Physics 44 Electro-Magneto-Dynamics M. Berrondo Physics BYU 1
Paravectors Φ= V + cα Φ= V cα 1 = t c 1 = + t c J = c + ρ J J ρ = c J S = cu + em S S = cu em S Physics BYU
EM Wave Equation Apply to Maxwell F = J Z F = Z J Separate scalar and vector parts F v = Z J v F = Z J v F = Z = tρ + J = s s J Physics BYU 3
Wave Equation (1-d) Initial disturbance g(z) v f( zt, = ) = gz ( ) f (,) z t = g( z vt) Propagating with speed v: after t: f( z, t) = g( z vt) = g( u) l l = vt Physics BYU 4
Partial differential equation f dg u dg = = v t du t du 1 f f + = f dg u dg v t z = = z du z du propagating to the right f (,) z t = g( z + vt) 1 f f + = v t z propagating to the left Physics BYU 5
Either way f ± v = ( ± vg ) ' = vg"( u) t t f z = g"( u) f (,) z t = g( z ± vt) g() u 1 f f = v t z wave equation in 1+1 dimensions f (,) z t = g( z vt) + h( z + vt) linear equation constant v Physics BYU 6
Harmonic waves g( u) = Acos( ku + δ ) u = z vt k wave number A, δ λ amplitude, phase f( z,) [ ] 1 k = m π λ = wave length k δ k 1 v ν = = T λ Physics BYU 7
Dispersion relation ω = kv ω = ω( k) dispersionless i( kz ωt) iδ (, ) cos( ) Re( ) f z t = A kz ωt + δ = Ae e f (,) z t = Be i( kz ωt) complex form B = i Ae δ includes phase Physics BYU 8
Reflection and Transmission in 1-d k 1 k I R T kv 1 1= kv ω Z dispersion relation Incident wave: 1 f( zt, ) = Acos( ωt kz) 1 Reflected: z < 1 A cos( ω t+ kz) R 1 z > Acos( ωt kz) Transmitted: T Physics BYU 9
Continuity: ] ] f( z =, t) = f( z =, t) 1 A+ AR = AT k1( A AR) = ka T and derivative 1 AR = v + v1 where δr = δt = δ for v > v 1 δr + π = δt = δ for v < v 1 Physics BYU 1 v v A reflected wave is 18 out of phase (upside down)
transverse waves: longitudinal waves: Polarization displacement displacement (compression propagation propagation (d) sound waves) long. transverse polarization f f -d vector Transverse vectors f f v h = Acos( ωt kz)ˆ e = Acos( ωt kz)ˆ e 1 left - right or up - down Physics BYU 11
define polarization vector transversality nˆ eˆ3 = ˆ nˆ = 1 arbitrary linear polarization: f = Acos( ω t kz) nˆ n polarization angle nˆ = cosϕeˆ + sinϕeˆ ϕ 1 y ϕ ˆn x ê 1 ϕ rotate by radians about z axis iˆ ϕ ˆ = e e ˆ ˆ 1 = 1 n e e e ieˆ ϕ 3 3 nˆ fixed linear polarization Physics BYU 1
Horizontal and Vertical Polarization Physics BYU 13
Case: Electromagnetic Waves in Vacuum apply ρ = J = J = to the left: F = 3-d wave equation for E and for B Maxwell s equations propagation of EM waves with speed: F c = = 1 εµ F = J = 1 c t = empty space supports Physics BYU 14
EM spectrum 1 Hz 1 1 1 m 1 1 1 15 3 1 6 6 γ visible RF Physics BYU 15
Physics BYU 16
Monochromatic waves: in a homogeneous medium ω = const k = const Consider a wave with the combination: ω s = ω t k r = ( k)( ct + r) = kx c ω where k = + k x = ct + r paravectors c and F = F( s) = F f( s) with f() = 1 s s instead of F (,) r t (for plane waves: f( s) = cos s) Physics BYU 17
Then 1 s = ( t )( ω t k r) = + k k c c F () s = kf f '() s ω no sources: F ( s) = kk F f ''( s) = ω k k = dispersion relation: c and we can write: ω k = (1 + kˆ ) c 1 P (1 ˆ k = + k) is a projector: Pk kk = ω = c k (1 + kk ˆ) ˆ = 1 + = P kˆ Physics BYU 18 k
