Electrodynamics Subject Exam Prep Quiz, Tuesday, May 2, 2017

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1 Electodynamics ubject Exam Pep Quiz, Tuesday, May, 017 a ( b c ) b( a c ) c ( a b), a ( b c ) b ( c a ) c ( a b), ( a b) ( c d) ( a c )( b d) ( a d)( b c ), ( ψ) 0, ( a ) 0, ( a ) ( a ) a, (ψ a) a ψ + ψ a, (ψ a) ψ a + ψ a, ( a b) ( a ) b +( b ) a + a ( b)+ b ( a ), ( a b) b ( a ) a ( b), ( a b) a ( b) b( a )+( b ) a ( a ) b, 3, 0, ˆ /, ˆ 0, ( a )ˆ 1 [ a ˆ( a ˆ)] a d 3 A d 3 ψ d 3 A d 3 (φ ψ + φ ψ) d 3 (φ ψ ψ φ) ( A) d (d) ψ C d A, ψd, d A, φ(d ) ψ, (φ ψ ψ φ) d, d l A, d lψ + ρ + 1 ρ ρ m, ( ) 1 δ( ) l(l + 1),

2 L α β γv γ 0 0 γ γv Fαβ 0 Ex Ey Ez Ex 0 Bz By Ey Bz 0 Bx Ez By Bx 0, F αβ 0 Ex Ey Ez Ex 0 Bz By Ey Bz 0 Bx Ez By Bx 0 m d dτ uα ef αβ uβ, d p dt e E + e v B, ωc eb γm E J 0, ( B) t E J, B 0, t B + E 0, αf αβ J β, α F αβ 0 e c , T αβ π α β φ g αβ L, π α L ( αφ) T π (E + B ), T 0i 1 ϵ ijkejbk, T ij T i j 1 8π (δ ij(e + B ) EiEj BiBj) Pl(cos θ) E A 0 t A Φ t A, B A, Y0,0 Y1,±1 Y,±1 l +1 Y lm0(θ), 1, Y1,0 3 cos θ, 3 8π sin θei±φ, Y,0 8π sin θ cos θe±iφ, Y,± 16π (3 cos θ 1), 3π sin θe ±iφ, Yl m(θ, φ) ( 1) m Y lm(θ, φ), δll δ mm dω Yl,m(θ, φ)yl,m (θ, φ), P0(x) 1, P1(x) x, P(x) 1 (3x 1), P3(x) 1 (x3 3x), Pl(x 1) 1, (3D) Φ lm 1 (D) Φ A0 ln(ρ)+ m Φ A0J0 m Φ q E 1 Φ(, θ, φ) lm q q1 q0 Qij dx Pl(x)Pl (x) 1 l +1 δ ll, ( A lm l + Blm l 1) Ylm(θ, φ)e imφ, p m 3π 3 p +3 p 8π 16π e imφ ( Am m + Bm m), e imφ (AmJm(kρ)+BmNm(kρ)) e ±kz, 3 q m()ym(θ, φ), +, (l + 1) l+1 q lm()ylm(θ, φ), d 3 ρ( )(x iy) d 3 ρ( )(x iy) d 3 ρ( )(3z ) 16π Q 33, d 3 (3ij δij)ρ( ), 88π (Q 11 iq1 Q), 7π (Q 13 iq3), U qφ0 p E 1 6 Q ij iej,

