Waveguides and Boundary Conditions in Electrodynamics

phy 712 electrodynamics 9 9 50 am olin 105 n.w
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Delve into the principles of waveguides, TEM, TE, and TM modes, as well as the justification for boundary conditions and the behavior of waves near conducting surfaces in the context of electromagnetism. Explore Maxwell's equations without sources, analyzing the solutions for both electric and magnetic fields under different wave equations. Understand the complex interplay of variables such as n and k in wave solutions while considering real values for n and k.

  • Electrodynamics
  • Waveguides
  • Boundary Conditions
  • Maxwells Equations
  • Electromagnetism
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  1. PHY 712 Electrodynamics 9-9:50 AM Olin 105 Plan for Lecture 19: Chap. 8 in Jackson Wave Guides 1. TEM, TE, and TM modes 2. Justification for boundary conditions; behavior of waves near conducting surfaces 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 1

  2. 4 PM in Olin 101 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 2

  3. 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 3

  4. 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 4

  5. = = D E B H For linear isotropic media and no sources: Coulomb's law: ; = E 0 E = B Ampere-Maxwell's law: 0 t 0 B + = E Faraday's law: 0 t = B No magnetic monopoles: 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 5

  6. Analysis of Maxwells equations without sources -- continued: : law s Coulomb' E = 0 E t = B Ampere - Maxwell' law s : 0 B t + = E Faraday' law s : 0 = B magnetic No monopoles : 0 ( ) t E E = 2 B B t 2 B = + = 2 B 0 2 t ) ( t B B + = + 2 E E t 2 E = + = 2 E 0 2 t 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 6

  7. Analysis of Maxwells equations without sources -- continued: Both E and B fields are solutions to a wave equation: 2 B 1 v = 2 B 0 2 2 t 2 E 1 v = 2 E 0 2 2 t 2 c 2 2 0 0 where v c wave to 2 n equation Plane wave solutions : ( ) ( ) ( ) ( ) = = k r k r i i t i i t B r B E r E , , t e t e 0 0 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 7

  8. Analysis of Maxwells equations without sources -- continued: wave to solutions wave Plane = e t B r B equation = t r : ( ) ( ) ( ) ( ) i i k r k r i t i t E E , , e 0 0 2 2 v c n 2 = = k where n 0 0 Note: , n, k can all be complex; for the moment we will assume that they are all real (no dissipation). E B Note that and independen not are t; 0 0 B t + = E from Faraday' law s : 0 For real , n, k k c k E E n = = B 0 0 0 k k = = E B also note : and 0 0 0 0 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 8

  9. Analysis of Maxwells equations without sources -- continued: electromag plane of Summary c v netic waves : k c ( ) E n ( ) ( ) i i = = k r k r i t i t B r E r E 0 , , t e t e 0 2 2 n k 2 = = = k E where and 0 n 0 0 0 Poynting vector and energy density: 2 E n 1 2 k k 2 = = 0 c S E 0 2 avg E0 1 2 2 = E u 0 avg B0 k 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 9

  10. Transverse electric and magnetic waves (TEM) , t e c = = k E n ( ) ( ) ( ) k r i t k r i t = = i i B r E r E , 0 t e 0 2 2 n k E 2 = k where and 0 n 0 v c 0 0 TEM modes describe electromagnetic waves in lossless media and vacuum E0 B0 k For real , n, k 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 10

  11. Effects of complex dielectric; fields near the surface on an ideal conductor medium isotropic an for Suppose = E = D and E J E : b : H Maxwell' equations s in terms of = = E H 0 0 H E = = + E H E b t t 2 Plane : = = 2 F F E H 0 , b 2 t t E wave form for ( ) ( ) ( ) k = = + k r i i t E r E k , whe re t e n in 0 R I c ( ) ( ) ( ) k k r = r / in c i t / E r E , t e e R 0 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 11

  12. Some details: E Plane wave form for : ( ) ( ) ( ) k k r i t = = + i E r E k , where t e n in 0 R I c 2 = 2 E 0 b 2 t t 2 c ( ) 2 + + + = 2 0 n in i c R I b 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 12

  13. Fields near the surface on an ideal conductor -- continued : system our For / 1 2 2 2 c = + + 1 1 b n R b / 1 2 2 2 c = + 1 1 b n I b c c e 2 ) i 1 For 1 n n R I ( ( ) k r k r i t = / / i E r E , t e 0 + 1 n ( ) ( ) ( ) k E r k E r = = H r , , , t t t c 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 13

  14. Some representative values of skin depth Ref: Lorrain2 and Corson 1 n n R I 2 c c ( ( S/m) ) ( (0.001m) ) at 60 Hz 10.9 8.5 1.0 0.4 15.1 ( (0.001m) ) at 1 MHz 84.6 66.1 10.0 3.0 117 Al Cu Fe Mumetal Zn 3.54 5.80 1.00 0.16 1.86 1 1 100 2000 1 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 14

  15. Relative energies associated with field Electric energy density: 2 E b 2 H Magnetic energy density: 2 E H 2 2 = = b Ratio inside conducting media: b + b 2 2 2 1 i 2 2 =2 b 2 0 0 2 E H b For 1 magnetic energy dominates 2 2 E H = 0 Note that in free space, 1 2 0 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 15

  16. Fields near the surface on an ideal conductor -- continued 1 For 1 n n R I 2 c c 1 c ( ) i = = + = + In this limit, 1 c n in R I 0 0 ( ) ( ) k r k r i t = / / i E r E , t e e 0 r|| + 1 n i ( ) ( ) ( ) k E r k E r = = H r , , , t t t c 0 z 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 16

