Maxwell' s Equations in vauum E ρ ε Physis 4 Final Exam Cheat Sheet, 7 Apil E B t B Loent Foe Law: F q E + v B B µ J + µ ε E t Consevation of hage: J + ρ t µ ε ε 8.85 µ 4π 7 3. 8 SI ms) units q eleton.6 9 m eleton 9. 3 SI ms) units Maxwell' s Equations in linea medium D ρ E B t B D εe ε E + P B µh µ H + M Geneal plane wave in linea medium: v [ ] ; B ˆ E P f v ˆ v t v P f v ˆ v t Hamoni plane wave in linea medium: [ ] ; B E P exp i ωt) H J + D t [ ] ; v εµ ; P v ; ˆ [ ] ; ω P exp i ωt ω v ; P Linea medium bounday onditions: D nomal, E paallel, B nomal, H paallal ae ontinuous v E ˆ B ˆ Kinemati bounday onditions: θ inident θ efleted ; v inident sinθ inident v tansmitted sinθ tansmitted Enegy: U field [ E D + B H ] Consevation of enegy: Poynting veto: S E B µ t U + U field patiles) + S E H Maxwell Stess Tenso: T ij ε E i E j δ ij E + µ H H i j δ ij H Momentum density: Consevation of momentum: p field volume εµs S εe B p field t volume + p patile volume T
Pefet onduto bounday onditions: E B inside E paallel B nomal E nomal σ ε B paallal µ I E nomal nomal B paallel nomal) θ inidene θ efletion v nomal hanges sign, P paallel hanges sign Retangula avity fields: E exp iωt), B exp iωt), ω π m a + n b + l d E x A EX os mπ x a sin nπ y b sin lπ d B A sin mπ x x BX a os nπ y b os lπ d E y A EY sin mπ x a os nπ y b sin lπ d B A os mπ x y BY a sin nπ y b os lπ d E A EZ sin mπ x a sin nπ y b os lπ d B A os mπ x BZ a os nπ y b sin lπ d A E A EX, A EY, A EZ ) A B A BX,A BY, A BZ ) M mπ a, nπ b, lπ d A E M A B M M A E iωa B M A B iω A E Retangula waveguide: same xy dependene, t dependene exp[ i ωt) ] TE E TM B ω + mπ + nπ v a b goup dω d TEM mode: E B, eletostati & magnetostati solutions in xy, v, B ˆ E Waves in ondutos: ondutivity σ J σe τ ε σ [ ] B B exp[ i ωt) ] B ω ˆ E E exp i ωt) E eal ω σ + ωε + imag ω σ + ωε good onduto: eal imag ωσµ d sin poo onduto: eal ω, imag σ ε
Kinemati bounday onditions: θ inident θ efleted ; v inident sinθ inident v tansmitted sinθ tansmitted Index of efation: n v εµ ε µ Total Intenal Refletion: sinθ I > v v n n Fesnel Equations: α osθ T osθ I β µ v µ v µ n µ n ε µ ε µ E in plane of inidene: E R E to plane of inidene: E R Model of medium: d y dt α β α +β αβ + αβ E T α + β E T + αβ + γ dy dt + ω y q m E exp iωt + γ j ω tanθ Bewste n n if µ µ ; ε ε + Nq fo low density, n + Nq f j ω j ω mε ω j ω ; α Nq ω mε Model of onduto: ω σ Nfq m γ iω Model of plasma: γ ω ω p q µεω + iσµω Nf ω ω p mε f j mε ω j ω ) iγ j ω ω j ω f j γ j + γ j ω Fields fom potentials: E V A B A t Potential Maxwell: V + A A ) ρ A A t ε t A + V µ t J Gauge Tansfomations: A A + V V t Coulomb Gauge: A V ρ A A ε t V µ t J Loent Gauge: A + V t V V ρ A A t ε t Loent in d' Alembetian fom: [ ] V ρ ε [ ] A µ J [ ] t µ J