Properties of EM waves in vacuum: Wave front: s F F ω kf ˆ = (1 k)( E + icb) = c = = ( s = ) kˆ E = kˆ B = Maxwell eqn. and E = ckˆ B B = 1 kˆ E c Magnitudes: E = cb and E B = Physics BYU 19
B E ˆk Physics BYU
Transversality: kˆ E = kˆ B = E ˆ, B, k form an orthogonal RHS Finally F = E c B + iceb = F = and F = E + icb = E + i kˆ E = (1 + ke ˆ) F = (1 + ke ˆ) = E(1 kˆ) in terms of E only! Physics BYU 1
B E ˆk Summary ω = kc kˆ E = kˆ B = E B = ˆ 1 E = cb E = ck B B = kˆ E c F ˆ ˆ = (1 + ke ) = E(1 k) F = with 1 (1 + kˆ ) = P P = P k k k projector Physics BYU
F = (1 + ke ˆ ) If E is in the ˆε in general for monochromatic waves direction Er (, t) = εˆ E cos( k r ωt+ δ) δ phase then Br (, t) = 1 E ˆ ˆ cos( k r ωt + δ) k ε c with the same phase!!!! i B= ke ˆ and c Physics BYU 3
Physics BYU 4
Energy momentum in EM waves Energy density: u(,) r t = ε E cos( ωt k r δ) = ε E (,) r t and ε ce ˆ S= k = k Z for high frequency oscillations take time average over a cycle: ε 1 u = E ave cos ω t = so ε r.m.s. c S = ˆ ave E k Intensity I values 1 E g ˆ ave = ε Physics BYU 5 c E kˆ
Intensity I = S Power area ave = I and so energy S = area t ε c = = I S E Momentum transfer ˆk absorber energy flux density I = E Z Physics BYU 6
g Transfer of momentum at surface of perfect absorber pressure on surface: 1 p 1 I = = = = A t c P ε E u Pressure: P = u = Perfect reflector ( mirror ) S c equals time average energy density twice that amount Physics BYU 7
EM waves in matter linear medium 1 D= ε E H = B µ E = B= v 1 c = = εµ n index of refraction with 1 v = speed of propagation n = εµ εµ Z t = µ ε Physics BYU 8
for dielectrics with µ µ n κ = ε ε 1 ( ) is the index of refraction u = ED + BH and intensity S= E H 1 1 I = ε ve = E Z Physics BYU 9
Reflection and Refraction θr θ I k k 1 k plane i nˆ θt k k k 1 incident reflected transmitted F = F cos( ωt k r) + F cos( ωt k r) 1 1 = F cos( ωt k r) at boundary z = match F Physics BYU 3
n i = the index of refraction of medium i k nˆ = k nˆ = k nˆ 1 k, k ˆ 1, k, n in same plane Magnitude k = k k sinθ = k 1 I θ I = θ R Snell s law sinθ k sinθ = k sinθ n sinθ = n sinθ I T 1 I T k i = ω n c R i Physics BYU 31
Physics BYU 3
Fresnel (Amplitudes) E I k I incident kˆ R reflected E R ˆn E T k T transmitted assuming planar polarization parallel to plane incidence B nˆ = Physics BYU 33
E k B ˆn cosθ = nˆ kˆ sinθ = En ˆ ˆ nˆ ( D + D D ) = 1 nˆ ( E + E E ) = 1 nˆ ( B + B B ) = 1 nˆ ( H + H H ) = 1 D= ε E plane ˆ i n B n ˆ = i for i =,1, 1 1 H = B= k ˆ E µ µ v Physics BYU 34
Snell: n1sinθi = nsinθt nˆ D for amplitudes and E ε ( E sinθ + E sin θ ) = ε ( E sin θ ) 1 I I R R T T nˆ E: E cosθ E cosθ E cosθ I I R R T T n = εµ /( ε µ ) + = 1 From 1 n ε ε β 1 1( EI ER) = ET ( EI ER) = ET n defining nε Z n β = = n ε Z n 1 1 1 1 Physics BYU 35
and : EI + ER =αet cosθt α = cosθ α β E = E E = E α + β α + β R I T I I cosθ cosθ 1 Fresnel α > α < β β R R in phase with I out of phase with I E R n n cosθ cosθ n + n cosθ cosθ 1 1 1 1 E I Normal incidence Z α = 1 β = Z n n 1 1 ˆn kˆ I = E R nˆ = θ I = n n n + n 1 1 E Physics BYU 36 I
