ¨. …᫨ ¯¥â«ï ­ á⮫쪮 ¬ « , çâ® Hs  const, â®

Z L

E~ s
I~e dl = ; j!s 
H~ s
~n S Ie :

’ ª¨¬ ®¡à §®¬, ¨­â¥­á¨¢­®áâì ¢®§¡ã¦¤¥­¨ï ¯à®¯®à樮­ «ì­  ¯«®é ¤¨ ¯¥â«¨ ¨ ¯¥à¯¥­¤¨ªã«ïà­®© ª ­¥© á®áâ ¢«ïî饩 ¢¥ªâ®à­®© ä㭪樨 H~ s . ®í⮬㠤«ï íä䥪⨢­®£® ¢®§¡ã¦¤¥­¨ï ¯¥â«ï ¤®«¦­  ¡ëâì ¯®¬¥é¥­  ¢ ¯ãç­®á⨠¬ £­¨â­®£® ¯®«ï ¯¥à¯¥­¤¨ªã«ïà­® ¬ £­¨â­ë¬ ᨫ®¢ë¬ «¨­¨ï¬. €­ «®£¨ç­® ¬®¦­® ¯®ª § âì, çâ® èâëàì ( ­â¥­­ã) á«¥¤ã¥â ¯®¬¥é âì ¢ ¯ãç­®áâ¨ í«¥ªâà¨ç¥áª®£® ¯®«ï â ª, çâ®¡ë ­ ¯à ¢«¥­¨¥ èâëàï ᮢ¯ ¤ «® á ­ ¯à ¢«¥­¨¥¬ í«¥ªâà¨ç¥áª¨å ᨫ®¢ëå «¨­¨©. Žâ¢¥àá⨥ ­¥®¡å®¤¨¬® à §¬¥é âì â ª, çâ®¡ë ®­® ¯¥à¥á¥ª «® «¨­¨¨ ⮪®¢ ­  ¯®¢¥àå­®áâ¨. 6.8.

¥§®­ â®à ª ª í«¥¬¥­â «¨­¨¨ ¯¥à¥¤ ç¨

¥§®­ â®à, ¯à¨á®¥¤¨­¥­­ë© ª «¨­¨¨ ¯¥à¥¤ ç¨, ¬®¦­® à áᬠâਢ âì ª ª ®ª®­¥ç­®¥ ãáâனá⢮ { ¤¢ã寮«îá­¨ª, ¤«ï ª®â®à®£® ¬®¦¥â ¡ëâì ­ ©¤¥­® ¢å®¤­®¥ ᮯà®â¨¢«¥­¨¥.

229

6.8.1. ‚室­®¥ ᮯà®â¨¢«¥­¨¥ १®­ â®à 

 áᬮâਬ १®­ â®à { ®¡ê¥¬, ®£à ­¨ç¥­­ë© ¨¤¥ «ì­® ¯à®¢®¤ï饩 ¯®¢¥àå­®áâìî S , á¢ï§ ­­ë© á S S1 ¢®«­®¢®¤®¬ ç¥à¥§ ®â¢¥àá⨥ (á¬. à¨á.). ®¬¥á⨬ ¢ ¢®«­®¢®¤¥ ª®z à®âª®§ ¬ëª îéãî ¯« á⨭㠢 á¥ç¥­¨¨ S1 .  ááâ®ï­¨¥ í⮩ ¯« áâ¨­ë ®â ®â¢¥àáâ¨ï ¢ë¡¥à¥¬ â ª, çâ®¡ë ­  ®â¢¥àá⨨ ¨¬¥«® ¬¥áâ® à ¢¥­á⢮ ­ã«î â ­£¥­æ¨ «ì­®© ª®¬¯®­¥­âë í«¥ªâà¨ç¥áª®£® ¯®«ï. ’®£¤  ®¤­®¢à¥¬¥­­® ¢ í⮬ ¦¥ ¬¥á⥠¡ã¤¥â à ¢­  ­ã«î ­®à¬ «ì­ ï á®áâ ¢«ïîé ï ¬ £­¨â­®£® ¯®«ï, â.¥. ¢ë¯®«­ïîâáï £à ­¨ç­ë¥ ãá«®¢¨ï ­  ¨¤¥ «ì­® ¯à®¢®¤ï饩 ¯®¢¥àå­®áâ¨. ®í⮬ã ç áâ®â  ᢮¡®¤­ëå ª®«¥¡ ­¨© á«®¦­®£® १®­ â®à , á®áâ®ï饣® ¨§ ᮡá⢥­­® १®­ â®à  ¨ ®â१ª  ¢®«­®¢®¤  á ª®à®âª®§ ¬ëª îé© ¯« á⨭®© ¢ á¥ç¥­¨¨ S1 , ¡ã¤¥â à ¢­  ç áâ®â¥ ᢮¡®¤­ëå ª®«¥¡ ­¨© १®­ â®à  ¡¥§ ®â¢¥àáâ¨ï. ‘®¡á⢥­­ë¥ ¢¥ªâ®à­ë¥ ä㭪樨 â ª®£® १®­ â®à  ®¡®§­ ç¨¬ E~ s ¨ H~ s . ‘¥ç¥­¨¥ S1 ¯à¨ í⮬ ¤®«¦­® ¡ëâì ­  â ª®¬ à ááâ®ï­¨¨ ®â ®â¢¥àáâ¨ï, çâ®¡ë ¢®§­¨ªè¨¥ 㠮⢥àáâ¨ï ¢ëá訥 ¬®¤ë ¡ë«¨ ¡«¨§ª¨ ª ­ã«î ¢ í⮬ á¥ç¥­¨¨. ‚ ¯à®æ¥áᥠ᢮¡®¤­ëå ª®«¥¡ ­¨© í«¥ªâ஬ £­¨â­®¥ ¯®«¥ ¯à®­¨ª ¥â ¨§ १®­ â®à  ¢ ¢®«­®¢®¤, ¯à¨ç¥¬ ¢¡«¨§¨ ®â ª®à®âª®§ ¬ëª î饩 ¯« áâ¨­ë ®­® ¯à ªâ¨ç¥áª¨ ᮢ¯ ¤ ¥â á ¯®«¥¬ ®á­®¢­®© ¬®¤ë ¢ ¢®«­®¢®¤¥. ‚ ᨫã ᪠§ ­­®£® ¤«ï á¥ç¥­¨ï S1 ᮡá⢥­­ë¥ ä㭪樨 à ¢­ë

E~ st = 0; H~ st = j1s
~h(x; y); £¤¥ ~h(x; y) { ¯®¯¥à¥ç­ ï ¢¥ªâ®à­ ï äã­ªæ¨ï ¤«ï ®á­®¢­®© ¬®¤ë ¢ ¢®«­®¢®¤¥, E~ st , H~ st { â ­£¥­æ¨ «ì­ë¥ á®áâ ¢«ïî騥 í«¥ªâà¨ç¥áª®£® ¨ ¬ £­¨â­®£® ¯®«ï ­  ª®à®âª®§ ¬ëª îé© ¯« á⨭¥, 1s { ¢¥«¨ç¨­ , å à ªâ¥à¨§ãîé ï á¢ï§ì १®­ â®à  á ¢®«­®¢®¤®¬. ‡ ¬¥â¨¬, çâ® 1s ¯à¥¤¯®« £ ¥âáï ¢¥é¥á⢥­­®© ¢¥«¨ç¨­®©. ’®£¤  H~ st { ¬­¨¬ ï ¢¥«¨ç¨­ . “¤ «¨¬ ⥯¥àì ¯« á⨭㠨 à áᬮâਬ ¢®§¡ã¦¤¥­¨¥ १®­ â®à  ç¥à¥§ ¢®«­®¢®¤. ‚ á¥ç¥­¨¨ S1 (¢ ¢®«­®¢®¤¥) ¯®«¥ ¨¬¥¥â ¢¨¤ E~ ¯®¯ = U1
~e(x; y); H~ ¯®¯ = I1
~h(x; y): 230

«¥ªâ஬ £­¨â­®¥ ¯®«¥ ¢ १®­ â®à¥ ¬®¦­® ⥯¥àì ¢ëç¨á«ïâì ¯® í⮬㠯®«î, § ¤ ­­®¬ã ­  ¯®¢¥àå­®á⨠á¥ç¥­¨ï S1 . „«ï íâ¨å ¢ëç¨á«¥­¨© ¯®«¥ á¯à ¢  ®â á¥ç¥­¨ï S1 ¬®¦¥â ¡ëâì § ¬¥­¥­® í«¥ªâà¨ç¥áª¨¬ ¨ ¬ £­¨â­ë¬ ¯®¢¥àå­®áâ­ë¬¨ ⮪ ¬¨ ­  S1 :

K~ e = H~ ¯®¯  ~n = ; H~ ¯®¯  ~z0 ; K~ m = ; E~ ¯®¯  ~n = E~ ¯®¯  ~z0 ; £¤¥ ~z0 { ¥¤¨­¨ç­ë© ¢¥ªâ®à ¢ ­ ¯à ¢«¥­¨¨ १®­ â®à . …᫨ ®¡à â¨âìáï ª à §¤¥«ã 6.7, â® ¢®§¡ã¦¤¥­­®¥ ¯®¢¥àå­®áâ­ë¬¨ ⮪ ¬¨ ¬ £­¨â­®¥ ¯®«¥ ¢ १®­ â®à¥ à ¢­®

H~ = Bs
H~ s : ‡¤¥áì Bs { ª®íää¨æ¨¥­â ¢ à §«®¦¥­¨¨ ¯®«ï ¯® ᮡá⢥­­ë¬ äã­ªæ¨ï¬:

Bs = aj (s!!2 ;; b!s !2 s) ; s

£¤¥

as =

R K~ E~  dS e s S ; R ~ ~  ; bs  H H dV 1

V

s s

=

R K~ H~  dS m s S : R  H~ H~  dV 1

V

s s

‡ ¬¥â¨¬, çâ® as = 0, â ª ª ª E~ s = E~ st = 0 ­  S1 . ‚ â® ¦¥ ¢à¥¬ï ¬ £­¨â­®¥ ¯®«¥ H ­  ¯®¢¥àå­®á⨠S1 ¬®¦¥â ¡ëâì ­ ©¤¥­® á ãç¥â®¬ ⮣®, çâ® H~ ¯®¯ = Bs H~ s = Bs H~ st = Bs jk1s ~h(x; y): ‘à ¢­¨¢ ï íâ® á ¯à¥¤ë¤ã騬 ¢ëà ¦¥­¨¥¬ ¤«ï H~ ¯®¯ , ­ ©¤¥¬

I1 = j Bs 1s : ‚ëç¨á«¨¢ Bs ç¥à¥§ U1 , ¬ë ­ ©¤¥¬ á¢ï§ì ¬¥¦¤ã ⮪®¬ ¨ ­ ¯à殮­¨¥¬, çâ® ¯®§¢®«¨â ­ ©â¨ ¢å®¤­ãî ¯à®¢®¤¨¬®áâì १®­ â®à  ¢ á¥ç¥­¨¨ S1 . —â®¡ë ­ ©â¨ Bs , ­ã¦­® ¯®¤áâ ¢¨âì ¢ ¢ëà ¦¥­¨¥ ¤«ï bs ¯®¢¥àå­®áâ­ãî ¯«®â­®áâì ¬ £­¨â­®£® ⮪  K~ m = E~ ¯®¯  ~z0 : 231

bs =

R K~ H~  dS m s S R  H~ H~  dV 1

V

s s

=

R (E~ ¯®¯  ~z )H~  dS 0 s S R  H~ H~  dV 1

V

=

s s

=

R (E~ ¯®¯  H~ )~z dS s 0 S R  H~ H~  dV 1

R jk1s U1 (~e  ~h)~z0 dS : RS  H~ H~  dV

V

s s

=

1

s s

V

‡¤¥áì ãç⥭®, çâ® H~ s = ;H~ s , â ª ª ª H~ s { ç¨áâ® ¬­¨¬ ï ¢¥«¨ç¨­ . “ç¨âë¢ ï ­®à¬¨à®¢ªã ~e ¨ ~h, ¤«ï bs ¯®«ã稬 bs = Rj1s U1 :

 H~ s H~ s dV V

®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï Bs , ¯®«ã稬 Bs = ; !U21 ;1s !!s2 R ~ 1~  : s  Hs H dV s

V

“ç¨âë¢ ï ¯à¨¢¥¤¥­­®¥ à ­¥¥ ¢ëà ¦¥­¨¥ ¤«ï ⮪ , ¨¬¥¥¬ 2 I1 = j Bs 1s = j (!2 1s;!s!2 ) R ~ 1~  U1 : s  Hs Hs dV V

„ ­­®¥ ᮮ⭮襭¨¥ ¯®§¢®«ï¥â ­ ©â¨ ¯à®¢®¤¨¬®áâì १®­ â®à , à áᬠâਢ ¥¬®£® ª ª ®ª®­¥ç­®¥ ãáâனá⢮ (¨«¨ ­ £à㧪 ) ¢®«­®¢®¤ : 2 Y1 = UI1 = j (!21s;!s!2 ) R ~ 1~  : 1 s  Hs Hs dV V

“ç¨âë¢ ï ¢ë᮪ãî ¤®¡à®â­®áâì, íâ® ¢ëà ¦¥­¨¥ ¬®¦­® ¯à¨¢¥á⨠ª ¢¨¤ã Qs 21s : Y1 = 1 + 1jQ x R ~ s H~ s dV s !s  H V