Retaded Time: t t - Retaded Potentials: φ,t) ρ, t 4πε d v A µ ρ,t Fields: E ) ρ +,t ) 4πε ˆ J,t ) d v B µ 4π q Lienad - Wiehet Potentials: φ,t) 4πε u q Fields: E 4πε [ ] B u ) v )u + u a 3 Lamo Fomula: S da µ 6π qa) qa) 3 3 4πε Radiation at high veloity: dp ad dω q 6π ε Eleti Dipole Radiation: p p ˆ os ωt E µ p ω 4π sin θ os ωt )ˆ θ S µ p ω 4 sinθ os ωt 6π ) ˆ [ u a )] ˆ u B µ p ω 4π ˆ Powe µ p 6π 4π J, t d v J J A,t) v φ,t) u ˆ v ˆ E S µ 5 P µ q ρ,t )d v sinθ ω 4 os ωt )ˆ φ os ωt Magneti Dipole Radiation: m m ˆ os ωt) I t) Aea E µ m ω 4π sinθ os ωt )ˆ φ B µ m ω 4π sin θ os ωt )ˆ θ S µ m ω 4 sin θ os ωt 6π 3 ) ˆ Powe µ m ω 4 os ωt 6π 3 i e Multipole Radiation: A,t) ωt ) µ i) n n! J n γ 6 6π ˆ ) n d v ˆ [ E E ˆ E )] a v a d v Fields: B A E i B Radiation Reation: F µ q 6π da dt
Loent Tansfomation: β v x γ t γ β x βγ Coodinate 4 - veto: x µ t, x ) x µ t, x ) x µ x µ x µ x µ x ν x ν Relativisti veloity 3 ν βγ γ t x t) + x + y + η µ γ,β γ) Relativisti momentum: p ν E, p ) Newtonian mehanis: F dp elativisti dt obseve Minowsian mehanis: K µ dp µ dτ 4 veto deivative: µ t, 4 - veto uent density: J µ ρη µ ρ γ,β γ d'alembetian µ µ [ ] 4 - veto potential: A µ φ, A ) Maxwell Equations: ν ν A µ µ J µ Loent ondition: µ A µ Field - Stength Tensos: E x E y E E x B F µν E y B B x E B y B x Maxwell in Tenso Fom: B x B E B x E y G µν E B E y x E B y E x µ F µν µ J ν µ G µν Loent Tansfomation of Fields: E x E x B x B x E y γ E y vb ) E γ E + v ) γ + v E B γ B v E y pio 9 nano 6 mio µ) 3 milli +3 ilo +6 Mega +9 Giga + Tea
Catesian oodinates : f f x x ˆ + f y y ˆ + f ˆ V V y V y + V x V x + V y x V x y V V x x + V y y + V f f x + f y + f Cylindial oodinates : f f ˆ + f φ ˆ + f φ ˆ V f φ V ) V φ Spheial oodinates : f f ˆ + f θ V ˆ + f + f φ + f sinθ θ ˆ + f sinθ φ θ sinθv φ V ) V ) V φ ˆ + V φ + φ ˆ V V φ V θ ˆ + sin θ V + sinθ φ V ) V φ f f + f sinθ sinθ θ θ + f sin θ φ x ˆ y ˆ ˆ A B det A x A y A B x B A B C φ V φ V φ) + V ) ˆ θ sin θv θ) + θ ˆ + V θ A y B S ) x ˆ + A B x A x B ) y ˆ + A x A y B x ) ˆ B C A ) C A B ) B A C ) C A B ) A B ) C B C A ) A C B ) A B C f g + g f fg A B ) + B A ) + A )B + B f A ) + A f B A ) A B ) f A ) + A f ) B )A A )B + A B ) B A ) A B fa A B fa A B A sinθ φ V φ) θ V φ ˆ
+ if i j δ ij if i j x i ε ij if i j, j, o i + if ij 3, 3, 3 if ij 3, 3, 3 ε ij ε lm δ il δ jm δ im δ jl A B δ ij B j A B ε ij B j A B C [ A B C )] B C i C [ fg) ] fg [ A B )] fa f g + g f f ε ij B + f + f ε ij B + ε ij B ε ij f + ε ij f ε ij B j C A B [ fa ] ε ij f [ A B ] ε ij ε lmj A l B m ε ij ε lmj A l B m δ l δ im δ m δ il A l B m A B A + A B B