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1 MIT OpenCuseWae hp://cw.m.edu 6.03/ESD.03J Elecmagnecs and Applcans, Fall 005 Please use he fllwng can fma: Makus Zahn, Ech Ippen, and Davd Saeln, 6.03/ESD.03J Elecmagnecs and Applcans, Fall 005. (Massachuses Insue f Technlgy: MIT OpenCuseWae). hp://cw.m.edu (accessed MM DD, YYYY). Lcense: Ceave Cmmns Abun- Nncmmecal-Shae Alke. Ne: Please use he acual dae yu accessed hs maeal n yu can. F me nfman abu cng hese maeals u Tems f Use, vs: hp://cw.m.edu/ems

2 6.03 Fmula Shee aached. Pblem (35 Pns) Massachuses Insue f Technlgy Depamen f Eleccal Engneeng and Cmpue Scence 6.03 Elecmagnecs and Applcans Quz, Ocbe 0, 005 J=J ( ) = J 0 ampees/mee z z z R z 0 < < R φ R R A caxal cable f vey lng lengh caes a z deced cuen densy ha vaes wh adal psn n he nne cylnde as: J=J ( ) = J 0 ampees/mee z z z 0 < < R R A pefecly cnducng ue cylnde f adus R caes all he eun cuen s ha H= 0 f > R. a) Fnd he H feld f 0< < R. b) Wha ae he magnude and decn f he suface cuen densy (ampees/mee) n he = R suface?

3 Pblem (35 Pns) V 0 z α φ ε, σ Deph d 0 R R Tw fla elecdes a angle α exend fm adus R R and have a deph d n he z decn (u f he pape). The elecdes enclse a lssy delecc medum wh pemvy ε and cnducvy σ. Thee s n fee vlume chage whn he lssy delecc. The elecc Φ φ = 0 = 0andΦ φ = α = V. penals n he elecdes ae ( ) ( ) 0 a) The scala elecc penal Φ s f he fm ( φ) Aφ B B sasfy he bunday cndns? Φ = +. Wha values f A and b) Fnd he elecc feld E(, φ) whn he lssy delecc. c) Wha s he fee suface chage densy, σ sf, n he elecde a φ = α? d) Wha s he capacance f hs devce? Yu may neglec fngng felds.

4 Pblem 3 (30 Pns) ε, µ x θ y z E H An elecmagnec wave s avelng a an angle θ wh espec he z axs whn a medum wh delecc pemvy ε and magnec pemeably µ. The magnec feld s gven as: j π 0 8 π ( x+ 3z ) H = H 0 Re e y ampees/mee. a) Fnd he fequency n Hez. b) Fnd he wavelengh n mees. c) Fnd he numecal value f he speed f lgh n he medum n mees/secnd. d) Fnd he angle θ.

5 6.03 Quz Fmula Shee Ocbe 0, 005 Caesan Cdnaes (x,y,z): Ψ = Ψ Ψ xˆ Ψ + ŷ + ẑ x y z A x A y A A = + + z x y z A = x A z A y A + ŷ A z A y A + ẑ z x ˆ x y z x x y Ψ = Ψ + Ψ + Ψ x y z Cylndcal cdnaes (,φ,z): Ψ = ˆ Ψ +φˆ Ψ + ẑ Ψ φ z (A ) + A φ A A = + z φ z ˆ φˆ ẑ A z A φ ˆ A A z (A φ ) A = ˆ φ +φ + ẑ A = de φ z z z φ A A φ A z ( Ψ ) + Ψ + Ψ = Ψ φ z Sphecal cdnaes (,θ,φ): Ψ = ˆ Ψ +θˆ Ψ +φˆ Ψ θ sn θ φ A = ( A ) (sn θa + θ ) + A φ sn θ θ sn θ φ (sn θa φ ) A θ A (A φ ) ˆ ˆ (A θ ) A A = ˆ +θ +φ sn θ θ φ sn θ φ θ = sn de θ ˆ θˆ sn θφˆ θ φ A A θ sn θa φ Ψ ( ) sn θ + Ψ sn θ θ θ sn θ φ Ψ = ( Ψ ) + Gauss Dvegence Theem: Vec Algeba: = xˆ x + ŷ y + ẑ z A B = A B + A y By + A z Bz ( A) = 0 ( A) = ( A) A G dv = V Gnˆ da A x x Skes Theem: ( G ) nˆ da = G d A C

6 Fundamenals f = q ( E + v µ H)[ N] E = B c E ds = d B da d A H = J + D c H ds = A J da + d D da d A Basc Equans f Elecmagnecs and Applcans D =ρ A D da = ρdv V E// H// B D E // H // = 0 B D = ρ s = 0 = Js nˆ 0 = f σ = Elecmagnec Quassacs nˆ E = Φ (), Φ() = V' (ρ( ) 4πε ' )dv' B= 0 B da = 0 A J = ρ E = elecc feld (Vm - ) H = magnec feld (Am - ) D = elecc dsplacemen (Cm - ) B = magnec flux densy (T) Tesla (T) = Webe m - = 0,000 gauss ρ = chage densy (Cm -3 ) J = cuen densy (Am - ) σ = cnducvy (Semens m - ) Js = suface cuen densy (Am - ) ρ s = suface chage densy (Cm - ) ε Fm - µ = 4π 0-7 Hm c = (ε µ ) ms - e = C η 377 hms = (µ /ε ) 0.5 ( µε ) E = 0 [Wave Eqn.] j E y (z,) = E + (z-c) + E - (z+c) = Re{E (z)e ω } y H x (z,) = η - [E + (z-c)-e - (z+c)] [(ω-kz) (-z/c)] A ( E H ) da d ) V (ε H ) + (d E +µ Meda and Bundaes D =ε E + P D =ρ f, τ=εσ ε E =ρ +ρ f p P = ρ, J =σ E p B =µh =µ ( H + M ) = V E J dv (Pynng Theem) 0.5 ε=ε ( ω ω p ), ω p =(Ne mε ) ε eff =ε( jσ ωε) skn deph δ = (/ωµσ) 0.5 [m] dv (Plasma) Φ = -ρ f ε C = Q/V = Aε/d [F] L = Λ/I () = C dv()/d v() = L d()/d = dλ/d w e = Cv ()/; w m = L ()/ L slend = N µa/w τ = RC, τ = L/R Λ = B da A F= I µ H [ Nm (pe un) - ] Elecmagnec Waves ( µε ) E = 0 [Wave Eqn.] ( +k ) E = 0, E = E e jk k = ω(µε) 0.5 = ω/c = π/λ k x + k y + k z = k = ω µε v p = ω/k, v g = ( k/ ω) - θ = θ sn θ sn θ = k k = n n θ =sn c (n n ) 0.5 θ = an B ( ε ε ) f TM x jk z z θ>θ c E = E Te +α k = k ' jk '' Γ = T T TE = ( + [η cs θ η cs T TM = ( +η [ cs θ η cs θ ]) θ ])