Œ®¦­® ¯®ª § âì, çâ® ¢¥«¨ç¨­  232

G1 =

2

RQs 1s !  H~ H~  dV s

V

s s

¢¥é¥á⢥­­  ¨ ¯®«®¦¨â¥«ì­ . ‚¥é¥á⢥­­®áâì ¨ ¯®«®¦¨â¥«ì­®áâì á«¥¤ãîâ ¨§ ⮣®, çâ® 1s { ¢¥é¥á⢥­­ ï ¢¥«¨ç¨­ . ®á«¥¤­¥¥ á«¥¤ã¥â ¨§ ᮮ⭮襭¨ï, ¯à¨¢¥¤¥­­®£® ¢ëè¥: H~ st = j1s
~h(x; y): ’ ª ª ª ¯à¨ ᢮¡®¤­ëå ª®«¥¡ ­¨ïå ¢ १®­ â®à¥ ¯®«¥ ¢® ¢á¥å â®çª å ª®«¥¡«¥âáï ¢ ®¤­®© ä §¥ (¨«¨ ¯à®â¨¢®ä §¥), â® ¬®¦­® áç¨â âì, çâ® Hst { ¬­¨¬ ï äã­ªæ¨ï, ~h { ¢¥é¥á⢥­­ ï äã­ªæ¨ï, ⮣¤  1s ¢¥é¥á⢥­­ . ˆâ ª, ¢ á¥ç¥­¨¨ S1 1 Y1 = 1 +GjQ : sx ’ ª®© ¯à®¢®¤¨¬®áâìî ®¡« ¤ ¥â ¯®á«¥¤®¢ â¥«ì­ë© १®­ ­á­ë© ª®­âãà, ª®â®àë© ¬®¦¥â ¡ëâì ¯à¨­ïâ ¢ ª ç¥á⢥ íª¢¨¢ «¥­â­®© á奬ë १®­ â®à  ¢ á¥ç¥­¨¨ S1 (á¬. à¨á). ¥à¥©¤¥¬ ⥯¥àì ª á¥ç¥­¨î S2 , ®âáâ®ï饬㠮â S1 ­  à ááâ®ï­¨¥ =4, â.¥. ç¥â¢¥àâì ¢®«­ë ¢ ¢®«­®¢®¤¥. ‚室­®¥ ᮯà®â¨¢«¥­¨¥ ¢ í⮩ ¯«®áª®áâ¨

C1 L1 G1

2 2 1 R2 : Z2 = ZZ0 = Z02 Y1 = 1 Z+0 G = jQs x 1 + jQs x 1

L2

C2

’ ª¨¬ ᮯà®â¨¢«¥­¨¥¬ ®¡« ¤ ¥â ¯ à ««¥«ì­ë© १®­ ­á­ë© ª®­âãà, ª®â®àë© ¬®¦¥â ¡ëâì ¯à¨­ïâ ¢ ª ç¥á⢥ íª¢¨¢ «¥­â­®© á奬ë १®­ â®à  ¢ á¥ç¥­¨¨ S2 (á¬. à¨á.). Œ®¦­® ­ ©â¨ ¡¥áª®­¥ç­®¥ ç¨á«® ç¥à¥¤ãîé¨åáï ¯«®áª®á⥩, ¢

R2

233

ª®â®àëå १®­ â®à ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­ íª¢¨¢ «¥­â­®© á奬®© ¯®á«¥¤®¢ â¥«ì­®£® ¨«¨ ¯ à ««¥«ì­®£® १®­ ­á­®£® ª®­âãà  ¢¡«¨§¨ ç áâ®âë !s0 . â¨ ¯«®áª®á⨠­ §ë¢ îâ ¯«®áª®áâﬨ íª¢¨¢ «¥­â­®£® ¯à¥¤áâ ¢«¥­¨ï. ®«ì§ãïáì íª¢¨¢ «¥­â­®© á奬®©, ¬®¦­® ¢ëïá­¨âì àï¤ á¢®©á⢠१®­ â®à , ¯à¨á®¥¤¨­¥­­®£® ª «¨­¨¨. …᫨ «¨­¨ï ᮣ« á®¢ ­ , â® ¤«ï ᢮¡®¤­ëå ª®«¥¡ ­¨© ¨¬¥¥â ¬¥áâ® á奬 , ¯à¥¤áâ ¢«¥­­ ï ­  à¨á.

L

C

R

Z0

à¨ í⮬ í­¥à£¨ï à áᥨ¢ ¥âáï ¢ ᮯà®â¨¢«¥­¨¨ R { ¯®â¥à¨ ¢­ãâਠ१®­ â®à , ¨ ¢ ᮯà®â¨¢«¥­¨¨ Z0 { ¯®â¥à¨ ­  ¨§«ã祭¨¥ ¢ «¨­¨¨. Žâ­®è¥­¨¥ íâ¨å ¬®é­®á⥩ à ¢­®

PZ0 = R = Y0 : PR Z0 G ‚ â® ¦¥ ¢à¥¬ï

PZ0 = Q0 : PR Q¢­ ‘«¥¤®¢ â¥«ì­®,

Q0 = R = Y0 : Q¢­ Z0 G

„ ­­®¥ ®â­®è¥­¨¥ ­ §ë¢ îâ ª®íää¨æ¨¥­â®¬ á¢ï§¨ :

= QQ0 = ZR = YG0 = PPZ0 : ¢­ 0 R  £à㦥­­ ï ¤®¡à®â­®áâì ®¯à¥¤¥«ï¥âáï ᮮ⭮襭¨¥¬ 1 1 1 Q0 1 1 Q­ = Q0 + Q¢­ = Q0 (1 + Q¢­ ) = Q0 (1 + ); 234

®âªã¤ 

Q­ = 1 Q+0 :  ©¤¥¬ ª®íää¨æ¨¥­â ®âà ¦¥­¨ï ¯ à ««¥«ì­®£® १®­ ­á­®£® ª®­âãà , ¯à¨á®¥¤¨­¥­­®£® ª «¨­¨¨ ¯¥à¥¤ ç¨ á ¢®«­®¢®© ¯à®¢®¤¨¬®áâìî Y0 : Yk ; ; = YY0 ; 0 + Yk £¤¥ ¯à®¢®¤¨¬®áâì ¯ à ««¥«ì­®£® ª®­âãà  Yk à ¢­ 

Yk = G (1 + jQ0 x); ¯à¨ç¥¬ G = 1=R. à¨ १®­ ­á¥ x = 0, ¨ ª®íää¨æ¨¥­â ®âà ¦¥­¨ï à ¢¥­ G = Y0 =G ; 1 = ; 1 : ;0 = YY0 ; Y0 =G + 1 +1 0 + G „ ­­ ï ä®à¬ã«  ¯®§¢®«ï¥â ®¯à¥¤¥«¨âì Š‘‚ ¢ «¨­¨¨ ¯à¨ १®­ ­á¥: + j; 0 j = + 1 + j ; 1 j : 0 = 11 ; j;0 j + 1 ; j ; 1j à¨ > 1 (ᨫ쭠ï á¢ï§ì) 1+ ;1 0 = + + 1 ; + 1 = : à¨ < 1 (á« ¡ ï á¢ï§ì) + 1 ; + 1 = 1: 0 = + 1+ ; 1 Œ®é­®áâì ¯ ¤ î饩 ¢®«­ë à áᥨ¢ ¥âáï ¯®«­®áâìî ¢ १®­ â®à¥ (¡¥§ ®âà ¦¥­¨©) ¯à¨ 0 = 1, â.¥. ¯à¨ = 1. à¨ í⮬ ¢­¥è­ïï ¤®¡à®â­®áâì à ¢­  ᮡá⢥­­®©. à¨ x 6= 0 ª®íää¨æ¨¥­â ®âà ¦¥­¨ï १®­ â®à  à ¢¥­ G (1 + jQ0 x) = 2Y0 ; [Y0 + G (1 + jQ0 x)] = ; = YY0 ; Y0 + G (1 + jQ0 x) 0 + G (1 + jQ0 x) 235

¨«¨

2Y0 = Y + G (1 ; 1 = Y + G 2+Y0 jGQ x) ; 1; + jQ x ) 0 0 0 0 ; = 1 ;+0 +jQ1 x ; 1; ­

Y0 , Y0 = , Q­ = Q0 { ­ £à㦥­­ ï ¤®¡à®â­®áâì. £¤¥ ;0 + 1 = Y 2+ 1+ 0 G G à¥¤áâ ¢¨¬ ⥯¥àì § ¢¨á¨¬®áâì ª®íää¨æ¨¥­â  ®âà ¦¥­¨ï ®â ç áâ®âë ¢ ª®¬¯«¥ªá­®© ¯«®áª®á⨠ª®íää¨æ¨¥­â  ®âà ¦¥­¨ï (á¬. à¨á.).

Γ+

ω

1

Γ x= 0

1 Γ0

P= ½P0 (x=x½ )

”®à¬ã«ã ¤«ï ª®íää¨æ¨¥­â  ®âà ¦¥­¨ï ¬®¦­® § ¯¨á âì á«¥¤ãî騬 ®¡à §®¬: ; + 1 = 1 ;+0 +jQ1 x : ­ ‹¥¢ ï ç áâì í⮣® à ¢¥­á⢠ ¯à®¯®à樮­ «ì­  ­ ¯à殮­¨î ­  § ¦¨¬ å १®­ â®à  ¢ ¯«®áª®á⨠S2 . ‡ ¢¨á¨¬®áâì ; + 1 ®â ç áâ®âë ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­  ­  ªà㣮¢®© ¤¨ £à ¬¬¥. …᫨ > 1, â® ;0 > 0, ¨ ®ªà㦭®áâì ®å¢ â뢠¥â ­ ç «® ª®®à¤¨­ â. …᫨ ¦¥ < 1, â® ;0 < 0, ¨ ®ªà㦭®áâì ­¥ ®å¢ â뢠¥â ­ ç «® ª®®à¤¨­ â. à¨ = 1 ®ªà㦭®áâì ¯à®å®¤¨â ç¥à¥§ ­ ç «® ª®®à¤¨­ â (; = 0).  ©¤¥¬ ¬®é­®áâì, à áᥨ¢ ¥¬ãî ¢ १®­ â®à¥: P = P¯ ¤ (1 ; j;j2 ): 236

1 ; jQ­ x = ;20 + Q­2 x2 : j;j2 = ;0 + 11 +; jQ 1 + Q2­ x2 ­x

®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï P , ­ å®¤¨¬ 2 2 2 2 P = P¯ ¤ (1 ; ;10 ++ QQ2­xx2 ) = P¯ ¤ 1 1+;Q;2 0x2 = ­ ­ 0 : = 1 +PQ 2­ x2 ‡¤¥áì P0 = P¯ ¤ (1 ; ;20 ) { ¬®é­®áâì, à áᥨ¢ ¥¬ ï ¢ १®­ â®à¥ ¯à¨ १®­ ­á¥. Œ®é­®áâì 㬥­ìè ¥âáï ¢¤¢®¥, ¥á«¨

Q­x =  1; 祬ã ᮮ⢥âáâ¢ã¥â ®â­®á¨â¥«ì­ ï à ááâனª  2
f x1=2 = f 1=2 = Q1 : 0 ­ 6.8.2. à¨¬¥­¥­¨¥ १®­ â®à®¢

¥§®­ â®àë ¯à¨¬¥­ïîâáï ¢ ª ç¥á⢥ í«¥¬¥­â®¢ ‘‚— í«¥ªâ஢ ªã㬭ëå ¯à¨¡®à®¢ ¢® ¢å®¤­ëå ¨ ¢ë室­ëå 楯ïå,   â ª¦¥ ¢ ª ç¥á⢥ 䨫ìâ஢ ¨ ¢®«­®¬¥à®¢. Š®£¤  १®­ â®à ¯à¨¬¥­ï¥âáï ¢ ª ç¥á⢥ 䨫ìâà , â® ®­ ¨¬¥¥â ¤¢¥ á¢ï§¨ (á¬. à¨á.).

1

‚ ¦­ë¬ âॡ®¢ ­¨¥¬ ¯à¨ í⮬ ¯à¨¬¥­¥­¨¨ ï¥âáï ¯¥à¥¤ ç  í­¥à£¨¨ ¨§ ®¤­®© «¨­¨¨ ¢ ¤àã£ãî ­  १®­ ­á­®© ç áâ®â¥ á ¬ «ë¬¨ ¯®â¥àﬨ. „«ï í⮣® १®­ â®à ¤®«¦¥­ ¡ëâì ᮣ« á®¢ ­ á® áâ®à®­ë ¢å®¤ , â.¥. ª®íää¨æ¨¥­â á¢ï§¨ á® áâ®à®­ë ¢å®¤  ¤®«¦¥­ ¡ëâì à ¢¥­

2

0 10 = QQ0 = 1: ¢­1

237

‚¥«¨ç¨­  Q00 ®¯à¥¤¥«ï¥âáï ¢¥«¨ç¨­ ¬¨ Q0 ¨ Q¢­2: 1 1 1 1 Q0 1 + 2 Q00 = Q¢­2 + Q0 = Q0 (1 + Q¢­2 ) = Q0 : Žâá ¯®«ãç ¥¬

Q0 : Q00 = 1 + 2 ’ ª¨¬ ®¡à §®¬, ãá«®¢¨¥ ᮣ« á®¢ ­¨ï ¢å®¤  ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¢¨¤¥ ¨«¨

Q0 Q¢­1 (1 + 2 ) = 1;

1 1 + 2 = 1: ‡¤¥áì 1 ¨ 2 { ª®íää¨æ¨¥­âë á¢ï§¨ ¤«ï ᮮ⢥âá⢥­­® ¯¥à¢®£® ¨ ¢â®à®£® ¢å®¤®¢. …᫨ 1  1 ¨ 2  1, â® ãá«®¢¨¥ ᮣ« á®¢ ­¨ï ¨¬¥¥â ¢¨¤ 1  2 : Ž¤­ ª® ¤ ¦¥ ¯à¨ ¢ë¯®«­¥­¨¨ ãá«®¢¨ï ᮣ« á®¢ ­¨ï ç áâì ¬®é­®á⨠à áᥨ¢ ¥âáï ¢ á⥭ª å á ¬®£® १®­ â®à . Žâ­®è¥­¨¥ í⮩ ¬®é­®á⨠ª ¬®é­®áâ¨, ¯¥à¥¤ ¢ ¥¬®© ç¥à¥§ ¢â®à®© ¢å®¤ ­  ¯®«¥§­ãî ­ £à㧪ã, à ¢­®

Q0 Q¢­2 = 2 : Žâá ª®íää¨æ¨¥­â ¯¥à¥¤ ç¨ ¯® ¬®é­®á⨠१®­ ­á­®£® 䨫ìâà  à ¢¥­  = 1 + 2 = 2 : 2 1 …᫨ ¢ë¯®«­¥­ë ãá«®¢¨ï 1  1, 2  1 ¨ 1  2 , â® ª®íää¨æ¨¥­â ¯¥à¥¤ ç¨ ¯® ¬®é­®á⨠ ¡«¨§®ª ª 1. ®«®á  ¯à®¯ã᪠­¨ï 䨫ìâà  ®¯à¥¤¥«ï¥âáï ­ £à㦥­­®© ¤®¡à®â­®áâìî, 㤮¢«¥â¢®àïî饩 ᮮ⭮襭¨î 238

®âªã¤ 

1 1 1 1 1 Q­ = Q¢­1 + Q¢­2 + Q0 = Q0 (1 + 1 + 2 );

Q­ = 1 + Q0 + : 1 2

¥§®­ â®àë ¯à¨¬¥­ïîâáï â ª¦¥ ¢ ª ç¥á⢥ ¢®«­®¬¥à®¢. à¨ í⮬ ®­¨ ¤®«¦­ë ¡ëâì ®¡®à㤮¢ ­ë ¬¥å ­¨§¬®¬ £à ¤ã¨à®¢ ­­®© ¯¥à¥áâனª¨. — é¥ ¢á¥£® ¢®«­®¬¥à ¨¬¥¥â ¤¢¥ á¢ï§¨, ¯à¨ç¥¬ ®¤­  á«ã¦¨â ¤«ï ¢®§¡ã¦¤¥­¨ï,   ¤à㣠ï { ¤«ï ¨­¤¨ª æ¨¨, ¢ ­¥¥ áâ ¢¨âáï ¤¥â¥ªâ®à. Ž¡¥ á¢ï§¨ ¢ í⮬ á«ãç ¥ ¢ë¡¨à îâ á« ¡ë¬¨, çâ®¡ë ®¡¥á¯¥ç¨âì ¬ ªá¨¬ «ì­ãî ¤®¡à®â­®áâì. 7.

