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Plane Electromagnetic Waves Plane Electromagnetic Waves EE142 Dr. Ray Kwok •reference: Fundamentals of Engineering Electromagnetics , David K. Cheng (Addison-Wesley) Electromagnetics for Engineers, Fawwaz T. Ulaby (Prentice Hall) Plane EM Wave - Dr. Ray Kwok Source-free Maxwell Equations r r ε∇ ⋅E = ρf r in source-free medium ∇ ⋅ E = 0 r r ∂H (homogeneous, r ∂H ∇× E = −µ linear, isotropic) ∇× E = −µ r ∂t ∂t ρ = 0, J = 0 r ∇ ⋅H = 0 r ∇ ⋅ H = 0 r r r ∂E r ∇× H = J + ε ∂E f ∇× H = ε ∂t ∂t Plane EM Wave - Dr. Ray Kwok Wave Equationr r ∂H ∇ × E = −µ r∂t r ∂E ∇ × H = ε ∂t r r r r 2 ∂H ∂ ∂ E ∇ × ()∇ × E = ∇×− µ = −µ ()∇× H = −µε 2 ∂t ∂t ∂t r r r r ∇ × ()()∇ × E = ∇ ∇ ⋅ E − ∇ 2E = −∇ 2E r r 2 2 ∂ E ∇ E = µε 2 ∂t 2 plane wave equation with v = 1/ µε Plane EM Wave - Dr. Ray Kwok Traveling Wave x(f ± vt ) ≡ )u(f reverse / forward traveling wave ∂f ∂u = )u('f = )u('f ∂x ∂x ∂ 2f ∂u note: = )u("f = )u("f 1 2π 2π ∂x 2 ∂x x ± vt = x ± vt k λ λ ∂f ∂u = )u('f = ±vf )u(' 1 = ()kx ± 2πft ∂t ∂t k ∂ 2f ∂u 1 = ±vf )u(" = v2 )u("f = ± ()ωt ± kx ∂t 2 ∂t k 2 2 x(f ± vt ) = f ()ωt ± kx ∂ f 1 ∂ f r r = wave equation f ()ωt − k ⋅ r ∂x 2 v2 ∂t 2 (3D) Plane EM Wave - Dr. Ray Kwok Plane Wave r r 2 2 ∂ E ∇ E = µε Similarly for B or H field ∂t 2 r r r r r r (j ωt−k⋅ )r r 2 )t,r(E = Eoe 2 ∂ H r r ∇ H = µε 2 2 2 ∂t ∇ E = −k E r r r r r r H )t,r( = H e (j ωt−k⋅ )r ∂ 2E r o = −ω2E ∂t 2 − k 2 = −ω2µε k = ω µε ω 1 = v = k µε Plane EM Wave - Dr. Ray Kwok Source-free EM wave r r (j ωt−kz ) )t,r(E = xˆE e V/m in vacuum with no current source. o r What is H ? (j ωt−kz ) ∇× E = yˆ{− jkE oe } r r kE xˆ yˆ zˆ H )t,r( = yˆ o e (j ωt−kz ) r ∂ ∂ ∂ ωµ ∇ × E = k 1 1 µε ε 1 ∂x ∂y ∂z = = = = ≡ (j ωt−kz ) ωµ fλµ vµ µ µ η Eoe 0 0 r r r r ηηη = wave Eo (j ωt−kz ) ∇ × E = −jωµ H H )t,r( = yˆ e Impedance r η (j ωt −kz ) ∇ × E = yˆ{}− jkE oe r r µE B = µH = yˆ o e (j ωt−kz ) r r kE vµ H )t,r( = yˆ o e (j ωt−kz ) ωµ r r E )t,r(B = yˆ o e (j ωt−kz ) v Plane EM Wave - Dr. Ray Kwok EM wave properties r (1) E & H are in phase. r (j ωt−kz ) )t,r(E = xˆEoe r r E r r r H )t,r( = yˆ o e (j ωt−kz ) (2) E ⊥ H ⊥ k η kˆ = Eˆ × Hˆ r r E B )t,r( = yˆ o e (j ωt−kz ) v Index of refraction (3) E c c o = v = = Bo n εrµr E µ o = η = Ho ε Plane EM Wave - Dr. Ray Kwok Electromagnetic wave Plane EM Wave - Dr. Ray Kwok Earlier exercise r r H )t,r( = xˆ 01.0 cos( 900 t + β )z A/m in vacuum with no current source. What is f, λ, direction of propagation, E ? ω = 2 πf = 900, f = 143 Hz (1) E & H are in phase. λ = c/f = (3 x 10 8)/(143) = 2094 km r r r propagate to the –z direction (2) E ⊥ H ⊥ k r r kˆ = Eˆ × Hˆ ∇× H = jωε E E E(r) = a 3.77 cos[900t + 3(10 -6)z] V/m o = v y (3) Bo E o = η Ho ηo = 377 Ω Plane EM Wave - Dr. Ray Kwok Example r r H )t,r( = (xˆ 01.0 − ˆ 02.0y )sin( 10 9 t +βz − )1.0 A/m. What is E ? β = ω/c = 10 9/(3 x 10 8) = 3.33 rad/m r r r (1) E & H are in phase. 9 V/m )t,r(E = Eo sin(10 t + 33.3 z− )1.0 r r r y (2) E ⊥ H ⊥ k E ˆ ˆ ˆ 2xˆ + yˆ k = E× H x Eˆ = = .0 894 xˆ + .0 447 yˆ 5 H Eo (3) = v 2 2 2 → Ho = 0.01 + 0.02 Ho = 0.022 = Eo/ηo = E o/377 Bo Eo = 8.3 V/m E o = η r r 9 Ho )t,r(E = (3.8 xˆ .0 894 + yˆ .0 447 )sin( 10 t + 33.3 z − )1.0 η = 377 Ω r r o )t,r(E = (xˆ 4.7 + yˆ )7.3 sin( 10 9 t + 33.3 z − )1.0 V/m Plane EM Wave - Dr. Ray Kwok Exercise r r 8 )t,r(E = xˆ 20 cos( 10 t + z3 − )y V/m in non-magnetic material. What is H ? r r r (1) E & H are in phase. 