Electrodynamics Lecture 36 Review and Problem Solving Strategies

phy 712 electrodynamics 9 9 50 am mwf olin 103 n.w
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Explore the content of Lecture 36 in the Electrodynamics course, covering a review of units, problem-solving strategies, examples, and course evaluation forms. Dive into example problems on radiated power from circular motion, with additional information on Jackson's work. Learn about SI and Gaussian relationships, energy and power in SI units, and the solution of Maxwell's equations. Discover insights on electromagnetic energy density, Poynting vector, and equations for time-harmonic fields. Delve into the introduction of vector and scalar potentials, providing a comprehensive overview of Maxwell's equations.

  • Electrodynamics
  • Review
  • Problem Solving Strategies
  • Maxwells Equations
  • Examples
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  1. PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 36: Review of Electrodynamics 1. Units 2. Problem solving strategies 3. Examples 4. Course evaluation forms 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 1

  2. 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 2

  3. 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 3

  4. 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 4

  5. Example problems -- Radiated power from circular motion (Jackson 14.46): 2 2 v e 2 2 2 3 v c ( ) 2 = 4 v For circular orbit P 4.8 10 10 stat C 2 e 3 c = 5 10 cm 2 2 E P P 7 10 J erg J J v c = 2.3 10 = 2.3 10 24 31 erg J 2 2 2 2 3 2 1 E E e c c 4 2 31 / 2.3 10 8 .2 10 e 3 2 c mc = = 2.8 10 18 2 1 4 mc 2 4 / 4 E E e = 2.8 10 3 18 16 6.4 1 0 2 3 0.75 3 mc 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 5

  6. Jackson pg. 783 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 6

  7. 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 7

  8. 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 8

  9. More SI relationships: = + D E P More Gaussian relationships: 4 = 4 ) for ferromagnet = D E = + = H D H B E P D E 0 ( ) H 1 = = M B B F ( ) H = = H B M B H B = ) F 0 for ferromagnet elementary charge: e=1.6021766208 x 10-19 C =4.80320467299766 x 10-10 statC 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 9

  10. Energy and power (SI units) 1 2 ( ) u + E D H B Electromagnetic energy density: S E H Poynting vector: Equations for time harmonic fields : ( ) ( ) 1 ~ ~ ~ = ( E r + * i t i t i t E ,t) r E r E r E r ( ( , ( )e ( , )e ( , )e ) 2 * ( , ) D r ) 1 4 ( ) = ( , ) + ( , ) B r ( , ) r * r H u ,t avg t ( ) 1 2 ( ) ( ) = ( , ) E r ( , ) H r * S r ,t avg t 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 10

  11. Solution of Maxwell's equations: E 1 c = = E B J / 0 0 2 t B + = = E B 0 0 t Introduction of vector and scalar potentials: 0 = + = + = = B B A A B + = E E 0 0 t t A A = E E or t t 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 11

  12. Scalar and vector potentials continued: / : = = E 0 ( ) A = 2 / 0 t E 1 c B J 0 2 t ( ) 2 A 1 c ( ) + + = A J 0 2 2 t t 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 12

  13. Analysis of the t scalar and vector potential equations : ( ) A = 2 / 0 ( ) t 2 A 2 t 1 c ( ) + + = A J 0 2 1 c + = A Lorentz gauge form - - require 0 L L 2 t t 2 1 c + = 2 / L 0 L 2 2 2 A t 1 c + = 2 A J L 0 L 2 2 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 13

  14. Solution methods for scalar and vector potentials and their electrostatic and magnetostatic analogs: 1 / L c t + = In your bag of tricks: Direct (analytic or numerical) solution of differential equations Solution by expanding in appropriate orthogonal functions Green s function techniques 2 + = 2 L 0 2 2 2 A 1 c 2 A J L 0 L 2 2 t 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 14

  15. How to choose most effective solution method -- In general, Green s functions methods work well when source is contained in a finite region of space Con = sider the electrostatic problem: / L G r 2 0 r = 2 3 r r Define: ( , ) ' 4 ( ) ' 1 = ( ) ( , ) G r + 3 ( ) r r r d r L 4 V 0 1 2 r ( , ) r r r ( ) r ( , ) r r ( ) . d r G G 4 S 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 15

  16. r For electrostat ic problems where ( ) is contained in a small 1 , ( , ) G = r r r r region of space a nd S ' l 1 4 r ( ) ( ) lm * = l ' ' Y Y , , lm lm + + 1 r r ' 2 1 l r 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 16

  17. 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 )and ( r J r For dynamic problems wh e re ( , , ) are , contained in a small region of space and S r r ' ic e r r = ( , ', ) G r r ' 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 17

  18. 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 ' 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 18

  19. 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 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 19

  20. 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 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 20

  21. Electromagnetic waves from time harmonic sources continued: ( ( ) ) ( ) ( ) = + E r r A r , , , i ( ) = B r A r , , Power radiated: 2 r dP d r ( ) ( ) ( ) = = 2 * r r S E r B r , , r avg 2 0 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 21

  22. 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 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 22

  23. Example of dipole radiation source -- continued R r ~ = z r A Evaluation for : 3 ikr 2 e R ( ) , 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 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 23

  24. Electromagnetic waves from time harmonic sources for dipole radiation --: ( ) ( ) ( , , i = + = = B r A r ) E r r A r , ( ) ( ) r ( ) r r p p 3 ikr 1 e ( ) ( ) ( ) ( ) + 2 r p r 1 k ikr 2 4 r 0 ( ) 1 ( ) , , ikr 1 e ( ) ( ) = 2 r p 1 k 2 4 c r ikr 0 Power radiated for 1: kr 2 r dP d r ( ) ( ) ( ) = = 2 * r r S E r B r , , r avg 2 0 2 4 c k ( ) 2 ( ) = r p r 0 PHY 712 Spring 2017 -- Lecture 36 0 2 32 04/21/2017 24

  25. Radiation from a moving charged particle Variables (notation) : ( ) r t R d ( ) r t R q v q dt z r ( ) r t ( ) r t R r R R q v. r-Rq(tr)=R q r Rq(t) 2 dP q 2 = 2 v sin 3 4 d c y x 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 25

  26. Linard-Wiechert potentials (Gaussian units) ( ) q r q r r dt t t R r R R R d t ( ) t R v ( ) r ( ) r q 2 v v v q v R R c v c R c = + E ( , ) r R R R 1 t 3 2 2 c R c R v v R 2 v R R v / q c v c c = + B ( , ) r 1 t 3 2 2 2 c v R R R c c R E r ( , ) R t = B ( , ) r . t 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 26

  27. Electric and magnetic fields far from source: v v q v R ( ) = E r R R , t 3 2 c c R R c , ( ) R E r t ( ) = B r , t R R v v R Let R c c ( ) q ( ) = E r R R , t ( 1 ) 3 R cR R ( ) ( ) t , = B r E r , t 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 27

  28. Poynting vector: ( , r S t c ) ( ) = E B 4 ( ) q ( ) = E r R R , t ( 1 ) 3 R cR ( ) ( ) = B r R E r , , t t ( ) 2 R R 2 c q ( ) ( ) 2 = = S r R E r R , , t t ( 1 ) 6 2 4 4 cR R ( ) 2 R R 2 dP d q = S R = 2 R ( ) 6 4 c R 1 04/21/2017 PHY 712 Spring 2017 -- Lecture 36 28

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