‹…Š’ŽŒ€ƒˆ’›… ‚Ž‹› ‚ ƒˆŽŒ€ƒˆ’Ž‰ ‘…„…

‚ ‘‚—-â¥å­¨ª¥ ­ å®¤ï⠯ਬ¥­¥­¨¥ ä¥àਬ £­¥â¨ª¨ { ä¥àà¨âë, ¨¬¥î騥 ¢ë᮪ãî ¬ £­¨â­ãî ¯à®­¨æ ¥¬®áâì ¨ ¢ â® ¦¥ ¢à¥¬ï ®ç¥­ì ¬ «ãî ¯à®¢®¤¨¬®áâì. « £®¤ àï í⮬㠢 ­¨å ¬®£ãâ à á¯à®áâà ­ïâìáï á ¬ «ë¬ § âãå ­¨¥¬ í«¥ªâ஬ £­¨â­ë¥ ¢®«­ë. 7.1.

‘¢®©á⢠ ä¥àà¨â®¢

ˆ§¢¥áâ­®, çâ® ¬ £­¨â­ë¥ ᢮©á⢠ ä¥àà¨â®¢ ®¡ãá«®¢«¥­ë ᯨ­®¢ë¬ ¬ £­¨â­ë¬ ¬®¬¥­â®¬ í«¥ªâà®­ . Š ¦¤ ï ï祩ª  ªà¨áâ ««¨ç¥áª®© à¥è¥âª¨ ä¥àà¨â  ᮤ¥à¦¨â ­¥áª®¬¯¥­á¨à®¢ ­­ë© ¬®¬¥­â í«¥ªâà®­ .  ¬ £­¨ç¥­­®áâì ä¥àà¨â  á¢ï§ ­  á ®à¨¥­â æ¨¥© ¬ £­¨â­®£® ¬®¬¥­â  í«¥ªâà®­ . Šà®¬¥ ¬ £­¨â­®£® ¬®¬¥­â  í«¥ªâà®­ ¨¬¥¥â â ª¦¥ ¬¥å ­¨ç¥áª¨© ¬®¬¥­â. ‚¥ªâ®àë ¬ £­¨â­®£® ¨ ¬¥å ­¨ç¥áª®£® ¬®¬¥­â®¢ í«¥ªâà®­  ­ ¯à ¢«¥­ë ¢ ¯à®â¨¢®¯®«®¦­ë¥ áâ®à®­ë ¢á«¥¤á⢨¥ ®âà¨æ â¥«ì­®£® § à鸞:

~ m ~ e = ; K;

£¤¥ = me  1:76
1011 Š«/ª£ { £¨à®¬ £­¨â­®¥ ®â­®è¥­¨¥ ¤«ï í«¥ªâà®­ . …᫨ í«¥ªâà®­ ­ å®¤¨âáï ¢ ¬ £­¨â­®¬ ¯®«¥ H~ , â® ­  ­¥£® ¤¥©áâ¢ã¥â ¢à é î騩 ¬®¬¥­â 239

T~ = 0
(m ~ e  H~ );

¯®¤ ¤¥©á⢨¥¬ ª®â®à®£® ¯à®¨á室¨â ¨§¬¥­¥­¨¥ ¬¥å ­¨ç¥áª®£® ¬®¬¥­â  ᮠ᪮à®áâìî

dK~ = T~ = 
(m 0 ~ e  H~ ): dt …᫨ ¯®¤áâ ¢¨âì á m ~ e = ; K~ ,â® ¯®«ã稬 (¯®á«¥ ¯¥à¥áâ ­®¢ª¨ á®-

¬­®¦¨â¥«¥©)

dK~ =  H~  K: ~ 0 dt „ ­­®¥ ᮮ⭮襭¨¥ ¯®ª §ë¢ ¥â, çâ® ¢¥ªâ®à K~ ¢à é ¥âáï ®ª®«® ­ ¯à ¢«¥­¨ï H~ á 㣫®¢®© ç áâ®â®© (ç áâ®â  ®¤­®à®¤­®© ¯à¥æ¥áᨨ) !0 = 0 jH~ j; ¯à¨ç¥¬ ¢à é¥­¨¥ ®¡à §ã¥â á ¬ £­¨â­ë¬ ¯®«¥¬ H~ ¯à ¢ë© ¢¨­â. …᫨ ⥯¥àì 㬭®¦¨âì «¥¢ãî ¨ ¯à ¢ãî ç á⨠ãà ¢­¥­¨ï ­  ;n , £¤¥ n { ç¨á«® í«¥ªâà®­®¢ { ¬ £­¨â­ëå ¤¨¯®«¥© ¢ ¥¤¨­¨æ¥ ®¡ê¥¬ , â® ¯®«ã稬 ãà ¢­¥­¨¥ ¤¢¨¦¥­¨ï ¢¥ªâ®à  ­ ¬ £­¨ç¥­­®á⨠M~ (M~ = n m ~ e) dM~ =  (H~  M~ ): 0 dt

’ ª¨¬ ®¡à §®¬, ¢ ¯à¨áãâá⢨¨ ¯®áâ®ï­­®£® ¬ £­¨â­®£® ¯®«ï ¢¥ªâ®à ­ ¬ £­¨ç¥­­®á⨠¯à¥æ¥áá¨àã¥â ¢®ªà㣠¬ £­¨â­®£® ¯®«ï á ç áâ®â®© !0 . …᫨ ¡ë ­¥ ¡ë«® § âãå ­¨ï, â® ¯à¥æ¥áá¨ï ¯à®¤®«¦ « áì ­¥®£à ­¨ç¥­­® ¤®«£®. Ž¤­ ª® ¡« £®¤ àï ¢§ ¨¬®¤¥©áâ¢¨î ¯à¥æ¥áá¨àãî饣® ᯨ­  á ªà¨áâ ««¨ç¥áª®© à¥è¥âª®© ¨ ¤à㣨¬¨ ᯨ­ ¬¨ ¨¬¥¥â ¬¥áâ® § âãå ­¨¥, ¨ ¢à é¥­¨¥ ¯à®¨á室¨â ¯® ᢥàâ뢠î饩áï ᯨࠫ¨. ’ ªãî ¬®¤¥«ì ä¥à஬ £­¥â¨ª  ¬®¦­® à áᬠâਢ âì ª ª ª®«¥¡ â¥«ì­ãî á¨á⥬ã á ᮡá⢥­­®© ç áâ®â®© !0 . „¥©á⢨⥫쭮, ¥á«¨ H~ ­ ¯à ¢¨âì ¢¤®«ì ®á¨ z , ¨§ ãà ¢­¥­¨ï ¤«ï K~ «¥£ª® ¯®«ãç¨âì

dKx = ; ! K ; 0 y dt dKy = ! K ; 0 x dt 240

®âªã¤ , ¨áª«îç ï ¯®®ç¥à¥¤­® Kx ¨ Ky , ¯®«ã稬

d2 Kx + !2 K = 0; 0 x dt2 d2 Ky + !2 K = 0; 0 y dt2 â.¥. ãà ¢­¥­¨ï ª®«¥¡ ­¨© ¡¥§ § âãå ­¨ï. €­ «®£¨ç­ë¥ ãà ¢­¥­¨ï ¨¬¥îâ ¬¥áâ® ¤«ï á®áâ ¢«ïîé¨å M~ .  áᬮâਬ á«ãç ©, ª®£¤  ä¥àà¨â ­ ¬ £­¨ç¥­ ®¤­®à®¤­®, â.¥. ¬ £­¨â­ë¥ ¬®¬¥­âë ¢á¥å í«¥ªâà®­®¢ ¯ à ««¥«ì­ë. à¨ í⮬ ¬®¦­® ­¥ ãç¨â뢠âì ®¡¬¥­­ë¥ á¨«ë ¬¥¦¤ã ᯨ­ ¬¨ í«¥ªâà®­®¢. ¥ «ì­® ­ ¬ £­¨ç¥­­®áâì ­¥ ¡ë¢ ¥â ®¤­®à®¤­®©, ­® ¥á«¨ ¤«¨­  ­¥®¤­®à®¤­®á⨠¬­®£® ¡®«ìè¥ à ááâ®ï­¨ï ¬¥¦¤ã  â®¬ ¬¨, â® ®¡¬¥­­ë¥ á¨«ë ¬®¦­® ­¥ ãç¨â뢠âì ¨ ¯à¨ ­¥®¤­®à®¤­®¬ ­ ¬ £­¨ç¥­¨¨. ‚ 㪠§ ­­ëå ãá«®¢¨ïå ¢¥ªâ®à ­ ¬ £­¨ç¥­­®á⨠M~ 0 ç¥à¥§ ­¥ª®â®à®¥ ¢à¥¬ï ¡ã¤¥â ®à¨¥­â¨à®¢ ­ ¢¤®«ì ¯à¨«®¦¥­­®£® ¯®áâ®ï­­®£® ¯®«ï H~ 0 , ç⮠ᮮ⢥âáâ¢ã¥â ­ ¬ £­¨ç¥­¨î ¤® ­ áë饭¨ï. …᫨ ⥯¥àì ¢®§¤¥©á⢮¢ âì ­  ä¥àà¨â á« ¡ë¬ ¯¥à¥¬¥­­ë¬ ¬ £­¨â­ë¬ ¯®«¥¬ ~h, â® ®­® ¢ë§®¢¥â ¬ «ë¥ ®âª«®­¥­¨ï ¬®¬¥­â®¢ ®â ¨å ¯®«®¦¥­¨ï à ¢­®¢¥á¨ï, çâ® ¯à¨¢¥¤¥â ª ¯®ï¢«¥­¨î ᮮ⢥âáâ¢ãî饩 ¯¥à¥¬¥­­®© ­ ¬ £­¨ç¥­­®á⨠m ~ . ‘¢ï§ì ¬¥¦¤ã m ~ ¨ ~h ®¯à¥¤¥«¨â ¬ £­¨â­ãî ¢®á¯à¨¨¬ç¨¢®áâì ä¥àà¨â  ­  ­¥ª®â®à®© ç áâ®â¥ !. ãáâì ®áì z ᮢ¯ ¤ ¥â á ­ ¯à ¢«¥­¨¥¬ H~ 0 . ’®£¤  १ã«ìâ¨àãî饥 ¬ £­¨â­®¥ ¯®«¥ ¡ã¤¥â à ¢­®

H~ = ~h + H~ 0 ; M~ = m ~ + M~ 0 ; ¯à¨ç¥¬ H~ 0 ¨ M~ 0 ¨¬¥îâ ⮫쪮 z -¥ á®áâ ¢«ïî騥. ’ ª ª ª ¢á¥ ¢¥«¨ç¨­ë

¨§¬¥­ïîâáï ᨭãᮨ¤ «ì­®, â® ¯à®¨§¢®¤­ ï ¯® ¢à¥¬¥­¨ ®§­ ç ¥â 㬭®¦¥­¨¥ ­  j!. ’®£¤ , ®â¡à á뢠ï ç«¥­ë ¢ëá襣® ¯®à浪  ¬ «®áâ¨, ¨§ ãà ¢­¥­¨ï ¤«ï M~ ¯®«ãç ¥¬

j!mx = ; 0 (my H0 ; hy M0 ); j!my = 0 (mxH0 ; hx M0 ); j!mz = 0: „ ­­ãî á¨á⥬ã ãà ¢­¥­¨© ¬®¦­® à¥è¨âì ®â­®á¨â¥«ì­® mx ¨ my : 241

2 2 0 H0 !0 M0 h ; mx = ; !20; M h ; j x 2 2 2 2 0 H0 ! ; 20 2 H02 y 0 M0 h ; 20 2 M0H0 h ; my = j !2 ! ; 20 2H02 x !2 ; 20 2H02 y mz = 0:

…᫨ ®¡®§­ ç¨âì

!0 = 0 H0 ; !M = 0 M0 ; â® í⨠ä®à¬ã«ë ¬®¦­® § ¯¨á âì ¢ ¢¨¤¥

mx = ; !2!0;!Œ!2 hx ; j !2! ;!Œ!2 hy ; 0 0 my = j !2! ;!Œ!2 hx ; !2!0;!Œ!2 hy ; 0 0 m z = 0: Žâá ¬®£ãâ ¡ëâì ­ ©¤¥­ë á®áâ ¢«ïî騥 ¢¥ªâ®à  ¬ £­¨â­®© ¨­¤ãªæ¨¨, ãç¨â뢠ï, çâ®