8 H )t,r( = Ho cos( 10 t + z3 − )y A/m r r r y r (2) E ⊥ H ⊥ k k k = −3zˆ + yˆ ˆ ˆ ˆ E − 3yˆ − zˆ k = E× H z Hˆ = = − .0 949 yˆ − .0 316 zˆ 10 H v = ω/k = 10 8/√10 = 3.16 x 10 7 m/s Eo √ → 2 (3) = v v = c/n = c/ εr εr = (c/v) = 90 √ √ √ Bo η = (µ/ε ) = ηo/ εr = 377/ 90 = 39.7 E H = E /η = 20/39.7 =0.5 o = η o o Ho r r 8 η = 377 Ω H )t,r( = (5.0 −yˆ .0 949 − ˆ .0z 316 ) cos( 10 t + z3 − )y o r r H )t,r( = −(yˆ .0 475 + ˆ .0z 158 )cos( 10 8 t + z3 − )y A/m Plane EM Wave - Dr. Ray Kwok EM wave properties - generalized (1) E & H are in phase. r 1 r r r r H = kˆ × E (2) E ⊥ H ⊥ k η kˆ = Eˆ × Hˆ Index of refraction r r (3) E c c E = ηH × kˆ o = v = = Bo n εrµr E µ o = η = Ho ε Plane EM Wave - Dr. Ray Kwok Earlier exercise (free space) r r H )t,r( =0(xˆ . 01−0yˆ . 02)sin( 10 9 t + βz − )1.0 A/m. What is E ? β = ω/c = 10 9/(3 x 10 8) = 3.33 rad/m r r E = ηH× kˆ r E = 377 0(xˆ . 01−0yˆ . 02)sin( ωt + βz − )1.0 ×(−ˆ)z r 9 E =3(yˆ . 77+7xˆ . 54)sin( 10 3t + . 33z − )1.0 V/m. Plane EM Wave - Dr. Ray Kwok Earlier exercise r r 8 )t,r(E = xˆ 20 cos( 10 t + z3 − )y V/m in non-magnetic material. r What is H ? k = −3zˆ + yˆ v = ω/k = 10 8/√10 = 3.16 x 10 7 m/s √ → 2 v = c/n = c/ εr εr = (c/v) = 90 √ √ √ η = (µ/ε ) = ηo/ εr = 377/ 90 = 39.7 r 1 r H = kˆ × E η r 1 − 3zˆ + yˆ H = ×[]xˆ 20 cos ()10 8 t + z3 − y 39 7. 10 r H = ()− yˆ .0 478 − ˆ .0z 159 cos ()10 8 t + z3 − y A/m Plane EM Wave - Dr. Ray Kwok the work-energy theorem Poynting Theorem for electrodynamics Work done on a charge δδδq by EM force: r r r r r ∂H r r r ∂E E ⋅ J = −µH ⋅ − ∇ ⋅ ()E × H − εE ⋅ r r r r r r ∂t ∂t l r r dW = F ⋅ d = δq(E + v × B)⋅ dtv r r 2 2 r r r r µ ∂H ε ∂E E ⋅ J = − − − ∇ ⋅ ()E × H dW = δqE ⋅ vdt 2 ∂t 2 ∂t r r r r r r dW = ρδ V E ⋅ dtv = δVE ⋅ dtJ 2 2 r r () dW 1 ∂H ∂E = −∫ µ + ε dV − ∫∇ ⋅ ()E × H dV dW r r dt 2 ∂t ∂t = ∫ E ⋅ dVJ r r dt 2 2 r r r r dW 1 ∂H ∂E r r = −∫ µ + εo dV − ∫ ()E × H ⋅ ad ∂E dt 2 ∂t ∂t ∇ × H = Jf + ε ∂t r dW ∂U r r r r r r r ≡ − EM − S⋅ ad ∂E dt ∂t ∫ E ⋅ J = E ⋅∇ × H − εE ⋅ Poynting Vector ∂t r r r r r r (math) Power flow out of the enclosed surface ∇ ⋅ ()E × H = H ⋅∇ × E − E ⋅∇ × H EM power stored in the volume r r r r ∂H r r E ⋅ ∇ × H = H ⋅− µ − ∇ ⋅ ()E × H The decrease of the EM energy stored is to do work on ∂t a charge or due to the net energy escape through the surface. Plane EM Wave - Dr. Ray Kwok Poynting Vector r r r S ≡ E × H Magnitude gives power intensity = power/area Direction gives direction of propagation r 1 r r S ≡ ℜ E[e × H*] Time average ave 2 2 E η 2 S = o = H ave 2η 2 o This is similar to P = ½ (V 2/Z) = ½ ( I2Z) Plane EM Wave - Dr. Ray Kwok Example If solar illumination is characterized by a power density of 1 kW/m 2 at Earth’s surface, find (a) the total power radiated by the sun, (b) the total power intercepted by Earth, and (c) the electric field of the power density incident upon Earth’s surface, assuming that all the solar illumination is at a single frequency.
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