3 A α J α, m 1 d 3 J I B m ( m ), e µe ge me, d l, U ( µ N µ e) 3 3( µ N )( µ e ) 8π 3 ( µ N µ e)δ 3 ( ) e ( µ N L) m 3, T00 1 ( E 8π + B ) a i + b i 8π T0i ϵijk ˆki EjBk a cos ( k ωt), a cos ( k ωt), T ij T i j 1 8π (δ ij(e + B ) EiEj BiBj) 1 { } a δ ij aiaj bibj cos ( k ωt) ωs ω 1 v 1+v (TM) Ez ψ(x, y)e iωt+ikzz, Ψ 0 at bounday, E t(x, y) B t(x, y) ikz (ω k z) e iωt+ik zz tψ(x, y), ( ) ω ẑ Et, kz (TE) Bz ψ(x, y)e iωt+ik zz, (ˆn t)ψ(x, y) 0, B t(x, y) E t(x, y) ikz (ω k z) e iωt+ikzz tψ(x, y), ( ) ω kz ẑ Bt A α (x) E e { B ˆn E d 4 x 1 x x J α (x )δ(x0 x 0 x x ), ˆn [(ˆn β) β] (1 β ˆn) 3 x }, P e 3c β (NonRel), dω e (1 β ˆn) 6 (ˆn β) β), P 3c e γ 6 [ β β β ], dω e (1 β cos θ) β sin θ (linea), P e β 3c γ 6 (linea), dω e (1 βnβ) β ( (1 βnβ) (1 β )n ) (cicula), P 3c e β γ 4 (cicula) dω 1 8π k ˆn p, P ω4 3 p, (Thomson) σ 8πe4 3m, λ ω λ m (1 cos θ s) electon ± muon ± poton ± neuton ±

4 LONG ANWER ECTION 1 A thin wie extends fom x, y, z (0, 0, a) to(0, 0,a) The cuent on the wie is wave-like I I 0 cos(kz) cos ω 0 t, k π/a (a) ( pts) Find the chage density on the wie as a function of time (b) ( pts) What is the aveage powe adiated by the wie (c) ( pts) What is the angula distibution of the adiated powe? a) TIE - zi( a H Rios in kzws not a in kz s in wot IPOs by p w FI, ft Powe a) date fz XCZ ) da twlotfgtzsinkz Isoz z No 41 ayes - taut Cnn PT + ' we at 16 Io out a- 16 since bained in '

5 Exta wokspace fo #1

6 a) E Ex A squae wave guide has tansvese dimensions 0 <x<a,0 <y<a Fo a tansvese magnetic wave (TM), + s (a) (10 pts) Fo a wave that popagates in the +z diection with fequency ω, find a solution fo the electic and magnetic fields, whee the maximum electic field stength is E 0 Choose the solution with the fewest tansvese nodes (b) ( pts) What is the lowest value of ω that leads to a popagating solution? And what is longitudinal velocity of the wave in that limit? III Eo with e- b) bwetw z - :# KI i " + + i s in tat ( s in s in s intza tzyz ) e- iattihtte Eas # sn#e-iwttiteo Is w E) KI in # that) K as #? eiottokst to

7 , ' HORT ANWER ECTION 3 A linea acceleato acceleates eithe electons o potons with an electic field of stength E 0 along the z diection In tems of the atio of masses M p /M e, (a) (4 pts) Find the atio of adiative powes, P p /P e, at the beginning of the acceleation cm#zdfh& when both paticles would be moving non-elativistically (b) (4 pts) Find the atio of adiative powes, P p /P e, at the end of the acceleation when both paticles would be moving ulta-elativistically with enegy K seaman ;ke IEF% 4 (4 pts) Two gases of fully ionized paticles ae stoed in identical containes with identical numbes of ions (and electons) In gas A, the gas is fully ionized hydogen, whee all the positive ions ae potons In gas B, the positive ions ae deuteons (one neuton and one poton), which have appoximately twice the mass of a poton The numbe of fee electons in each containe is thus equal The mean fee paths, l A and l B, of light taveling though the containes vay by appoximately what facto? Cicle the coect answe: l A /l B (a) 1 16 (b) 1 8 (c) 1 4 (d) 1 (e) 1 (f) (g) 4 (h) 8 (i) 16 scatteing dominated 0 by electons!

DO NOT WRITE YOUR NAME OR STUDENT NUMBER ON ANY SHEET!

DO NOT WRITE YOUR NAME OR STUDENT NUMBER ON ANY SHEET! tatistical Mechanics ubject Exam, Wednesday, May 3, 2017 DO NOT WRITE YOUR NAME OR TUDENT NUMBER ON ANY HEET! a ( b c) = b( a c) c( a b), a ( b c) = b ( c a) = c ( a b), ( a b) ( c d) = ( a c)( b d) (