~b = 0 (~h + m ~ ): Ž¡®§­ ç¨¢ ¨

 = 0 (1 ; !2!0;!Œ!2 ) 0 k = 0 !2! ;!Œ!2 ; 0

¤«ï á®áâ ¢«ïîé¨å ¬ £­¨â­®© ¨­¤ãªæ¨¨ ¯®«ã稬

bx =  hx ; jk hy ; by = jk hx +  hy ; bz = 0 hz : 242

‚ ¯®á«¥¤­¥© áâப¥ ¢¬¥áâ® 0 ®¡ëç­® ¯¨èãâ z , ª®â®à®¥ ¬®¦¥â ¡ëâì ®â«¨ç­® ®â 0 . ‘¢ï§ì ¬¥¦¤ã ~b ¨ ~h ¬®¦¥â ¡ëâì § ¯¨á ­  ç¥à¥§ ⥭§®à ¬ £­¨â­®© ¯à®­¨æ ¥¬®á⨠^:

~b = ^ ~h; ¯à¨ç¥¬ ⥭§®à ¬ £­¨â­®© ¯à®­¨æ ¥¬®á⨠^ à ¢¥­

 ;jk 0

^ =

jk  0

:

0 0 z ‘®áâ ¢«ïî騥 ⥭§®à  ¬ £­¨â­®© ¯à®­¨æ ¥¬®á⨠§ ¢¨áïâ ®â ç áâ®âë १®­ ­á­ë¬ ®¡à §®¬. ¥§®­ ­á ¨¬¥¥â ¬¥áâ® ¯à¨ ! = !0 = 0 H0 . Ž¤­ ª® ¯à¨ ®¯à¥¤¥«¥­­ëå ãá«®¢¨ïå ¬ £­¨â­ãî ¯à®­¨æ ¥¬®áâì ¬®¦­® ¯à¥¤áâ ¢¨âì ¢ ¢¨¤¥ ᪠«ïà . â® ¬®¦­® ᤥ« âì, ¥á«¨ ¯®¯¥à¥ç­®¥ ¢ë᮪®ç áâ®â­®¥ ¯®«¥ ï¥âáï ¢à é î騬áï. Œ®¦­® à §«¨ç âì ¯à ¢®¥ ¢à é¥­¨¥, ®¡à §ãî饥 á H~ ¯à ¢ë© ¢¨­â, ¨ «¥¢®¥, ®¡à §ãî饥 á H~ «¥¢ë© ¢¨­â. z

„«ï ¯à ¢®£® ¢à é¥­¨ï, ª ª ­¥âà㤭® ¢¨¤¥âì (á¬. à¨á.),

Ho

hy = ; jhx (¨«¨ hx = jhy ):

hy hx

y

®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï bx ¨ by , ¯®«ã稬

x

bx = hx ; jkhy = hx ; khx = ( ; k) hx ; by = jkhx + hy = ; khy + hy = ( ; k) hy : ’ ª¨¬ ®¡à §®¬, ¢ ¤ ­­®¬ á«ãç ¥ ¬ £­¨â­ ï ¯à®­¨æ ¥¬®áâì ï¥âáï ¤¨ £®­ «ì­ë¬ ⥭§®à®¬

 ; k 0 0

^+ =

0  ; k 0

;

0 0 z 243

  ¥á«¨ ¢ë᮪®ç áâ®â­®¥ ¬ £­¨â­®¥ ¯®«¥ ­¥ ¨¬¥¥â z -© á®áâ ¢«ïî饩, â® ^+ áâ ­®¢¨âáï ᪠«ï஬: + =  ; k = 0 (1 ; !2!0;!Œ!2 ; !2! ;!Œ!2 ) = 0 (1 ; ! !;Œ ! ): 0 0 0 ‚ á«ãç ¥ «¥¢®£® ¢à é¥­¨ï

hy = jhx (¨«¨ hx = ; jhy ): ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï bx ¨ by , ¯®«ã稬 bx = hx ; jkhy = hx + khx = ( + k) hx ; by = jkhx + hy = khy + hy = ( + k) hy ; ®âªã¤ 

^; = ¨

 + k 0 0

0  + k 0

0 0 z





; =  + k = 0 (1 ; !2!0;!Œ!2 + !2! ;!Œ!2 ) = 0 (1 + ! !+Œ ! ): 0 0 0 ‚ § ¢¨á¨¬®á⨠®â ç áâ®âë + ¨ ; ¬®¦­® ¯à¥¤áâ ¢¨âì ¢ ¢¨¤¥ £à ä¨ª®¢, ¯®ª § ­­ëå ­  à¨á. µ+ , µ m µ 0 (1 + ω ω ) 0

µ-

µ0

ω ω0

Š ª ¢¨¤­®, + ¨¬¥¥â à §àë¢, ª®£¤  ç áâ®â  ¢à é¥­¨ï ¯®«ï ᮢ¯ ¤ ¥â á ç áâ®â®© ᢮¡®¤­®© ¯à¥æ¥áᨨ. ‚ ¤¥©á⢨⥫쭮á⨠ࠧàë¢ ®£à ­¨ç¥­ 244

¯®â¥àﬨ ¨ ­¥«¨­¥©­®áâìî. « £®¤ àï ¯®â¥àï¬ ¬ £­¨â­ ï ¯à®­¨æ ¥¬®áâì ¯à¨®¡à¥â ¥â ¬­¨¬ãî á®áâ ¢«ïîéãî. ®â¥à¨ ¢ ä¥àà¨â¥ ¬®£ãâ ¡ëâì ®¯¨á ­ë ¤®¯®«­¨â¥«ì­ë¬ ç«¥­®¬ ¢ ãà ¢­¥­¨¨ ¤«ï ­ ¬ £­¨ç¥­­®áâ¨

dM~ =  (H~  M~ ) + R~ : 0 dt

‘ãé¥áâ¢ã¥â ­¥áª®«ìª® ä®à¬ ¤¨áᨯ â¨¢­®£® ç«¥­ . à®á⥩è ï ¡ë«  ¯à¥¤«®¦¥­  ‹ ­¤ ã ¢ 1935 £.: R~ = ; 0  (M~  (H~  M~ )):

jM~ j

’ ª ï ä®à¬  § ¯¨á¨ ãç¨â뢠¥â ¯®â¥à¨ ç¨á⮠䥭®¬¥­®«®£¨ç¥áª¨. à¨ í⮬ ¢¥ªâ®à ­ ¬ £­¨ç¥­­®á⨠¯à¨¡«¨¦ ¥âáï ¯® ­ ¯à ¢«¥­¨î ª ¢¥ªâ®àã ­ ¬ £­¨ç¨¢ î饣® ¯®«ï (ª à ¢­®¢¥á¨î) ᮠ᪮à®áâìî, ¯à®¯®à樮­ «ì­®© ᪮à®á⨠¢à é¥­¨ï ¨ 㣫㠮⪫®­¥­¨ï ®â à ¢­®¢¥á¨ï. ‚¥«¨ç¨­   { 䥭®¬¥­®«®£¨ç¥áª ï ª®­áâ ­â , ®¯¨á뢠îé ï § âãå ­¨¥. ‚¥«¨ç¨­  ¢¥ªâ®à  ­ ¬ £­¨ç¥­­®á⨠¯à¨ í⮬ ¯à¥¤¯®« £ ¥âáï ­¥¨§¬¥­­®©. …᫨ à¥è¨âì ãà ¢­¥­¨¥ á ãç¥â®¬ § âãå ­¨ï, â® á â®ç­®áâìî ¤® ¬ «ëå ¯®à浪  ¢ëè¥ 2 ¯®«ã稬

+ = 0+ ; j00+ ; £¤¥ 2 2 0+ = 0 (1 ; (!!Œ2 (!; !;2 )2!0+)(!4!+2 !!2 0 )2 ); 0 0 2 !! ( ! + ! ) 00+ = 0 (!2 ; !Œ2 )2 + 4!02 !2 2 : 0

„«ï «¥¢®£® ¢à é¥­¨ï

; = 0; ; j00; ; £¤¥

245

0

2 2 0; = 0 (1 ; (!!Œ2 (!; !;2)2!0+)(!4!;2!!2 0 )2 ); 0 0 2 !! ( ! ; ! ) 00; = 0 (!2 ; !Œ2 )2 + 4!02!2 2 : 0

0

ƒà ä¨ç¥áª¨ íâ® ¯à¥¤áâ ¢«¥­® ­  à¨á. µ+ , µ+ µ+ µ+

0

ω ω0

1

µ-

µ− 0

1

ω ω0

’ ª¨¬ ®¡à §®¬, ⮫쪮 + ­®á¨â १®­ ­á­ë© å à ªâ¥à,   ; { ­¥à¥§®­ ­á­ë©. 7.2.

 á¯à®áâà ­¥­¨¥ ¯«®áª¨å ¢®«­ ¢ ­ ¬ £­¨ç¥­­®¬ ä¥àà¨â¥

7.2.1. à®¤®«ì­® ­ ¬ £­¨ç¥­­ë© ä¥àà¨â

ã¤¥¬ ¢ ¤ «ì­¥©è¥¬ ¯à¥¤¯®« £ âì, çâ® ä¥àà¨â ­ ¬ £­¨ç¥­ ¢¤®«ì ®á¨ z .  áᬮâਬ à á¯à®áâà ­¥­¨¥ ¯«®áª®© ¢®«­ë ¢ ­ ¯à ¢«¥­¨¨ ®á¨ z . ®«¥ ¢®«­ë ¯à¨¬¥â ¢¨¤

~e = ~e0
e; z ; ~h = ~h0
e; z ; ~b = ~b0
e; z ; ¯à¨ç¥¬ ¤«ï ¯«®áª®© ¢®«­ë ¢ ­¥®£à ­¨ç¥­­®¬ ¯à®áâà ­á⢥ ¨¬¥¥¬ 246

@ = 0; @ = 0; @ = ; : @x @y @z  á¯à®áâà ­¥­¨¥ ®¯¨á뢠¥âáï ãà ¢­¥­¨ï¬¨ Œ ªá¢¥«« : rot ~e = ; j! ~b; rot ~h = j!"~e: ˆáª«îç ï ~e, ¯®«ã稬 ãà ¢­¥­¨¥ rot rot ~h ; !2 "
~b = 0: ‡ ¯¨è¥¬ íâ® ãà ¢­¥­¨¥ ¤«ï á®áâ ¢«ïîé¨å ¢ ¤¥ª à⮢®© á¨á⥬¥ ª®®à¤¨­ â: grad div ~h ;
h ; !2 "
~b = 0: „ «¥¥,

@hy @hz x div ~h = @h @x + @y + @z : ® ¢ ¯«®áª®© ¢®«­¥ ¯®í⮬ã

@ @ @ @x = 0; @y = 0; @z = ; ; div ~h = ; hz :

’ ª ª ª grad ¨¬¥¥â ⮫쪮 z -î á®áâ ¢«ïîéãî, â® ãà ¢­¥­¨¥ ¬®¦¥â ¡ëâì à á¯¨á ­® ¯® ¤¥ª àâ®¢ë¬ á®áâ ¢«ïî騬 ¢ á«¥¤ãî饬 ¢¨¤¥:

2 hx + !2 "bx = 0;

2 hy + !2 "by = 0; ; 2hz + 2 hz + !2 "bz = 0: 247

ˆ§ ¯®á«¥¤­¥£® ãà ¢­¥­¨ï á«¥¤ã¥â, çâ® bz = z hz = 0, ®âªã¤  hz = 0, â.¥. ¢á¥ ¯à®¤®«ì­ë¥ á®áâ ¢«ïî騥 ¯®«ï à ¢­ë ­ã«î. ‘«¥¤®¢ â¥«ì­®, ¢ á«ãç ¥ à á¯à®áâà ­¥­¨ï ¢ ¯à®¤®«ì­® ­ ¬ £­¨ç¥­­®¬ ä¥àà¨â¥ ¯®«¥ ï¥âáï ¯®¯¥à¥ç­ë¬. …᫨ ¢ ¯¥à¢ë¥ ¤¢  ãà ¢­¥­¨ï ¯®¤áâ ¢¨âì ¢ëà ¦¥­¨ï ¤«ï bx ¨ by ç¥à¥§ hx ¨ hy :

bx = m hx ; jk hy ; by = jk hx +  hy ; â® ¯®«ã稬 ãà ¢­¥­¨ï ( 2 + !2") hx ; j!2 "k hy = 0; j!2 "k hx + ( 2 + !2 ") hy = 0: „ ­­ ï ®¤­®à®¤­ ï á¨á⥬  «¨­¥©­ëå ãà ¢­¥­¨© ®â­®á¨â¥«ì­® hx , hy ¨¬¥¥â ­¥âਢ¨ «ì­ë¥ à¥è¥­¨¨ï ¯à¨ ãá«®¢¨¨ à ¢¥­á⢠ ­ã«î ®¯à¥¤¥-

«¨â¥«ï

( 2 + !2 ")2 ; (!2 "k)2 = 0; ®âªã¤ 

2 + !2 " =  !2 "k; ¨«¨

2 = ; !2 " (  k); â.¥.

p

=  j! "(  k) =  j  ; £¤¥

p

p

+ = ! ( ; k)"; ; = ! ( + k)": „«ï + ¨§ ¯¥à¢®£® ãà ¢­¥­¨ï ¯®«ãç ¥¬ ᮮ⭮襭¨¥ 248

!2 "k hx ; j!2 "k hy = 0; ®âªã¤  hx = j hy . ’ ª®¥ ᮮ⭮襭¨¥ ᮮ⢥âáâ¢ã¥â ¢®«­¥ á ªà㣮¢®© ¯®«ïਧ æ¨¥©, ®¡à §ãî饩 ¯à ¢ë© ¢¨­â á Hz . à¨ í⮬

p

+ =  ; k:

„«ï ; = !
( + k)" ¨¬¥¥¬

hx = ; j hy ; â.¥. íâ® { «¥¢®¯®«ïਧ®¢ ­­ ï ¢®«­ . ’ ª¨¬ ®¡à §®¬, à¥è¥­¨ï¬¨ ãà ¢­¥­¨© Œ ªá¢¥««  ïîâáï ¯«®áª¨¥ ¢®«­ë á ªà㣮¢®© ¯®«ïਧ æ¨¥©, ¯à¨ç¥¬ ᪮à®á⨠à á¯à®áâà ­¥­¨ï (¨ § âãå ­¨ï) ¢®«­ á ¯à®â¨¢®¯®«®¦­ë¬ ­ ¯à ¢«¥­¨¥¬ ¢à é¥­¨ï à §«¨ç­ë. „«ï ¨áá«¥¤®¢ ­¨ï à á¯à®áâà ­¥­¨ï «¨­¥©­® ¯®«ïਧ®¢ ­­®© ¢®«­ë ¥¥ á«¥¤ã¥â à §«®¦¨âì ­  ¤¢¥ á ªà㣮¢®© ¯®«ïਧ æ¨¥© á à §«¨ç­ë¬¨ ¯®áâ®ï­­ë¬¨ à á¯à®áâà ­¥­¨ï. ‚ á¢ï§¨ á í⨬ ­ ª ¯«¨¢ ¥âáï à §­®áâì ä §
' = ; l ( + ; ; ): ‡  áç¥â í⮣® ¯à®¨á室¨â ¯®¢®à®â ¯«®áª®á⨠¯®«ïਧ æ¨¨ ­  㣮«  =
2' = ; 2l ( + ; ; ): …᫨ !0 < !, !Œ < !, ç⮠ᮮ⢥âáâ¢ã¥â á« ¡®¬ã ­ ¬ £­¨ç¥­¨î, â®

+  0 (1 ; !!Œ ) ; p+  p0 (1 ; !2!Œ ) ; ;  0 (1 + !!Œ ) ; p;  p0 (1 + !2Œ! ) : ’®£¤ 

h i   ; 2l !p0 "(1 ; !2Œ! ) ; !p0 "(1 + !2!Œ ) = = l ! p "; 2 Œ

249

0

â.¥. ­¥ § ¢¨á¨â ®â ç áâ®âë. Žâ¬¥â¨¬, çâ® §¤¥áì " { í«¥ªâà¨ç¥áª ï ¯à®­¨æ ¥¬®áâì ä¥àà¨â®¢®© á।ë. ®¢®à®â ¯«®áª®á⨠¯®«ïਧ æ¨¨ ¯à¨ á« ¡®¬ ­ ¬ £­¨ç¥­¨¨ á®áâ ¢«ï¥â ¯à ¢ë© ¢¨­â á ­ ¯à ¢«¥­¨¥¬ ­ ¬ £­¨ç¨¢ î饣® ¯®«ï ­¥§ ¢¨á¨¬® ®â ­ ¯à ¢«¥­¨ï à á¯à®áâà ­¥­¨ï. …᫨ ¨§¬¥­¨âì ­ ¯à ¢«¥­¨¥ ¬ £­¨â­®£® ¯®«ï, â® ¯®¢®à®â ¯«®áª®á⨠¯®«ïਧ æ¨¨ ¨§¬¥­¨â §­ ª, ­® ¯®-¯à¥¦­¥¬ã á®áâ ¢«ï¥â á ­ ¬ £­¨ç¨¢ î騬 ¯®«¥¬ ¯à ¢ë© ¢¨­â. â® ᢮©á⢮ ¨­®£¤  ¨á¯®«ì§ã¥âáï ¤«ï ᮧ¤ ­¨ï ­¥¢§ ¨¬­ëå í«¥¬¥­â®¢ ‘‚—-â¥å­¨ª¨. …᫨ ãç¥áâì § âãå ­¨¥, â® ¯® ¬¥à¥ à á¯à®áâà ­¥­¨ï «¨­¥©­® ¯®«ïਧ®¢ ­­ ï ¢®«­  ¯à¥¢à é ¥âáï ¢ ¯®«ïਧ®¢ ­­ãî ¯® í««¨¯áã, â ª ª ª ¯à ¢®¯®«ïਧ®¢ ­­ ï ¢®«­  § âã堥⠡ëáâ॥. ‚ ª®­æ¥ ª®­æ®¢ «¨­¥©­® ¯®«ïਧ®¢ ­­ ï ¢®«­  ¬®¦¥â ¯à¥¢à â¨âìáï ¢ ¯®«ïਧ®¢ ­­ãî ¯® ªàã£ã. Žá®¡¥­­® ¡ëáâà® íâ® ¯à®¨á室¨â ¢¡«¨§¨ १®­ ­á . ‚ ¬ £­¨â­®¬ ¯®«¥, ¡®«ì襬 १®­ ­á­®£® §­ ç¥­¨ï, ¢à é¥­¨¥ ¯«®áª®á⨠¯®«ïਧ æ¨¨ ¯à®¨á室¨â ¢ ®¡à â­®¬ ­ ¯à ¢«¥­¨¨. 7.2.2. ®¯¥à¥ç­® ­ ¬ £­¨ç¥­­ë© ä¥àà¨â

z  áᬮâਬ ⥯¥àì à á¯à®áâà ­¥­¨¥ ¢ ­ ¯à ¢«¥­¨¨ ®á¨ y, â.¥. ¯¥à¯¥­¤¨ªã«ïà­® ­ ¬ £­¨ç¨¢ î饬㠯®«î (á¬. à¨á.). ‚ í⮬ á«ãç ¥

Ho y x

~e = ~e0
e; y ; ~h = ~h0
e; y ; ~b = ~b0
e; y : @ @ @ @x = 0; @z = 0; @y = ; :

“à ¢­¥­¨¥ í«¥ªâ஬ £­¨â­®£® ¯®«ï ¨¬¥¥â ¯à¥¦­¨© ¢¨¤ rot rot ~h ; !2 "
~b = 0 ¨«¨ 250


~h + !2 "
~b ; grad div ~h = 0: Žâ¬¥â¨¬, çâ® grad div ~h ¢ í⮬ á«ãç ¥ ¨¬¥¥â y-î á®áâ ¢«ïîéãî, à ¢­ãî

2 hy . “à ¢­¥­¨¥ § ¯¨á뢠¥âáï ¢ ¯àאַ㣮«ì­ëå ª®®à¤¨­ â å

2 hx + !2 "bx =

2 hy + !2 "by ;

2 hz + !2 "bz =

0;

2 hy = 0; 0:

®¤áâ ¢¨¬ ⥯¥àì á®áâ ¢«ïî騥 ~b. ®«ã稬

hx ( 2 + !2 ") ; j!2"k hy = 0; jk hx +  hy = 0; hz ( + !2 "z ) = 0: ®¤áâ ¢«ïï hy ¨§ ¢â®à®£® ãà ¢­¥­¨ï ¢ ¯¥à¢®¥, ¯®«ã稬 2 2 hx ( 2 + !2 " ; ! "k ) = 0; hz ( 2 + !2 "z ) = 0: …᫨ ¯®«®¦¨âì hx 6= 0, â® ¨§ ¯¥à¢®£® ãà ¢­¥­¨ï á«¥¤ã¥â 2 2

2 + !2" ; ! "k = 0;

¨«¨

s

2 2

1 =  j! "  ; k : ˆ§ ¢â®à®£® ãà ¢­¥­¨ï ¯à¨ í⮬ á«¥¤ã¥â, çâ® hz = 0, â.¥. ¢®«­  ¯®«ïਧ®¢ ­  «¨­¥©­® ¢ ¯«®áª®á⨠H~ 0 (í«¥ªâà¨ç¥áª¨© ¢¥ªâ®à). ‡ ¬¥â¨¬, çâ® ¢

í⮬ á«ãç ¥ íä䥪⨢­ ï ¬ £­¨â­ ï ¯à®­¨æ ¥¬®áâì à ¢­  2 2 ? =  ; k : 251

‚ ¤à㣮¬ á«ãç ¥ hz 6= 0, hx = hy = 0. ®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï ¢ í⮬ á«ãç ¥ ®¯à¥¤¥«ï¥âáï ãà ¢­¥­¨¥¬

2 + !2 "z = 0; ®âªã¤ 

2 =  j!pz ":

„ ­­ ï ¢®«­  ¯®«ïਧ®¢ ­  ¯¥à¯¥­¤¨ªã«ïà­® H~ 0 . ˆâ ª, ¢ á«ãç ¥ ¯®¯¥à¥ç­®£® ­ ¬ £­¨ç¥­¨ï ä¥àà¨â  ¨¬¥¥â ¬¥áâ® íä䥪⠤¢®©­®£® «ã祯५®¬«¥­¨ï. Ž¤­  { ­¥®¡ëª­®¢¥­­ ï { ¢®«­  ­¥ ï¥âáï ç¨áâ® ¯®¯¥à¥ç­®©, â ª ª ª ªà®¬¥ hx ¨¬¥¥âáï ¯à®¤®«ì­ ï á®áâ ¢«ïîé ï ­ ¯à殮­­®á⨠¬ £­¨â­®£® ¯®«ï hy , ᤢ¨­ãâ ï ¯® ä §¥ ­  90o ®â­®á¨â¥«ì­® hx. „«ï ­¥®¡ëª­®¢¥­­®© ¢®«­ë 2 2 ? =  ; k = ( ; k)( + k) = +; : „à㣠ï { ®¡ëª­®¢¥­­ ï { ¢®«­  ¯®¯¥à¥ç­  ¨ ­¨ç¥¬ ­¥ ®â«¨ç ¥âáï ®â ¯«®áª¨å ¢®«­ ¢ ¤à㣨å á। å. „«ï í⮩ ¢®«­ë ¨¬¥¥¬

k = z : 7.3.

 á¯à®áâà ­¥­¨¥ ¢®«­ ¢ ¢®«­®¢®¤¥, ç áâ¨ç­® § ¯®«­¥­­®¬ ä¥àà¨â®¬

A S

11 00 00 11 00 11

à¥¤¯®«®¦¨¬, çâ® ­¥ª®â®à ï ç áâì á¥ç¥­¨ï ¢®«­®¢®¤  § ¯®«­¥­  ­ ¬ £­¨ç¥­­ë¬ ä¥àà¨â®¬ (á¬. à¨á.). ‚á«¥¤á⢨¥ í⮣® § ¯®«­¥­¨ï ¯®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï ¨ ᮡá⢥­­ë¥ ¢¥ªâ®à­ë¥ ä㭪樨 ¢®«­®¢®¤  ­¥áª®«ìª® ¨§¬¥­ïîâáï. ãáâì E~ m , H~ m , 0 { ¢¥ªâ®à­ë¥ ä㭪樨 ¨ ¯®áâ®ï­­ ï à á¯à®áâà ­¥­¨ï ­¥¢®§¬ã饭­®£® ¢®«­®¢®¤ , E~ , H~ , { â® ¦¥ ¤«ï ¢®§¬ã饭­®£® ¢®«­®¢®¤ . …᫨ ¯à¥­¥¡à¥çì ¯®â¥àﬨ, â® 0 = j 0 ,

= j . 252

®¤áâ ¢«ïï íâ® ¢ ãà ¢­¥­¨ï Œ ªá¢¥««  ¨ ãç¨âë¢ ï ¢¥ªâ®à­®¥ ⮦¤¥á⢮ rot (
F~ ) = 
rot F~ + grad   F~ ;

¯®á«¥ ᮪à é¥­¨ï íªá¯®­¥­â e j z ¨ e;j z ¯®«ã稬

rot E~ m + j 0
~z0  E~ m = j!0 H~ m ; rot H~ m + j 0
~z0  H~ m = ; j!"0E~ m ; ~ rot E~ ; j
~z0  E~ = ; j!
H; ~ rot H~ ; j
~z0  H~ = j!"
E: à¨ í⮬

 = " =

 ; 0 ^ ; " ;

¢­¥ á¥ç¥­¨ï ä¥àà¨â  A; ­  A; ¢­¥ A; 0 "^ ; ­  A:

„«ï ¯®«ã祭¨ï ä®à¬ã«ë, 㤮¡­®© ¤«ï ¨á¯®«ì§®¢ ­¨ï ¢ ⥮ਨ ¢®§¬ã饭¨©, á®áâ ¢¨¬ ¢ëà ¦¥­¨¥ div (E~ m  H~ ) + div (E~  H~ m ) = H~
rot E~ m ; E~ m
rot H~ + + H~ m
rot E~ ; E~
rot H~ m : ®¤áâ ¢«ïï á rot ¨§ ­ ¯¨á ­­ëå ¢ëè¥ ãà ¢­¥­¨©, ¯®«ã稬 div (E~ m  H~ ) + div (E~  H~ m ) = H~ (j!0 H~ m ; j 0~z0  E~ m ) ;

; E~ m (j!"E~ + j ~z0  H~ ) + H~ m (;j!H~ + j ~z0  E~ ) ; ; E~ (;j!"0E~ m ; j 0~z0  H~ m ) = ; j!H~ m ( ; 0 )H~ ; j!E~ m (" ; "0 )E~ ; ; j ( ; 0 )E~ m (~z0  H~ ) ; j ( ; 0 )E~ (~z0  H~ m ):

„ ­­®¥ à ¢¥­á⢮ ¯à®¨­â¥£à¨à㥬 ¯® á¥ç¥­¨î ¢®«­®¢®¤ . à¨ í⮬ ¨­â¥£à « ®â ¤¨¢¥à£¥­æ¨© ¯à¥®¡à §ã¥âáï ¢ ¨­â¥£à « ¯® ¯¥à¨¬¥âàã á¥ç¥­¨ï ¨ ®¡à é ¥âáï ¢ ­ã«ì ¢á«¥¤á⢨¥ £à ­¨ç­ëå ãá«®¢¨©. ‚ १ã«ìâ â¥ ¯®«ãç ¥¬ à ¢¥­á⢮ 253

Z

0 = ; ! [H~ m
( ; 0 )
H~ + E~ m
(" ; "0 )
E~ ] dS + S

Z

+ ( ; 0 ) [E~ m  H~ + E~  H~ m ]
~z0 dS; S

®âªã¤ , á ãç¥â®¬ ⮣®, çâ®  ; 0 6= 0 ¨ " ; "0 6= 0 «¨èì ­  á¥ç¥­¨¨ ä¥àà¨â  A, ­ å®¤¨¬ R[H~ 
(^ ;  )
H~ + E~ 
(^" ; " )
E~ ] dS 0 0 m m = 0 + ! A : R[E~   H~ + E~  H~  ]
~z dS m m 0 S

