# c2 - Forces dQ dU V dx dx 0 E E U= dV 2 1 Ucap = CV 2 2...

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Forces F x = - dU dx + V dQ dx U = ε 0 E · E 2 dV U cap = 1 2 CV 2 Electromagnetic Power-Energy Continuity Equation S ( r, t ) = E ( r, t ) × H ( r, t ) W ( r, t ) = μ 0 H ( r, t ) · H ( r, t ) 2 m-field en. density + εE ( r, t ) · E ( r, t ) 2 e-field en. density -∇ S ( r, t ) = ∂W ( r, t ) ∂t + J ( r, t ) · E ( r, t ) - S ( r, t ) = ∂t W ( r, t ) dV + J ( r, t ) · E ( r, t ) dV { net power flow } = rate of increase of total energy in closed volume + rate of total energy loss thru diss. Electromagnetic Waves f = λv, c = 1 / ε 0 μ 0 3 × 10 8 m/s, η 0 = μ 0 0 377Ω Time Harmonic Fields E ( r, t ) = ˆ nE 0 cos( ωt - k · r ) H ( r, t ) = ( k × ˆ n ) E 0 η 0 cos( ωt - k · r ) k · ˆ n = 0 , k 2 = k · k, ω = kc E ( r, t ) = E ( r ) e jωt , E ( r ) = ˆ nE 0 e - jk · r Maxwell’s Equations (Complex Phasor Notation) (Gauss, Gauss, Faraday, Ampere) ∇ · εE ( r ) = ρ ( r ) (1) ∇ · μ 0 H ( r ) = 0 (2) ∇ × E = - jωμ 0 H ( r ) (3) ∇ × H = J ( r ) + jωεE ( r ) (4) Complex Poynting Vector E ( r ) = ˆ nE 0 e - jk · r H ( r ) = ˆ k × ˆ n E 0 η 0 e - jk · r S ( r ) = E ( r ) × H * ( r ) S ( r ) = 1 2 S ( r ) = 1 2 ˆ n × ˆ k × ˆ n * E 2 0 η 0 Wave Propagation in Dielectric Medium n = ε/ε 0 k = ω μ 0 ε = ω n c = 2 π λ λ = 2 πc ωn η = μ 0 ε Wave Propagation in Conductive Medium ε eff ( ω ) = ε 1 - j σ ωε n eff ( ω ) = ε eff ( ω ) ε 0 = ε ε 0 1 - j σ ωε k = ω μ 0 ε 0 ε eff ( ω ) ε 0 = ω n eff ( ω ) c k = k - jk η eff ( ω ) = μ 0 ε eff ( ω ) 1 η eff ( ω ) = k ωμ 0 = k - jk ωμ 0 S ( r, t ) = ˆ z E 2 0 2 k ωμ 0 e - 2 k z W.P. in Cond. Medium (Lossy Dielectric) σ ωε 1 n eff ε ε 0 1 - j σ 2 ωε k = k - k λ = 2 π k k = σ 2 μ 0 ε E ( r ) = ˆ xE 0 e - jkz = ˆ xE 0 e - jk z e - k z W.P. in Cond. Medium (Non-perfect Metals) σ ωε 1 n eff ε ε 0 - j σ 2 ωε = σ 2 ωε 0 (1 - j ) k = k - jk = σωμ 0 2 (1 - j ) = 1 δ (1 - j ) k = σωμ 0 2 = 1 δ λ = 2 π k = 2 πδ E ( r ) = ˆ xE 0 e - jkz = ˆ xE 0 e - jk z e - z δ

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