Physics 401 Midterm Example Cheat Sheet Maxwell' s Equations in vacuum r r r ρ r r r ∂B ∂E ∇⋅ E = ∇×E = − ∇⋅B = 0 ∇ × B = µ 0 J + µ 0ε 0 ε0 ∂t ∂t r r r r r ∂ρ Lorentz Force Law: F = q E + v × B Conservation of charge: ∇ ⋅ J + =0 ∂t
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General plane wave in linear medium: r r r v r r r r r ˆ r r r 1 ˆ ˆ ˆ ˆ ˆ E = P f [v ⋅ (r − v t )] ; B = × P f [v ⋅ (r − v t )] ; v = ; P ⋅v = 0 ; v = E × B v εµ Harmonic plane wave in linear medium: r r r r r r r r k r E = P exp i k ⋅ r − ωt ; B = × P exp i k ⋅ r − ωt ω r r ω ; r = v ; P⋅k = 0 k
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Energy: Ufield
1 r r r r = E ⋅ D + B⋅ H 2
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Conservation of energy:
r r r E ×B r r Poynting vector: S = = E ×H µ r ∂ (Ufield + Uparticles ) + ∇ ⋅ S = 0 ∂t
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1 1 Maxwell Stress Tensor: Tij = ε Ei E j − δ ij E 2 + µ Hi H j − δij H2 2 2 r r 1 r r pfield Momentum density: = εµS = 2 S = εE × B volume c r r t pparticle ∂ pfield Conservation of momentum: + − ∇⋅ T = 0 ∂t volume volume r r Perfect conductor boundary conditions: E = B = 0 inside σ I Eparallel = 0 Bnormal = 0 Enormal = Bparallal = µ0 ε0 l ∂Enormal =0 ∂( normal ) θ incidence = θ reflection ∂Bparallel =0 ∂( normal ) r ⇒ vnormal changes sign,
r Pparallel changes sign
r r m2 n2 l 2 Rectangular cavity fields: E exp (−iωt ) , B exp( −iωt ) , ω = π c 2 + 2 + 2 a b d x x y z y z Ex = AEX cos mπ sin nπ sin lπ Bx = ABX sin mπ cos nπ cos lπ a b d a b d x x y z y z Ey = AEY sin mπ cos nπ sin lπ By = ABY cos mπ sin nπ cos lπ a b d a b d x y z x y z Ez = AEZ sin mπ sin nπ cos lπ Bz = ABZ cos mπ cos nπ sin lπ b d b d a a r r r mπ nπ lπ AE = ( AEX , AEY , AEZ ) AB = ( ABX ,ABY , ABZ ) M = , , a b d r r r r r r r r r iω r AE ⋅ M = AB ⋅ M = 0 M × AE = −iω AB M × AB = 2 AE c Rectangular waveguide: same xy dependence, zt dependence ⇒ exp[i (kz − ωt )] TE ⇔ Ez = 0 TM ⇔ Bz = 0
2 2 mπ nπ ω= c k + + a b 2
v group =
dω dk
TEM mode: Ez = Bz = 0 , electrostatic & magnetostatic solutions in xy,
r z r ˆ v=c , B= ×E c