®«ã祭­ ï ä®à¬ã«  ï¥âáï â®ç­®©, ­® âॡã¥â §­ ­¨ï ¢®§¬ã饭­ëå ä㭪権 E~ ¨ H~ . ˆå ¬®¦­® ®¯à¥¤¥«¨âì ¯à¨¡«¨¦¥­­®, ¯®í⮬ã ä®à¬ã«  ¯à¨£®¤­  ¢ ª ç¥á⢥ ¯¥à¢®£® ¯à¨¡«¨¦¥­¨ï ¤«ï á« ¡®§ ¯®«­¥­­®£® ¢®«­®¢®¤ . ®«ì§ãïáì í⮩ ä®à¬ã«®©, à áᬮâਬ ¢ ¦­ë© ç áâ­ë© á«ãç ©: à á¯à®áâà ­¥­¨¥ ¬®¤ë ⨯  H11 ¢ ªà㣫®¬ ¢®«­®¢®¤¥, ­  ®á¨ ª®â®à®£® à á¯®«®¦¥­ â®­ª¨© ªàã£«ë© ä¥àà¨â®¢ë© áâ¥à¦¥­ì, ­ ¬ £­¨ç¥­­ë© ¯à®¤®«ì­®. Ž¡®§­ ç¥­¨ï ¯à¨¢¥¤¥­ë ­  à¨á. 2b

Žâ¬¥â¨¬, çâ® ¢ §­ ¬¥­ â¥«¥ ¯®«ã祭­®© ä®à¬ã«ë ¢ ¯¥à¢®¬ ¯à¨¡«¨¦¥­¨¨ ¬®¦­® ¯à¥­¥¡à¥çì ®â«¨ç¨¥¬ ¢®§¬ã饭­®£® ¯®«ï ®â ­¥¢®§¬ã饭­®£®, â.¥. ¯®«®¦¨âì H~  H~ m , E~  E~ m , â ª ª ª ¨­â¥£à « ¢ §­ ¬¥­ â¥«¥ ¡¥à¥âáï ¯® ¢á¥¬ã á¥ç¥­¨î ¢®«­®¢®¤ , â ª çâ® ¯®¯à ¢ª  ¡ã¤¥â ¬ «®©. ’®£¤  §­ ¬¥­ â¥«ì ¯à¥®¡à §ã¥âáï á«¥¤ãî騬 ®¡à §®¬:

111 000 000 111 2a

Z S

(E~ m  H~ + E~  H~ m )
~z0 dS 

Z

Z S

(E~ m  H~ m + E~ m  H~ m )
~z0 dS =

Z = 2Re (E~ m  H~ m )~z0 dS = Z2
jEm¯®¯ j2 dS; m S

S

254

£¤¥

!0 0 Zm = j!

0 = 0 |

¢¥é¥á⢥­­ ï ¢¥«¨ç¨­ . „«ï ­ å®¦¤¥­¨ï ¢®§¬ã饭­®£® ¯®«ï ¢ ä¥àà¨â¥ ¯à¨ ¬ «®¬ ¯® áà ¢­¥­¨î á ¤«¨­®© ¢®«­ë ¯®¯¥à¥ç­®¬ à §¬¥à¥ ä¥àà¨â  ¬®¦­® ¢®á¯®«ì§®¢ âìáï ª¢ §¨áâ â¨ç¥áª¨¬ ¯à¨¡«¨¦¥­¨¥¬. à¨ ¢­¥á¥­¨¨ ¬ £­¥â¨ª  ¢® ¢­¥è­¥¥ ®¤­®à®¤­®¥ ¬ £­¨â­®¥ ¯®«¥ ¢ ¬ £­¥â¨ª¥ ãáâ ­ ¢«¨¢ ¥âáï ¯®«¥, ®¯à¥¤¥«ï¥¬®¥ à §¬ £­¨ç¨¢ î騬 ä ªâ®à®¬:

~hi = ~he ; N^
m ~: ‡¤¥áì ~hi { ¢­ãâ७­¥¥ ¯®«¥ ¢ ¬ £­¥â¨ª¥, ~he { ¯®«¥, ª®â®à®¥ ¡ë«® ¤® ¢­¥á¥­¨ï ¬ £­¥â¨ª , m ~ { ­ ¬ £­¨ç¥­­®áâì, N^ { à §¬ £­¨ç¨¢ î騩 ä ªâ®à

{ ᨬ¬¥âà¨ç­ë© ⥭§®à. ‚ ç áâ­®áâ¨, ¤«ï 樫¨­¤à  ¢ ¯®¯¥à¥ç­®¬ ¯®«¥ ⥭§®à à §¬ £­¨ç¨¢ î饣® ä ªâ®à  ¤¨ £®­ «¥­, ¯à¨ç¥¬ Nxx = Nyy = 12 , Nzz = 0.  ¬ £­¨ç¥­­®áâì á¢ï§ ­  á ¢­ãâ७­¨¬ ¯®«¥¬ ¢ ¬ £­¥â¨ª¥ ç¥à¥§ ¢®á¯à¨¨¬ç¨¢®áâì:

m ~ = ^
~hi ; ¯à¨ç¥¬ ¢®á¯à¨¨¬ç¨¢®áâì ¢ ¤ ­­®¬ á«ãç ¥ â ª¦¥ ¨¬¥¥â ⥭§®à­ë© å à ªâ¥à: ^ = ^ ; ^0 : 0 ®¤áâ ¢«ïï í⨠ᮮ⭮襭¨ï ¢ ¨á室­ãî ä®à¬ã«ã, ¯®«ãç ¥¬

~hi = ~he ; N^
^~hi : ¥è ï í⮠ᮮ⭮襭¨¥ ®â­®á¨â¥«ì­® ~hi , ¯®«ã稬 ~hi = (1 + N^
^);1
~he : ‚ ­ è¥¬ á«ãç ¥ ¢ ª ç¥á⢥ ¢­¥è­¥£® ¯®«ï ¬ë ¤®«¦­ë ¯à¨­ïâì ­¥¢®§¬ã饭­®¥ ¯®«¥ H~ m ,   ¢ ª ç¥á⢥ ¢­ãâ७­¥£® { ¢®§¬ã饭­®¥ ¯®«¥ H~ , â.¥. ¢­ãâਠä¥àà¨â  ¯®«¥ à ¢­® 255

H~ = (1 + N^
^);1
H~ m : „«ï ã¯à®é¥­¨ï ¡ã¤¥¬ ¨áª âì ¯®áâ®ï­­ë¥ à á¯à®áâà ­¥­¨ï ¤«ï ¢®«­ á ªà㣮¢®© ¯®«ïਧ æ¨¥© (¯à ¢®© ¨ «¥¢®©), â ª ª ª ¤«ï ­¨å ⥭§®àë ¯à®­¨æ ¥¬®á⨠¤¨ £®­ «ì­ë:

^ =

 0 0

0  0

0 0  z



; ^



=

 0 0

0  0

0 0  z



;

¯à¨ç¥¬

^ = ^ ; ^0 : 0 ’ ª ª ª ®¡  ⥭§®à  ¤¨ £®­ «ì­ë, â®

1  0 0

2 

N^
^ =

0 12  0

:

0 0 0 Žâá ¨¬¥¥¬ ᮮ⭮襭¨ï 1 1 Hx = H ; H = Hmy ; Hz = Hmz : mx y 1 + 21  1 + 12  ’ ª ª ª 1 1 0 ; = =  2+  ;  1 1 0  1 + 2  1 + 2  0 0 â® 0
H ; H = 20
H ; H = H : Hx =  2+ mz 0 mx y  + 0 my z  €­ «®£¨ç­® ¤«ï í«¥ªâà¨ç¥áª®£® ¯®«ï ¯®«ã稬 Ex = " 2+"0 " Emx ; Ey = " 2+"0 " Emy : 0 0 256

„«ï ­¥¢®§¬ã饭­®© ¬®¤ë H11 ¢ ªà㣫®¬ ¢®«­®¢®¤¥ ¯®¯¥à¥ç­ë¥ á®áâ ¢«ïî騥 ¬ £­¨â­®£® ¨ í«¥ªâà¨ç¥áª®£® ¯®«ï ¨¬¥îâ ¢¨¤ 0 J 0 (g0 r) cos ' ; Hr = ; j 0 g11 1 11 sin ' 0 r) sin ' ; H' =  j r 0 J1 (g11 cos ' sin ' 0 0 Er =  j! r J1 (g11 r) cos ' ; 0 J 0 (g0 r) cos ' : E' = j!0
g11 1 11 sin '

 á ¨­â¥à¥áãîâ ¯®«ï ¢¡«¨§¨ 業âà , £¤¥ ®­¨ ¯à ªâ¨ç¥áª¨ ®¤­®à®¤0 r) ¨ J10 (g11 0 r) ­ë. â¨ ¯®«ï ¬®£ãâ ¡ëâì ­ ©¤¥­ë ¯ã⥬ à §«®¦¥­¨ï J1 (g11 ¯® á⥯¥­ï¬ r á ®áâ ¢«¥­¨¥¬ ¢ ¢ëà ¦¥­¨ïå ¤«ï ª®¬¯®­¥­â ¯®«ï ­¥ § ¢¨áïé¨å ®â r ç«¥­®¢: 0 cos ' ; Hr = ; 12 j 0 g11 sin ' 1 0 sin ' ; H' =  2 j 0 g11 cos ' 0 sin ' ; Er =  12 j!0 g11 cos ' 0 cos ' : E' = ; 21 j!0 g11 sin '

’ ª ª ª ¬ë å®â¨¬ ¨¬¥âì ¤¥«® á ¢à é î騬¨áï ¯®«ï¬¨, â® á«¥¤ã¥â ᪮¬¡¨­¨à®¢ âì § ¯¨á ­­ë¥ ¢ëè¥ ¯®«ï. ’®£¤  ¯®«ãç ¥¬ 0 ej' ; Hr = ; 21 j 0 g11 0 ej' ; H' =  12 0 g11

0 ej' ; Er =  21 j!0g11 0 ej' : E' = ; 12 !0 g11

257

¥à¥å®¤ ª ¤¥ª àâ®¢ë¬ á®áâ ¢«ïî騬 ¬®¦¥â ¡ëâì ¯à®¨§¢¥¤¥­ ¯® ä®à¬ã« ¬

Hx Hy Ex Ey

= = = =

Hr cos ' ; H' sin '; Hr sin ' + H' cos '; Er cos ' ; E' sin '; Er sin ' + E' cos ':

â® ¤ ¥â 0 ; Hx = ; 12 j 0 g11 0 ; Hy =  21 0 g11

0 ; Ex =  12 j!0 g11 0 : Ey = ; 12 !0 g11

Œë ­ è«¨, â ª¨¬ ®¡à §®¬, ¯à¨¡«¨¦¥­­ë¥ ¢ëà ¦¥­¨ï ¤«ï ­¥¢®§¬ã饭­®£® ¯®«ï ¢ 業âà «ì­®© ®¡« á⨠ªà㣫®£® ¢®«­®¢®¤ . ‡ ¬¥â¨¬, çâ® §­ ª¨ \+" ¨ \;" ᮮ⢥âáâ¢ãîâ «¥¢®¬ã ¨ ¯à ¢®¬ã ¢à é¥­¨ï¬. Žâá á ¯®¬®éìî ¯®«ã祭­ëå ¢ëè¥ ä®à¬ã« ­ å®¤¨¬ ¯®«¥ ¢ ä¥àà¨â¥: 0 Hx = ; j0 +0 g11 ;  0 0  g Hy =   0 +0 11 ;  0 0 0 g11 Ex =  j""0 ! + "0 ; 0 0 g11 Ey = ; ""0 ! + "0 :

’¥¯¥àì ¬ë ¨¬¥¥¬ ¢®§¬®¦­®áâì ¢ëç¨á«¨âì ¨­â¥£à «ë, ¢å®¤ï騥 ¢ ä®à¬ã«ã ¤«ï ­®¢®£® §­ ç¥­¨ï ¯®áâ®ï­­®© à á¯à®áâà ­¥­¨ï . “ç¨âë0 = 1:84=a ¨ ¯®¤áâ ¢«ïï ¨­â¥£à «ë, ¯®«ã稬 ¢ ï, çâ® g11 258





2 0 + k02 " ; "0 ; k2 = !2  " :   0 + 2:1
0 ab 2  ; 00 0 02 " + "0  + 0

’ ª¨¬ ®¡à §®¬, ä §®¢ë¥ ᪮à®á⨠¤«ï ¯à ¢®£® ¨ «¥¢®£® ¢à é¥­¨ï à §«¨ç­ë, ¯®í⮬㠯«®áª®áâì ¯®«ïਧ æ¨¨ «¨­¥©­® ¯®«ïਧ®¢ ­­®© ¢®«­ë ¢à é ¥âáï ¯à¨ à á¯à®áâà ­¥­¨¨. “£®« ¯®¢®à®â  ¬®¦¥â ¡ëâì ¢ëç¨á«¥­ ­  ®á­®¢ ­¨¨ ¯à¥¤ë¤ã饩 ä®à¬ã«ë: 2  = ; l ( + 2; ; ) = 4:2
0 l ab 2 ( + k)(0 +  ) : + 0 ; 0 ‡ ¬¥â¨¬, çâ® + + 0 = 0 (2 ; ! !;Œ ! ); ; + 0 = 0 (2 + ! !+Œ ! ); k = !20 !;!Œ!2 : 0 0 0 à¨ á« ¡® ­ ¬ £­¨ç¥­­®¬ ä¥àà¨â¥ (!  !0 ) 2 k0  20 !!Œ ; (+ + 0 ) (; + 0 )  20 (4 ; !!Œ2 ): ®¤áâ ¢«ïï íâ® ¢ ¢ëà ¦¥­¨¥ ¤«ï , ¤«ï á« ¡® ­ ¬ £­¨ç¥­­®£® ä¥àà¨â  ¯®«ã稬

!Π2 b2 !Π: b !  l   4:2
0 l a2 0 2 a2 ! 20 (4 ; !!Œ2 ) ’ ª ª ª 0 = v! , â® ¢¤ «¨ ®â ªà¨â¨ç¥áª®© ç áâ®âë  á« ¡® § ¢¨á¨â ®â ä ç áâ®âë, ­® ¯®ç⨠«¨­¥©­® ®â ­ ¬ £­¨ç¥­­®á⨠(!Œ ). 7.4.

