Delve into the fascinating realm of electromagnetic waves and Maxwell's equations as we explore the sources of radiation, dipole radiation patterns, and the formulation of Maxwell's equations in terms of vector and scalar potentials. Discover the intricacies of Lorentz gauge, solutions for the electromagnetic field, and the generation of electromagnetic waves from time-harmonic sources. Gain insights into charge and current densities, continuity conditions, and general source equations in this comprehensive lecture series.
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PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 23: Sources of radiation Start reading Chap. 9 A. Electromagnetic waves due to specific sources B. Dipole radiation patterns 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 1
Comment on Mid-term exam n ( ) = + k x y sin cos i i 0 i c E ( ) = + + E x y z cos sin 0 2 n i i i 0 i ( ) = k x y sin cos i i 0 R c ( ) = + + s p E z x y cos sin E E i i 0 0 0 R R R ' c n ( ) = + k x y sin cos 0 T ( ) = + + s p T E z x y cos sin E E 0 0 0 T T 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 3
= = P M Microscopi or vacuum c form ( E 0; 0) : = Coulomb' law s : / 0 E 1 c = B J Ampere - Maxwell' law s : 0 2 t B + = E Faraday' law s : 0 t = B magnetic No monopoles : 0 1 = 2 c 0 0 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 4
Formulation of Maxwells equations in terms of vector and scalar potentials -- continued 1 c + = A require - - form gauge Lorentz : 0 L L 2 t 2 1 c + = 2 / L 0 L 2 2 t A 2 1 c + = 2 A J L 0 L 2 2 t General equation form : 2 1 c = 2 4 f 2 2 t 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 6
Solution of Maxwells equations in the Lorentz gauge -- continued 1 ( ( ) ) ( ) = r ' , ' r r / ' r , ; ' G t t t t c r r ' ( ) + r Solution for t field = , : t ( ) ( ) r r , , t = 0 f 1 1 c ( ) 3 r r ' , ' r ' ' ' ' d r dt t t f t r r ' 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 7
Electromagnetic waves from time harmonic sources ( ( continuity that the Note r J t t ( ( ) ) ) ) ( ( ) ) ~ i = t r r Charge density : , , t e ~ J i = t J r r Current density : , , t e condition : ( ) r , t ~ J ( ) ( ) ( ) ~ + = i + = r r , 0 , , 0 ( ) ~ f ( ( ) ( ) i = , t r r General source : , f t e ~ f 1 ) ( ) ~ , = , r r For 4 0 ~ f ~ J ( ) ( ) , = , r r 0 or i 4 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 8
Electromagnetic waves from time harmonic sources continued: ( ) ( ) = + r r , , t t = 0 f 1 1 c ( ) 3 r r ' , ' r ' ' ' ' d r dt t t f t r r ' ~ ~ ( ) ( ) = + i t i t r r , , e e = 0 f ~ f 1 1 c ( ) 3 ' i t r r , ' r ' ' ' ' d r dt t t e r r ' r r ' i ~ f e ~ ( ) ( ) c = + 3 i t i t r , ' r , ' e d r e = 0 f r r ' 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 9
Electromagnetic waves from time harmonic sources continued: c For scalar potential (Lorentz gauge, ) k r r ' ik 1 e ~ ~ ( ) ( ) ( ) ~ = + 3 r r , ' r , , ' d r 0 r r 4 ' 0 For vector potential (Lorentz c gauge, ) k r r ' ik ~ A ~ A e ~ J ( ) ( ) ( ) = + 3 r r , ' r 0 , , ' d r 0 r r 4 ' 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 10
Electromagnetic waves from time harmonic sources continued: Useful expansion : r r ' ik e ( ) ( h ) ( ) r ( ) ' r = * ik j kr kr Y Y lm l l lm r r 4 ' lm ( ) ( ) kr h l Spherical Bessel function : j kr l ( ) kr ( ) kr = + Spherical Hankel function : j in l l , r ~ 0 ) ~ ~ ( ) ( ( ) ( ) lm Y lm = + r r , , r lm ~ ik ( ) ( ) ( j ) ( h ) ( ) ' r ~ = 3 * , ' r , ' r d r kr kr Y lm lm l l 0 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 11
Electromagnetic waves from time harmonic sources continued: Useful expansion : r r ' ik e ( ) ( h ) ( ) r ( ) ' r = * ik j kr kr Y Y lm l l lm r r 4 ' lm ( ) ( ) kr h l Spherical Bessel function : j kr l ( ) kr ( ) kr = + Spherical Hankel function : j in l l ~ A ) , r ( 0 A ) + ~ ( ( ) ( ) lm Y ~ a lm = r r , , r lm ~ J ( ) ( ) ( j ) ( h ) ( ) ' r ~ a = 3 * , ' r , ' r ik d r kr kr Y lm 0 lm l l 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 12
