Electromagnetic Waves in Wave Guides and Near Ideal Conductors
Explore the propagation of electromagnetic waves near ideal conductors and within rectangular wave guides. Understand Maxwell's equations, field behavior, waveforms, surface conditions, and skin depth values. Analyze the relationship between electric and magnetic energy densities in conducting media.
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PHY 712 Electrodynamics 9-9:50 AM Olin 103 Plan for Lecture 21: Chap. 8 in Jackson Wave Guides 1. Electromagnetic waves near an ideal conductor 2. Electromagnetic waves within an ideal rectangular wave guide 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 1
03/01/2017 PHY 712 Spring 2017 -- Lecture 21 2
03/01/2017 PHY 712 Spring 2017 -- Lecture 21 3
Fields near the surface on an ideal conductor isotropic an for Suppose = E = D and E J E medium : 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 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 4
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 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 5
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 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 6
Some representative values of skin depth Ref: Lorrain2 and Corson ( ( 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 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 7
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 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 8
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 i ( ) ( ) ( ) = = D r E r E r , , , t t t r|| + 1 n i ( ) ( ) ( ) k E r k E r = = H r , , , t t t c + 1 n c i ( ) ( ) ( ) ( ) k E r k E r = = = B r H r 0 z , , , , t t t t 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 9
Fields near the surface on an ideal conductor -- continued ( i t t + = = H r k E r ) ( ) k r k r i t = / / i E r E , t e e 0 ( ) ( ) ( ) = = D r E r E r , , , t r|| 1 n i ( ) ( ) ( ) k E r , , , t t t c + 1 E n c i 0 z ( ) ( ) ( ) ( ) k E r k E r = = = B r H r , , , , t t t t Note that the field is larger than field so we can write: H ( i ) ( ) k r k r i t = / / i H r H , t e e 0 1 ( ) ( ) k H r = E r , , t t 2 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 10
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 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 11
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) 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 12
Analysis of rectangular waveguide Boundary conditions at surface of waveguide: Etangential=0, Bnormal=0 y x z Cross section view b a 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 13
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: 0 = + = E B 0 B E = E B =0 0 t t 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 14
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 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 15
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 03/01/2017 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 2017 -- Lecture 21 16
TE modes for retangular 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 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 17
( ) 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 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 18
Solution for m=n=1 2 2 m n = + 2 2 mn 2 k k a b k 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 19
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 = 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 k d 20 20
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 03/01/2017 PHY 712 Spring 2017 -- Lecture 21 21