‚¥­â¨«¨ ¨ æ¨àªã«ïâ®àë

‚¢¥¤¥­¨¥ ­ ¬ £­¨ç¥­­®£® ä¥àà¨â  ¢ ¢®«­®¢®¤ ¯®§¢®«ï¥â ᮧ¤ âì ãáâனá⢠, ­¥ 㤮¢«¥â¢®àïî騥 ¯à¨­æ¨¯ã ¢§ ¨¬­®áâ¨. à®á⥩訥 ¨§ ­¨å ¯à®¯ã᪠îâ ¢®«­ë ¢ ®¤­®¬ ­ ¯à ¢«¥­¨¨ ¨ ¯®£«®é îâ { ¢ ¤à㣮¬. ’ ª®¥ ãáâனá⢮ ­ §ë¢ îâ ¢¥­â¨«¥¬ (¨«¨ ¨§®«ïâ®à®¬). ®«¥¥ á«®¦­ë¥ ¯¥à¥¤ îâ ᨣ­ « ¨§ ®¤­®£® ¢å®¤  ¢ ¤à㣮© ¯® ªàã£ã (æ¨àªã«ïâ®à). ‡¤¥áì ¬ë à áᬮâਬ ­¥ª®â®àë¥ ¢¨¤ë íâ¨å ãáâனáâ¢. 259

7.4.1. ”¥àà¨â ¢ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥

„«ï ⮣® ç⮡ë ä¥àà¨â ¯à®ï¢¨« ᢮¨ ­¥¢§ ¨¬­ë¥ ᢮©á⢠, ¥£® á«¥¤ã¥â ¯®¬¥é âì ¢ â ª®© ®¡« á⨠¢®«­®¢®¤ , £¤¥ ¢ë᮪®ç áâ®â­®¥ ¬ £­¨â­®¥ ¯®«¥ ¨¬¥¥â ªà㣮¢ãî ¯®«ïਧ æ¨î. ‚ ¯àאַ㣮«ì­®¬ ¢®«­®¢®¤¥ ¬ £­¨â­®¥ ¯®«¥ ®á­®¢­®© ¬®¤ë H10 ¨¬¥¥â ¤¢¥ á®áâ ¢«ïî騥:

  2

x ;
cos a a j x Hx = a
sin a :

Hz =

â¨ á®áâ ¢«ïî騥 ®à¨¥­â¨à®¢ ­ë ¯®¤ 㣫®¬ 90o ¤à㣠ª ¤àã£ã ¨ ᤢ¨­ãâë ­  90o ¯® ä §¥. â® §­ ç¨â, çâ® ¬ £­¨â­®¥ ¯®«¥ ¯®«ïਧ®¢ ­® ¯® í««¨¯áã. ®«ïਧ æ¨ï áâ ­®¢¨âáï ªà㣮¢®© ¢ â®çª¥, £¤¥  ¬¯«¨âã¤ë ®à⮣®­ «ì­ëå á®áâ ¢«ïîé¨å à ¢­ë   2 x  sin x : cos =  a a a a Žâá ­ å®¤¨¬

  tg x a =  a =  2a ; £¤¥  { ¤«¨­  ¢®«­ë ¢ ¢®«­®¢®¤¥. Š®®à¤¨­ â  x, ¤«ï ª®â®à®© ¬ £­¨â­®¥ ¯®«¥ ¯®«ïਧ®¢ ­® ¯® ªàã£ã, à ¢­   = a arctg  ; x = a arctg a  2a ¨«¨ ¢â®à®¥ §­ ç¥­¨¥ x  ) = a ( ; arctg  ): x = a ( ; arctg a  2a ‡¤¥áì §­ ç¥­¨¥ arctg ¡¥à¥âáï ¢ ¯à¥¤¥« å ®â 0 ¤® =2. ‚ í⮬ ¬¥á⥠¢®«­®¢®¤  ¯®¬¥é ¥âáï ä¥àà¨â®¢ ï ¯« á⨭ , ­ ¬ £­¨ç¥­­ ï ¯®¯¥à¥ç­® ®â­®á¨â¥«ì­® ¢®«­®¢®¤ . à ¢®¥ ¨«¨ «¥¢®¥ ¢à é¥­¨¥ ¬ £­¨â­®£® ¯®«ï ¢ ä¥àà¨â¥ § ¢¨á¨â ®â ­ ¯à ¢«¥­¨ï à á¯à®áâà ­¥­¨ï 260

¢®«­ë, ¯®í⮬㠢¥«¨ç¨­  ¬ £­¨â­®© ¯à®­¨æ ¥¬®á⨠ § ¢¨á¨â ®â ­ ¯à ¢«¥­¨ï à á¯à®áâà ­¥­¨ï. …᫨ ¯®áâ®ï­­®¥ ¬ £­¨â­®¥ ¯®«¥ ¢ë¡à ­® â ª, ç⮡ë

!0 = 0 H0  !; â® ¯®â¥à¨ ¤«ï ¢®«­ë á ¯à ¢ë¬ ¢à é¥­¨¥¬ १ª® ¢®§à áâ îâ, ¢ â® ¢à¥¬ï ª ª ¤«ï ¢®«­ë á «¥¢ë¬ ¢à é¥­¨¥¬ ¯®â¥à¨ ®áâ îâáï ¬ «ë¬¨. ®â¥à¨ ¢ ¯àאַ¬ ­ ¯à ¢«¥­¨¨ ®¡ëç­® á®áâ ¢«ïîâ 0.5{0.7 ¤,   ¢ ®¡à â­®¬ ¤®á⨣ îâ 25{30 ¤. „«ï à áè¨à¥­¨ï ¯®«®áë ­ ¬ £­¨ç¨¢ î饥 ¯®«¥ ¤¥« îâ ­¥®¤­®à®¤­ë¬ ¢ ¯à®¤®«ì­®¬ ­ ¯à ¢«¥­¨¨. Šà®¬¥ ⮣®, íä䥪⠭¥¢§ ¨¬­®á⨠ãᨫ¨¢ ¥âáï, ¥á«¨ à冷¬ á ä¥àà¨â®¬ ¯®¬¥áâ¨âì ¤¨í«¥ªâà¨ç¥áªãî ¯« á⨭ã á ¡®«ì让 ¢¥«¨ç¨­®© ". ’ ª®© ¢¥­â¨«ì ­ §ë¢ îâ १®­ ­á­ë¬. ‚ ¤àã£¨å ¢¥­â¨«ïå ¨á¯®«ì§ãîâ íä䥪â ᬥ饭¨ï í«¥ªâà¨ç¥áª®£® ¯®«ï ¢ ãáâனá⢥ á ä¥àà¨â®¬. „«ï í⮣® ¯®¤¬ £­¨ç¨¢ î饥 ¯®«¥ ¤®«¦­® ¡ëâì §­ ç¨â¥«ì­® ¬¥­ìè¥ à¥§®­ ­á­®£® §­ ç¥­¨ï. ®¤¬ £­¨ç¨¢ î饥 ¯®«¥ ®¯à¥¤¥«ï¥âáï ãá«®¢¨¥¬ + = 0 (1 ; ! !;Œ ! )  0; 0 ®âªã¤ 

!0 = ! ; !Œ; â.¥. §­ ç¨â¥«ì­® ¬¥­ìè¥, 祬 !0 = ! ¤«ï १®­ ­á­®£® ¢¥­â¨«ï. Ey

H0

11 00 00 11 00 11 00 11

X

„«ï ¯àאַ© ¢®«­ë + à ¢­® ­ã«î, ¯®í⮬㠬 £­¨â­®¥ ¨ í«¥ªâà¨ç¥áª®¥ ¯®«ï ¢¡«¨§¨ ä¥àà¨â  ®ç¥­ì ¬ «ë. „«ï ®¡à â­®© ¢®«­ë, ­ ®¡®à®â, ¯®«¥ ª®­æ¥­âà¨àã¥âáï ¢¡«¨§¨ ä¥àà¨â  (á¬. à¨á.).

…᫨ ­  ä¥àà¨â ­ ª«¥¨âì ¤¨í«¥ªâà¨ç¥áªãî ¯« á⨭ã, ¯®ªàëâãî ¯®«ã¯à®¢®¤ï騬 ¯®ªàë⨥¬, â® § âãå ­¨¥ ®¡à â­®© ¢®«­ë ¡ã¤¥â áãé¥á⢥­­® ¡®«ìè¥ § âãå ­¨ï ¯àאַ© ¢®«­ë. àאַ¥ § âãå ­¨¥ ¢ â ª¨å ¢¥­â¨«ïå á®áâ ¢«ï¥â 0.5{1 ¤, ®¡à â­®¥ { 20{30 ¤. ‚¥á â ª¨å ¢¥­â¨«¥© ¬ « ¡« £®¤ àï ¬ «®© ¢¥«¨ç¨­¥ ­¥®¡å®¤¨¬®£® ¯®«ï. 261

7.4.2. Š® ªá¨ «ì­ë© ¢¥­â¨«ì

Š® ªá¨ «ì­ë¥ ¢¥­â¨«¨ ¬®£ãâ ¡ëâì ᮧ¤ ­ë, ¥á«¨ ¤®¡¨âìáï ¢ ­¨å ¯®«ã祭¨ï ¢à é î饣®áï ¬ £­¨â­®£® ¯®«ï. â® ¤®á⨣ ¥âáï ¯ã⥬ ¢¢¥¤¥­¨ï ¢ ª® ªá¨ «ì­ãî «¨­¨î ¤¨í«¥ªâਪ , § ¯®«­ïî饣® ­¥ª®â®àë© á¥ªâ®à á¥ç¥­¨ï. ’®£¤  ¢ «¨­¨¨ ¯®ï¢«ï¥âáï ¯à®¤®«ì­ ï á®áâ ¢«ïîé ï ¬ £­¨â­®£® ¯®«ï, ª®â®à ï ¢¬¥áâ¥ á ¯®¯¥à¥ç­®© á®áâ ¢«ïî饩 ®¡à §ã¥â ¢à é î饥áï ¬ £­¨â­®¥ ¯®«¥. ”¥àà¨â®¢ãî ¯« á⨭㠯®¬¥é îâ ­  ¡®ª®¢®© ¯®¢¥àå­®á⨠¤¨í«¥ªâਪ  ¢ ¯®¯¥à¥ç­®¥ ¯®¤¬ £­¨ç¨¢ î饥 ¯®«¥. H0

111 000 000 111 000 111 00 11 00 11 7.4.3. –¨àªã«ïâ®àë

ˆ¤¥ «ì­ë© æ¨àªã«ïâ®à ®¯à¥¤¥«ï¥âáï ª ª ­¥¢§ ¨¬­®¥ ãáâனá⢮ ¡¥§ ¯®â¥àì á ç¨á«®¬ ¢å®¤®¢ ¡®«ìè¥ ¤¢ãå, ¢ ª®â®à®¬ ᨣ­ «, ¯®¤ ­­ë© ­  «î¡®© ¨§ ¢å®¤®¢, ¯¥à¥¤ ¥âáï ¯®«­®áâìî ­  ®¤¨­ ¨§ á®á¥¤­¨å (­  ¤à㣨¥ ¢å®¤ë ¯à¨ í⮬ ­¥ ¯®áâ㯠¥â ­¨ç¥£®). ‘ãé¥áâ¢ãîâ à §«¨ç­ë¥ ⨯ë æ¨àªã«ïâ®à®¢. — é¥ ¢á¥£® ¢áâà¥ç îâáï âà¥å- ¨ ç¥âëà¥å¯«¥ç­ë¥ æ¨àªã«ïâ®àë. ’à¥å¯«¥ç­ë¥ ¯®áâ஥­ë ­  ®á­®¢¥ ä¥àà¨â®¢®£® १®­ â®à , ç¥âëà¥å¯«¥ç­ë¥ ¢®«­®¢®¤­ë¥ { ­  ®á­®¢¥ ­¥¢§ ¨¬­®£® ä §®¢à é â¥«ï.  áᬮâਬ âà¥å¯«¥ç­ë© ¢®«­®¢®¤­ë© æ¨àªã«ïâ®à ­  ®á­®¢¥ १®­ â®à  á ­ ¬ £­¨ç¥­­ë¬ ä¥àà¨â®¬. …£® ãáâனá⢮ ¯à¥¤áâ ¢«¥­® ­  à¨á. Žá­®¢®© æ¨àªã«ïâ®à  ï¥âáï ªàã£«ë© à¥§®­ â®à ⨯  E110 , ã ª®â®à®£® í«¥ªâà¨ç¥áª®¥ ¯®«¥ ¨¬¥¥â «¨èì ®¤­ã á®áâ ¢«ïîéãî Ez .   ®á¨ í«¥ªâà¨ç¥áª®¥ ¯®«¥ à ¢­® ­ã«î, ­® «¨­¥©­® ¢®§à á⠥⠯ਠ㤠«¥­¨¨ ®â ®á¨. Œ £­¨â­®¥ ¯®«¥ ¨¬¥¥â ¤¢¥ á®áâ ¢«ïî騥: Hr ¨ H' . ‚ 業âà «ì­®© ®¡« á⨠¬ £­¨â­®¥ ¯®«¥ ¡«¨§ª® ª ®¤­®à®¤­®¬ã. –¥­âà «ì­ ï ®¡« áâì १®­ â®à  § ¯®«­¥­  ä¥àà¨â®¬, ­ ¬ £­¨ç¥­­ë¬ ¢ ­ ¯à ¢«¥­¨¨ ®á¨ १®­ â®à . 262