Forms of spherical Bessel and Hankel functions: ( ) x cos sin ix sin x e ( ) x ( ) x = = j h 0 0 ix + 1 ( ) x 2 ( ) x ix x i e ( ) x ( ) x = = j h 1 1 x 3 x x ( ) x ix 1 3 cos 3 3 i e ( ) x ( ) x ( ) x = = + sin 1 j h i 2 2 3 2 2 x x x x x x Assymptoti behavior c : ( ) l 2 l x + ( ) x 1 x j ( ) ! ! l 1 ix e ( ) ( ) x + 1 l 1 x h i l x 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 13
Electromagnetic waves from time harmonic sources continued: ( ) ( ) , , 0 lm ik r ( ) ( ) , , 0 lm r d ik r ~ l lm d kr h r ~ ~ ~ ( ) ( ) lm Y = + r r r , r lm ~ ( ) ( ) ( j ( lm ) ( h ) ( ) lm Y ) ( ) ' r ~ = 3 * , ' r , ' d r kr kr Y lm lm ~ A l l 0 ~ A ~ a = + r r r , r ~ J ( ) ( ) ( j ) ( h ) ( ) ' r ~ a = 3 * , ' r , ' kr kr Y lm 0 lm l l For (extent ik of source) r ) ( ( ) ( ) ( j ) ( ) ' r ~ 3 * , ' r , ' ' r kr Y lm l 0 ~ J ( ) ( ) kr ( ) ( j ) ( ) ' r ~ a 3 * , ' r , ' ' r ik h d r kr Y lm 0 lm l l 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 14
Electromagnetic waves from time harmonic sources continued: ~ 3 l lm d kr h r For (extent ik of source) r ( ) ( ) ( ) ( j ) ( ) ' r ~ * , ' r , ' ' r kr Y lm l 0 ~ J ( ) ( ) kr ( ) ( j ) ( ) ' r ~ a ~ J 3 * , ' r , ' ' r ik h d r kr Y lm 0 lm l l ( ) ( ) ~ ~ , ' r , ' r Note that and connected are via the ~ J ( ) ( ) + = r r continuity condition ik : , , 0 i ~ ( ) ( ) kr ( ) ( j ) ( ) ' r ~ 3 * , ' r , ' ' r h d r kr Y lm lm l l 0 ( ) k ~ J ( ) kr ( ) ( ) ( ) ' r = 3 * , ' r ' ' ' h d r j kr Y lm l l 0 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 15
Electromagnetic waves from time harmonic sources continued: ions approximat Various : ikr e ( ) ( ) kr + 1 l 1 kr h i l kr ( ) 1 l ' kr ( ) ' 1 ' kr j kr ( ) ! ! l + 2 l Lowest (non - trivial) contributi ons in expansions : l ikr ' ~ ik e kr ( ) ( ) ( ) ' r ~ 3 * , ' r , ' r d r Y 1 m 1 m 3 kr 0 ikr e ~ J ( ) ( ) ( ) ( ) ' r ~ a 3 * , ' r , ' r ik i d r Y 00 00 0 kr 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 16
Electromagnetic waves from time harmonic sources continued: Lowest order contributi dipole on; radiation : Define dipole moment frequency at : 1 ~ J ( ) ( ) ( ) ~ = 3 3 p r r r , , d r d r i ikr ~ A i e ( ) ( ) = r p 0 , 4 r ikr + 1 i i e ~ ( ) ( ) = r r p , 4 kr r 0 Note: in this case we have assumed a restricted extent of the source such that kr <<1. 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 17
Electromagnetic waves from time harmonic sources continued: ( ) ( ) ( 1 p r r ~ A ~ E ~ ) = + r r r , , , i ( ( ) r ) ( )( ikr 3 r r p p e ( ( ( ) ) ) ) = + 2 r 1 k ikr 2 4 0 ~ A ~ B ( ) ( ) = r r , , ikr 1 1 e ( ( ) ) = 2 r p 1 k 2 4 c r ikr 0 Power radiated for : 1 kr ( ) 2 r dP r ~ E ~ B ( ) ( ) = = 2 * r r S r r , , r 2 avg d 0 2 4 c k ( ( ) ) 2 = r r p 0 2 32 0 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 18
Example of dipole radiation source ( ) e J 0 , = z r J J ~ ( ) ~ = / / r R r R r 0 , cos e i R ~ A ( ) ( ) ( ) ( 0 j ) = 2 / ' r R r z , ' ' J ik r dr e h kr kr 0 0 0 0 J k ~ ( ) ( ) ( 1 j ) = 2 / ' r R r 0 , cos ' ' r dr e h kr kr 1 for R 0 0 R Evaluation : r 3 ikr 2 ~ A e R ( ) = r z , J ( ikr ) 0 0 2 r + 2 2 1 k R 3 + 2 J k e i R ~ ( ) = r 0 , cos 1 ( 1 ) 2 r kr + 2 2 k R 0 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 19
Example of dipole radiation source -- continued R r : for Evaluation 3 ikr 2 ~ A e R ( ) = r z , J ( ikr ) 0 0 2 r + 2 2 1 k R 3 + 2 J k e i R ~ ( ) = r 0 , cos 1 ( 1 ) 2 r kr + 2 2 k R 0 Relationship to pure dipole approximation (exact when kR 0) ( ) ( ) i 3 8 1 R J ~ J ( ) ~ = = 3 3 p r r r z 0 , , d r d r i ikr ~ A i e ( ) ( ) = r p 0 Correspond dipole ing fields : , 4 r ikr + 1 i i e ~ ( ) ( ) = r r p , 4 kr r 0 03/20/2017 PHY 712 Spring 2017 -- Lecture 23 20
