3/19/2018. PHY 712 Electrodynamics 9-9:50 AM MWF Olin 105

Transcription

1 PHY 7 Eletodynamis 9-9:5 A WF Olin 5 Plan fo Letue 4: Complete eading of Chap. 9 & A. Supeposition of adiation B. Satteed adiation PHY 7 Sping 8 -- Letue 4 PHY 7 Sping 8 -- Letue 4 Eletomagneti waves fom time hamoni soues eview: Fo sala potential (Loentz gauge, ) 4 i ' e ' 3,, d ' ', Fo veto potential (Loentz gauge, ) A 4 3, A, d ' J', i ' e ' PHY 7 Sping 8 -- Letue 4 3

2 Conside antenna soue (ente-fed) Note these notes diffe fom pevious fomulation d/ d z d y d x J, zˆ I sin d z x y fo d z d PHY 7 Sping 8 -- Letue 4 4 Conside antenna soue -- ontinued J fo, zˆ I sin d z x y n ; n,,3... d d z d n n n3 PHY 7 Sping 8 -- Letue 4 5 Conside antenna soue -- ontinued zˆ J, I sin d z x y fo d z d Veto potential fom soue: i ' e 4 ' 3, d ' J ', A i e Fo d A, d ' e ', 4 3 iˆ ' J i d e iz 'os, zˆ I dz ' e sin d z ' A 4 d PHY 7 Sping 8 -- Letue 4 6

3 Conside antenna soue -- ontinued i d e iz os A, zˆ I dz e sin d z 4 d i e os zˆ I 4 In the adiation zone : B, A, iˆ A, E, iˆ ˆ A, d ˆ os I d 8 d os osd sin * E, B, A, ˆ A, d os osd sin PHY 7 Sping 8 -- Letue 4 7 Conside antenna soue -- ontinued d os d os osd I 8 sin n n n3 PHY 7 Sping 8 -- Letue 4 8 Conside antenna soue -- ontinued d os d os osd I 8 sin Fo d n : n n n3 PHY 7 Sping 8 -- Letue 4 9 3

4 Radiation fom antenna aays z y d x a N, ˆ sin J zi d z x N j a y n ; d j n,,3... fo d z d PHY 7 Sping 8 -- Letue 4 A J Radiation fom antenna aays -- ontinued Veto potential fom aay soue : iˆ ', d ' J', d ' e J', N, zˆ I sin d z x N ja y A d e 4 j N d os sin a N sin os sin asin os i N iajsin os izos, zˆ e I dz e sind z N e j N iajsin i ' e ' j e 4 i fo d z d PHY 7 Sping 8 -- Letue 4 Radiation fom antenna aays -- ontinued In the adiation zone : B E, A, iˆ A,, iˆ ˆ A, d ˆ os I d 8 * E, B, A, ˆ A, d os osd sin an sin sin asin sin os N;ad/; N N;ad; os PHY 7 Sping 8 -- Letue 4 4

5 os d os os d sin a N sin os I 8 sin sin asin os d Example fo, N, d a Additional amplitude pattens an be obtained by ontolling elative phases of antennas. PHY 7 Sping 8 -- Letue 4 3 Bief intodution to multipole expansion of eletomagneti fields (Chap. 9.7) Soueless axwell's equations in tems of E and H fields with time dependene e it : E iz H H ie / Z E H whee / and Z / Deoupled equations: E H i iz H E E H Z PHY 7 Sping 8 -- Letue 4 4 ultipole expansion of eletomagneti fields -- ontinued Note that: E H Define: L i Note that L L Y Convenient opeatos fo angula momentum analysis Eigenfuntions: L (, ) sin sin sin Y (, ) l( l ) Y (, ) PHY 7 Sping 8 -- Letue 4 5 5

6 ultipole expansion of eletomagneti fields -- ontinued agneti multipole field: l l H gl ( ) Y(, ) E L E l l Z g ( ) Y (, ) l Eleti multipole field: l l E Z E H E l f ( ) Y (, ) E L H l l f ( ) Y (, ) l spheial Bessel funtion spheial Bessel funtion PHY 7 Sping 8 -- Letue 4 6 ultipole expansion of eletomagneti fields -- ontinued Veto spheial hamonis: (fo l ) X(, ) LY(, ) l( l ) Othogonality onditions: d X (, ) X (, ) * l ' m' ll mm * l ' m', ), ) d X ( X ( Geneal expansion of fields: E i H a fl ( ) X(, ) a gl ( ) Xl m(, ) i E E a fl ( ) X(, ) a gl ( ) X(, ) PHY 7 Sping 8 -- Letue 4 7 ultipole expansion of eletomagneti fields -- ontinued Time aveaged powe distibution of adiation fa fom soue: Z d X ˆ X ( i) a (, ) a (, ) l E E Fo a pue multipole adiation with eithe a o a : Z d a X (, ) X (, ) m Y l ml m Y l ml m l m Y l( l ) PHY 7 Sping 8 -- Letue 4 8 6

7 Fo example: l 3 3 X(, ) sin X(, ) X (, ) os 8 6 PHY 7 Sping 8 -- Letue 4 9 Fo example: l X(, ) sin os X(, ) 3os 4os X(, ) os PHY 7 Sping 8 -- Letue 4 7