Quiz 2 - Basic Equations of Electrodynamics Mathematical Identities Electromagnetic Variables Maxwell's Equations, Force Constants v(t) = Re{V 4/13/09 ejωt}where V = |V|ejφ E = electric field (V/m) ∇×=EB−∂/∂t εo = 8.85×10-12 F/m ∇ = xˆ∂/∂x + yˆ∂/∂y + zˆ∂/∂z H = magnetic field (A/m) v∫∫dEd⋅=sˆˆ− B⋅nda μo = 4π×10-7 H/m CSdtAB⋅=AxxB +AyBy+AzBz D = electric displacement (C/m2) ∇×=HJ+∂D/∂t c =1/εooμ ≅ 3×108m/s ∇2φ = (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2)φ B = magnetic flux density (T) v∫∫dH ⋅=ds J ⋅ndaˆˆ+∫D ⋅nda h = 6.624×10-34 Js CSdtSsin2θ + cos2θ = 1 Tesla (T) = Weber/m2 = 104 gauss ∇⋅D =ρ→v∫∫D ⋅ndaˆ= ρdv e = 1.60×10-19 [C] sv3∇⋅(A∇× )=0 ρ = charge density (C/m ) ∇⋅=B 0 →v∫B⋅ndaˆ=0 kSB = 1.38×10-23 J/K ∇×(A∇× )=∇(∇⋅A)−∇2A J = current density (A/m2) ∇ ·⎯J = -∂ρ/∂t ηo ≅ 377ohms =μoo/ ε ∫∫(G∇⋅ )dv=vG⋅nˆda σ = conductivity (Siemens/m) ⎯f = q(⎯E +⎯v × μo⎯H) [N] me = 9.1066×10-31 kg VS∫∫(∇×G) ⋅ndaˆˆ= G ⋅dsSCv Js = surface current density (A/m) Waves Media ejωt = cos ωt + j sin ωt ρs = surface charge density (C/m2) (t∇22−με∂ ∂2)E=0 DE= ε=εoE+P cosα + cosβ = 2 cos[(α+β)/2] cos[(α-β)/2] (∇2 + k2)⎯E = 0, EE=jk roe− ⋅ ∇ ⋅=D ρf ∫+∞H(ω=) h(t)e−ωjtdt Boundary Conditions −∞ωπ2kk=ω με= = =222xy+k+kz c λ∇⋅εofE =ρ +ρp ex = 1 + x + x2/2! + x3/3! + … n(ˆ× E12−=E) 0 vp = ω/k, vg = ∂ω/∂k DE= ε=, JσE sinα = (ejα – e-jα)/2j n(ˆ× H12−=H) Js Ex(z,t) = E+(z-ct) + E-(z+ct) [or (ωt-kz) or (t-z/c)] BH=μ =μo(H+M) cosα = (ejα + e-jα)/2 n(ˆ⋅ B12−=B) 0 Hy(z,t) = (1/ηo)[E+(z-ct) - E ε= (122-(z+ct)] εop−ω/ω ) n(ˆ⋅ D12−=D) ρs Ex(z,t) = Re{Ejx(z)eωt} ω Ne2p=ε/mo ⎯E =⎯H = 0 if σ = ∞ 1<×*EH>=Re{E×H} ε2eff = ε(1 – jσ/ωε) d1v221∫∫(E × H) ⋅=ndaˆ− ( εE +μH )dv −∫E ⋅Jdv Δ =σ2/( η) svdt 2 2v δ = 2/ωμσPlanar Interfaces Quasistatics Circuit Elements Electromagnetic Forces θ=irθ 4/13/09 E = −∇Φ QC = Vfq=+(Ev×μoH) [N] sin θtknεμ==iiii sin θitkεμttnt∇Φ2ρ=− εoΛL = IFI= ×μoH [N/m] θ=1ctsin−(n / ni) Φ=(r)∫{ρ(r ')/4πεr'−r }dv' v'dv(t)i(t) = C dtEeo=−v×μ H (inside conductor) ⎛⎞ta1nθ=tBn−⎜⎟ ⎝⎠niμ=oHA∇× di(t)v(t) = L dtdw dzvi =+f dt dt1T+Γ= ∇=2AJ−μo Λ =⋅∫Bdaˆ (per turn)⋅N Adwf|ex=−Q=const. dxTE TEΓ=TM TMTE(Zn−1) (Zn+1) TMA(r) =μ∫{oJ(r ')/ 4πr '−r }dv' v'1w(2et)= Cv(t) 2dwf|mx=−Λ=const. dxηθZTEticosn= TEM Transients ηθitcos1w(2mt)= Li(t) 2Tr= × f Pe=μH2/2, εE2/2 [N/m2] ηθZTMttcosn= ηθiicosdv(z, t) di(z, t)=−L dz dtLτ =τRC, = RdwTθΛ=−Q or =const dθkk=+'jk" dv22(z,t) dv(z,t)= LC TEM Sinusoidal Steady State RLC Resonators dz22dt2P2ds≅σJ /2 δ [W/m ] zzv(z, t) = f+−(t −+) f (t +) ccV(z) =+V e−+jkzV ejkz+ − Zseries = R + jωL +1/jωC Waveguides ⎛⎞zzi(z,t) =−Yo⎜⎟f+−(t ) −f (t +) ⎝⎠ccI(z) =−Yjkz jkzo(V e−++−V e ) Yparallel = G + jωC + 1/jωL EyˆjkzzTE=⋅Eosinkxxe− c1==/LC1/με k2= π/λ=ω/c=ω με ωoTw ωω=oo1/ LC , Q = = PdissΔωEyˆzTE=⋅Eosinkxxe−α ZLo= /C Z(z) = V(z) / I(z) =⋅ZonZ (z) EM Resonators kk222 2xz+=ko=ωμε RZ−LoR1Ln−Γ=L= RZLo+ RLn+1Γ(z) ==(V / V )e2jkz−+(Znn(z) −1) /(Z (z) +1) At ,ωoe 〈〉w =〈wm〉 1 λ=22gz1 λ=1/λo−1/λx v2Th= f+(t) , RTh= Zo Znn(z) =[1+Γ(z)]/[1−Γ(z)] =R +jXn ∫2〈〉we=(εE /4)dv Vωωdv,pg== v kdkZ(z) =ZoL⋅−(Z jZotankz)/(Zo−jZLtankz) ∫2〈〉wm=(μH /4)dv V 1+ΓVSWR ==Vmax/ Vmin=R n max1−Γcf222mnp= (m/a) +(n/b) +(p/d) 2 ωQ,owTω ω 111=== =+ P2dissαΔω QLQEQIMIT OpenCourseWare http://ocw.mit.edu 6.013 Electromagnetics and Applications Spring 2009 For information about citing these materials or our Terms of Use, visit:
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