Ho 2a b

Š १®­ â®àã ¯à¨á®¥¤¨­¥­ë âਠ¯àאַ㣮«ì­ëå ¢®«­®¢®¤ .   à¨á. (á¬. á«¥¤. áâà.) ¯®ª § ­  áâàãªâãà  í«¥ªâ஬ £­¨â­®£® ¯®«ï à ¡®ç¥© (¤¨¯®«ì­®©) ¬®¤ë E110 १®­ â®à . ‚ ¨§®âய­®¬ ¯à¨¡«¨¦¥­¨¨ íâ  ¬®¤  ¤¢ ¦¤ë ¢ë஦¤¥­ . ‚ ç áâ­®áâ¨, «¨­¥©­® ¯®«ïਧ®¢ ­­ ï ¬®¤  ¬®¦¥â ¡ëâì à §«®¦¥­  ­  ¤¢¥ ¬®¤ë á ªà㣮¢®© ¯®«ïਧ æ¨¥©, ¢à é î騥áï ¢ ¯à®â¨¢®¯®«®¦­®¬ ­ ¯à ¢«¥­¨¨. „«ï íâ¨å ¬®¤ ¬ £­¨â­®¥ ¯®«¥ ¢ 業âॠ१®­ â®à  ¨¬¥¥â ªà㣮¢ãî ¯®«ïਧ æ¨î, ª®â®à ï ¯à¥®¡à §ã¥âáï ¢ í««¨¯â¨ç¥áªãî á à®á⮬ à ¤¨ãá .   £à ­¨æ¥ १®­ â®à  ¯®«ïਧ æ¨ï «¨­¥©­ , â ª ª ª ­ ¯à ¢«¥­¨¥ ¬ £­¨â­®£® ¯®«ï 䨪á¨à®¢ ­® £à ­¨ç­ë¬¨ ãá«®¢¨ï¬¨. ‚ १®­ â®à¥, ç áâ¨ç­® § ¯®«­¥­­®¬ ­ ¬ £­¨ç¥­­ë¬ ä¥àà¨â®¬, ᮡá⢥­­ë¥ ç áâ®âë æ¨àªã«ïà­® ¯®«ïਧ®¢ ­­ëå ¬®¤ E110 áâ ­®¢ïâáï à §«¨ç­ë¬¨ ¡« £®¤ àï à §«¨ç¨î ¬ £­¨â­®© ¯à®­¨æ ¥¬®á⨠¤«ï ¯à ¢®£® ¨ «¥¢®£® ¢à é¥­¨ï. à¨­æ¨¯ ¤¥©á⢨ï æ¨àªã«ïâ®à  á â ª¨¬ १®­ â®à®¬ ¬®¦¥â ¡ëâì ®¡êïá­¥­ ¯ã⥬ à áᬮâ७¨ï ¢ë­ã¦¤¥­­ëå ª®«¥¡ ­¨© ¢ í⮬ १®­ â®à¥. à¥¤¯®«®¦¨¬, çâ® ¢®«­®¢®¤ë, á¢ï§ ­­ë¥ á १®­ â®à®¬, ¯à¨á®¥¤¨­¥­ë á 㣫®¢ë¬ à ááâ®ï­¨¥¬ 120o ¬¥¦¤ã ­¨¬¨, ¨ ¢®§¡ã¦¤ î饥 ¯®«¥ ¯à¨«®¦¥­® ª ®¤­®¬ã ¨§ ­¨å. ‚ ¨§®âய­®¬ ¯à¨¡«¨¦¥­¨¨ «¨­¥©­® ¯®«ïਧ®¢ ­­®¥ ¯®«¥ ¬®¤ë E110 ¡ã¤¥â ¢®§¡ã¦¤¥­® ¢ १®­ â®à¥ â ª, çâ® ¢®§¡ã¦¤ î騩 ¢®«­®¢®¤ ­ å®¤¨âáï ¢ ¬ ªá¨¬ã¬¥ í«¥ªâà¨ç¥áª®£® ¯®«ï. ‘âàãªâãà  í⮣® ¯®«ï ¯®ª § ­  ­  à¨á. â® ¯®«¥ ¢®§¡ã¦¤ ¥â «¨­¨¨ II ¨ III  ­ «®£¨ç­ë¬ ®¡à §®¬.

263

III

III

II

II

I

I

Š®£¤  ¯®áâ®ï­­®¥ ¬ £­¨â­®¥ ¯®«¥ ¢ª«î祭®, ¯«®áª®áâì ¯®«ïਧ æ¨¨ ¢®§¡ã¦¤¥­­®© ¬®¤ë E110 ¯®¢®à ç¨¢ ¥âáï.   ¤à㣮¬ à¨á. ¯®ª § ­  áâàãªâãà  ¯®«ï í⮩ ¬®¤ë ¯à¨ ¯®¢®à®â¥ ¯«®áª®á⨠¯®«ïਧ æ¨¨ ­  30o . ‚ í⮬ á«ãç ¥ ¢®«­®¢®¤ II ®ª §ë¢ ¥âáï ¢ 㧫¥ í«¥ªâà¨ç¥áª®£® ¯®«ï ¨ ­¥ ¢®§¡ã¦¤ ¥âáï. …᫨ «¨­¨ï ᮣ« á®¢ ­ , â® ¢áï ¬®é­®áâì ¨§ ¢®«­®¢®¤  I ¡ã¤¥â ¯¥à¥¤ ­  ¢ ¢®«­®¢®¤ II . “áâனá⢮ ¡ã¤¥â à ¡®â âì ª ª æ¨àªã«ïâ®à. Ž¡á㤨¬ ãá«®¢¨ï ¯®¢®à®â  ¯«®áª®á⨠¯®«ïਧ æ¨¨. Š®£¤  ¯®áâ®ï­­®¥ ¬ £­¨â­®¥ ¯®«¥ ¢ª«î祭®, ¯à®¨á室¨â à á饯«¥­¨¥ ç áâ®â ¢ë஦¤¥­­ëå ¬®¤ á ªà㣮¢®© ¯®«ïਧ æ¨¥©. ¥§®­ ­á­ ï ç áâ®â  ®¤­®© ¬®¤ë 㢥«¨ç¨¢ ¥âáï, ¤à㣮© { 㬥­ìè ¥âáï. ‚ ᮮ⢥âá⢨¨ á í⨬ ¢®§­¨ª îâ ᤢ¨£¨ ¯® ä §¥ ¢®§¡ã¦¤¥­­ëå ¢à é îé¨åáï ¬®¤ ®â­®á¨â¥«ì­® ¯®«ï ¢ ¢®§¡ã¦¤ î饩 «¨­¨¨ '. Œ®¤ë, ¢à é î騥áï ¢ ¯à®â¨¢®¯®«®¦­®¬ ­ ¯à ¢«¥­¨¨, ᪫ ¤ë¢ ïáì, ¤ îâ áâ®ïçãî ¢®«­ã, â.¥. ä ªâ¨ç¥áª¨ «¨­¥©­® ¯®«ïਧ®¢ ­­ãî áâàãªâãàã ¯®«ï. ® ¯«®áª®áâì ¯®«ïਧ æ¨¨ ®ª §ë¢ ¥âáï ¯®¢¥à­ã⮩ ­  㣮« '. “£®« ' ¢ १®­ â®à¥ à ¢¥­, ª ª ¨§¢¥áâ­®,

' = ; arctg Qx; £¤¥ Q { ­ £à㦥­­ ï ¤®¡à®â­®áâì १®­ â®à , x = 2
! ! { ®â­®á¨â¥«ì­ ï à ááâனª ,
! = ! ; !0 , !0 { १®­ ­á­ ï ç áâ®â . ‡ ¬¥â¨¬, çâ® ­ £à㦥­­ ï ¤®¡à®â­®áâì ®¯à¥¤¥«ï¥âáï ¢ ®á­®¢­®¬ á¢ï§ìî á ¢®«­®¢®¤ ¬¨, â ª ª ª ¯®â¥à¨ ¢ ­ ¬ £­¨ç¥­­®¬ ä¥àà¨â¥ ¬ «ë. ‘奬  æ¨àªã«ïâ®à  ­  ®á­®¢¥ ­¥¢§ ¨¬­®£® ä §®¢à é â¥«ï ¯®ª § ­  ­  à¨á. ‚ ª ç¥á⢥ ¯à¨¬¥à  §¤¥áì ¯à¨¢¥¤¥­ æ¨àªã«ïâ®à ­  ¤¢®©­ëå âன­¨ª å. 264

H1

1

E1

E2

2

1

H2

2

‚®«­ , ¯®¤ ­­ ï ­  ¢å®¤, ¢ ¯«¥ç® H1 ¯¥à¢®£® âன­¨ª , ¤¥«¨âáï ¯®à®¢­ã ¢ ¡®ª®¢ë¥ ¯«¥ç¨, ᮥ¤¨­¥­­ë¥ ¤¢ã¬ï ¢®«­®¢®¤ ¬¨ á ¡®ª®¢ë¬¨ ¯«¥ç ¬¨ ¢â®à®£® ¤¢®©­®£® âன­¨ª . ‚ ®¤¨­ ¨§ ¢®«­®¢®¤®¢ ¯®¬¥é¥­ ­¥¢§ ¨¬­ë© ä §®¢à é â¥«ì (ᤢ¨£ ä §ë ¢ ¯àאַ¬ ­ ¯à ¢«¥­¨¨ ­  180o ¬¥­ìè¥, 祬 ¢ ®¡à â­®¬). ‚ ¤à㣮© ¯®¬¥é¥­ ®¡ëç­ë© ä §®¢à é â¥«ì (¢­®á¨¬ë© ᤢ¨£ ä §ë à ¢¥­ ᤢ¨£ã ä §ë ­¥¢§ ¨¬­®£® ä §®¢à é â¥«ï ¢ ¯àאַ¬ ­ ¯à ¢«¥­¨¨). à¨ à á¯à®áâà ­¥­¨¨ ¢®«­ ¢ ¢®«­®¢®¤ å ¢ ¯àאַ¬ ­ ¯à ¢«¥­¨¨ ᤢ¨£¨ ä §ë ¢ ®¡®¨å ¢®«­®¢®¤ å ®¤¨­ ª®¢ë, ¯®í⮬㠢®«­ë, ᪫ ¤ë¢ ïáì, ¯à®å®¤ïâ ¢ ¯«¥ç® H2 ¢â®à®£® âன­¨ª . à¨ ¯®¤ ç¥ ¢®«­ë ­  ¢å®¤ H2 ¢â®à®£® âன­¨ª  ¢®«­ë, à á¯à®áâà ­ïïáì ¯® ¢®«­®¢®¤ ¬ ¢ ®¡à â­®¬ ­ ¯à ¢«¥­¨¨, ¯à¨®¡à¥â îâ à §­®áâì ä § 180o. ®í⮬㠮­¨, ᪫ ¤ë¢ ïáì, ¯à®å®¤ïâ ¢ ¯«¥ç® E1 ¯¥à¢®£® âன­¨ª . ¥âà㤭® ¢¨¤¥âì, çâ® æ¨àªã«ïæ¨ï ¯à®¨á室¨â ¯® á奬¥ H1 ; H2 ; E1 ; E2 ; H1 ; : : : ¨ â.¤. ‚ ॠ«ì­ëå æ¨àªã«ïâ®à å á ­¥¢§ ¨¬­ë¬ ä §®¢à é â¥«¥¬ ¢¬¥áâ® ¤¢®©­ëå âன­¨ª®¢ ¤«ï ¤¥«¥­¨ï ¢®«­ë ç áâ® ¨á¯®«ì§ãîâ 3-¤ ­ ¯à ¢«¥­­ë¥ ®â¢¥â¢¨â¥«¨, ª®â®àë¥ ¡®«¥¥ è¨à®ª®¯®«®á­ë ¨ ª®¬¯ ªâ­ë. ‘¯¨á®ª «¨â¥à âãàë

[1] ‚ ©­è⥩­ ‹.€. «¥ªâ஬ £­¨â­ë¥ ¢®«­ë. Œ.:  ¤¨® ¨ á¢ï§ì, 1988. [2] ’¥®à¨ï «¨­¨© ¯¥à¥¤ ç¨ ᢥàå¢ë᮪¨å ç áâ®â. Œ.: ‘®¢. à ¤¨®, 1951. [3] Š æ¥­¥«¥­¡ ã¬ .‡. ‚ë᮪®ç áâ®â­ ï í«¥ªâத¨­ ¬¨ª . Œ.:  ãª , 1966. 265

[4] Œ¨ª í«ï­ €.‹. ’¥®à¨ï ¨ ¯à¨¬¥­¥­¨¥ ä¥àà¨â®¢ ­  ᢥàå¢ë᮪¨å ç áâ®â å. Œ.; ‹.: ƒ®áí­¥à£®¨§¤ â, 1963. [5] ‘âàíââ®­ „¦. ’¥®à¨ï í«¥ªâ஬ £­¥â¨§¬ . Œ.; ‹.: Žƒˆ‡ { ƒ®áâ¥å¨§¤ â, 1948. [6] ‹ ­¤ ã ‹.„., ‹¨äè¨æ ….Œ. «¥ªâத¨­ ¬¨ª  ᯫ®è­ëå á।. Œ.: ƒ®á¨§¤ â, 1959. [7] €«ì⬠­ „¦. “áâனá⢠ ‘‚—. Œ.: Œ¨à, 1968. [8] ‹¥¢¨­ . ‘®¢à¥¬¥­­ ï ⥮à¨ï ¢®«­®¢®¤®¢. Œ.: ˆ§¤-¢® ¨­®áâà. «¨â., 1954.

266

Œ à«¥­ Œ®¨á¥¥¢¨ç Š à«¨­¥à ‹…Š’Ž„ˆ€ŒˆŠ€ ‘‚— Šãàá «¥ªæ¨©

 ¡®â  ¢ë¯®«­¥­  ¯à¨ ᮤ¥©á⢨¨ ”– \ˆ­â¥£à æ¨ï" ®¤¯¨á ­® ¢ ¯¥ç âì 01.99 ”®à¬ â 60x84, 1/16 ¥ç âì ®äá¥â­ ï. “ç.-¨§¤. «. 6,5 ’¨à ¦ íª§. ‡ ª § ü . ‹¨æ¥­§¨ï ‹ 02/285 ®â 6 ¬ ï 1998 £.; ˆ§¤ â¥«ì᪨© 業âà ƒ“; 630090, ®¢®á¨¡¨àáª{90, ã«. ¨à®£®¢ , 2.