  17. Fields near the surface on an ideal conductor -- continued ( ) ( ) k r k r i t = / / i E r E , t e e r|| 0 + 1 n i ( ) ( ) ( ) k E r k E r = = H r , , , t t t c 0 z Note that the field is larger than field so we can write: H E ( i ) ( ) k r k r i t = / / i H r H , t e e 0 1 ( ) ( ) k H r = E r , , t t 2 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 17

  18. Boundary values for ideal conductor At the boundary of an ideal conductor, the E and H fields decay in the direction normal to the interface. conductor the Inside : ( i ) ( ( ) ) k k = r r / / i i t H r H , t e e 0 1 ( ) t , k = E r H r , t 2 Ideal conductor boundary condit ions: k H0 = = n E n H 0 0 S S n 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 18

  19. Wave guides dielectric media with one or more metal boundary Ideal conductor boundary condit ions: k H0 = = n E n H 0 0 S S n Waveguide terminology TEM: transverse electric and magnetic (both E and H fields are perpendicular to wave propagation direction) TM: transverse magnetic (H field is perpendicular to wave propagation direction) TE: transverse electric (E field is perpendicular to wave propagation direction) 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 19

  20. Analysis of rectangular waveguide Boundary conditions at surface of waveguide: Etangential=0, Bnormal=0 y x z Cross section view b a 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 20

  21. Analysis of rectangular waveguide y x z + ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) = + ikz i t B x y z , , , B x y B x y B x y e x y z = + + ikz i t E x y z , , , E x y E x y E x y e x y z Inside the dielectric medium: (assume to be real) 0 = + = E B 0 B E = E B =0 0 t t 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 21

  22. Solution of Maxwells equations within the pipe: Combining Faraday's Law and Ampere's Law, we find that each field component must satisfy a two-dimensional Helmholz equation: + + 2 2 = 2 2 ( , ) 0. k E x y x 2 2 x y For the rectangular wave guide discussed in Section 8.4 of your text a solution for a TE mode can hav e: m x a n y b = ( , ) E x y 0 and ( , ) B x y cos cos , B 0 z z 2 2 m n = + 2 2 mn 2 wit h k k a b 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 22

  23. Maxwells equations within the pipe in terms of all 6 components: B B ikB x y E E ikE x y E ikE i B y E ikE i B x E E i B x y 02/27/2019 For TE mode with k B E 0 E y z + + = 0. x z = x y k = B E y + + = 0. x y x z B y = = . z . ikB i E z y x y x B x B y = = . z . ikB i E z x y x y B x y y = = . x . x i E z z PHY 712 Spring 2019 -- Lecture 19 23

  24. TE modes for rectangular wave guide continued: m x a i n y b n b = ( , ) E x y 0 and ( , ) B x y cos cos , B 0 z z B y i m x a n y b = = = cos sin , E B B z 0 x y 2 2 2 2 k k m n + a b B x i i m m x a n y b = = = sin cos . E B B z 0 y x 2 2 2 2 k k a m n + a b Check boundary conditions: = E = = 0 because: ( ,0) E x ( , ) E x b 0 = tangential and = B x x = (0, ) ( , ) E a y 0 . E y y y 0 normal 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 24

  25. ( ) y Solution for m=n=1 , Bz x m x n y ( ) = , cos cos B x y B 0 z a b / n 2 b m x n y ( ) = , cos sin iE x y B ( ) a ( ) b 0 x 2 + a b m n / m 2 a m x n y ( ) = , sin cos iE x y B ( ) a ) y ( ) b 0 y 2 + a b m n ( ) y , iEy x ( , iEx x 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 25

  26. Solution for m=n=1 2 2 m n = + 2 2 mn 2 k k a b k 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 26

  27. Resonant cavity 0 x a 0 y b y 0 z d x z cos ( ( ) ) ( ( ( ( : ) ) , ( ( ) ) y ( ( ) ) = + + i t B x y z , , , , , , B x y z B x y z B x y z e x y z = + + i t E x y z , , , , , , E x y z E x y z E x y z e x y = z ( ( ) ) ( ( ) ) ( ) ( ) kz ) ) ( ) ( ) kz general In , , sin or , E x y z E x kz E x y kz i i i = , , , sin or , cos B x y z B x y B x y i i i p = 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 k d 27 27

  28. Resonant cavity 0 x a 0 y b y 0 z d x z 2 2 2 p m n = = 2 2 k d a b 2 2 2 1 m n p = + + 2 a b d 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 28

  29. Wave guides dielectric media with one or more metal boundary Coaxial cable TEM modes Simple optical pipe TE or TM modes k k E H Waveguide terminology TEM: transverse electric and magnetic (both E and H fields are perpendicular to wave propagation direction) TM: transverse magnetic (H field is perpendicular to wave propagation direction) TE: transverse electric (E field is perpendicular to wave propagation direction) 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 29

  30. Wave guides Top view: z Inside medium, assumed to be real a b Coaxial cable TEM modes (following problem 8.2 in Jackson s text) Maxwell' B 02/27/2019 equations s inside medium B for : a b = = E B i E 0 i = E = 0 PHY 712 Spring 2019 -- Lecture 19 30

  31. Electromagnetic waves in a coaxial cable -- continued Example Top view: Find = solution for t a b : E a i = ikz E 0 e B k B a = 0 E i = ikz t B 0 e a 0 + b = x y cos sin = cos + x y sin medium cable thin Poynting vector wi (with , : ) 2 2 B ( ) 1 a 0 = = * S E B z 2 avg 2 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 31

  32. Electromagnetic waves in a coaxial cable -- continued Top view: Time averaged power cable in material : ( ) 2 2 2 b B a b a 0 0 = S z ln d d avg a b a 02/27/2019 PHY 712 Spring 2019 -- Lecture 19 32

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