Grazing angle: ET θ I π and E α R E I Opposite: E R = when α = β sin θ B = 1 β ( n / n ) β 1 tanθ B n n 1 Brewster angle polarized θ B reflected beam is totally Physics BYU 37
Physics BYU 38
n > n going from 1 water air 1 1 sin T = sin I > sin I n Total reflection for critical angle n θ θ θ fiber optics Total Reflection θ I θ c sinθ c = 4 for diamond 6 for fiber glass n n 1 Physics BYU 39
Incident Intensity: 1 II = ε1ve 1 cosθ interface I I R E R α β R I E = = = I I α + β while T T 1 T T = = = I ε n E cosθ αβ II ε1n EI cosθ I α + β Physics BYU 4
Absorption and Dispersion Conductors: σ = conductivity Ohm s laws so Maxwell from continuity equation differential equation: J f = σ E for free currents 1 F = ρ f µσ ve ε 1 = t + v ρ t f ρ t σ = ρ f ε f = J = σ E f Physics BYU 41
ρ f α t ( t) = e ρf () where α = σ ε ρ t > ε / σ f is dissipated for very small and F + µσν E = homogeneous equation with ρ f Physics BYU 4
Modified dispersion relation Apply : F + µσ ( v ) E = t where E= + E= B i i t ( µσ ) t + F = with dissipation term wave diffusion equation t dissipation out of phase Physics BYU 43
Look for F ( iv )e i( ωt kz) = E + B = I = ( E + ivb )e e k z i( ωt k z) R with k complex k = k + ik R I complex dispersion relation: + = µ ε ω k iµσω k R εµ σ = ω 1+ + 1 εω 1/ Physics BYU 44
k I I e kz εµ σ = ω 1+ 1 εω polar form: 1/ attenuation factor while k = i Ke ϕ Ke iϕ we get B = E ω tanϕ = 1 k I n k k I R = = skin depth ck R ω kz I E( z, t) = E e cos( kz ωt) εˆ, kz I B( z, t) = B e cos( kz ωt + ϕ) kˆ εˆ out of phase! Physics BYU 45
Physics BYU 46
Evanescent Wave Physics BYU 47
Reflection at conducting surface B I dielectric kˆ I 1 incident wave z direction ' 1 β R ' I x conductor ' k, v, σ z α = = E E E 1 + T = E ' β 1 + β 1 I Physics BYU 48
where ' β is complex: β = µ v k µω ' 1 1 since k = µ ε ω + iµ σω is complex: skin depth term for a perfect conductor: ' β E E E R = I T = Physics BYU 49
Microscopic Model of Dispersive Dielectric Media ω phase velocity v = k dω vg = dk for ω = ck v = v = c group velocity v and v g are independent of g non dispersive medium more realistic model: ε= εω ( ) even if the medium is linear and isotropic. ω Physics BYU 5
Model: electron bound to molecule small excursions SHO model with damping in 1-d ω = natural frequency interaction with EM wave with frequency ω ω F = mω x mγx bind EM interaction resonance F = qe = q E e ω drive Re i t ω Physics BYU 51
Newton: steady state solution: x dipole moment: i t m ( x + γx + ω x) = qe e ω i t e ω same EM ω q/ m p() t = qx = q E e ω ω iγω polarization with f j = oscillator strength : P Nq f = m j ω ω γω E E j ε ( κ 1) j i j complex dielectric constant: Nq f κ = 1+ mε j ω ω iγω j j j iω t Physics BYU 5
E ˆ I = E kz ε e cos( kr z ωt) intensity I E absorption coefficient where k I α = k I ckr = Im εµ ω and n = ω dilute gases: 1 1+ r 1+ r index refract. n f j( ωj ω ) j j + j ckr Nq = 1+ ω mε ( ω ω ) γω Physics BYU 53
α = k I Nq ω f γ mεc ω ω γω j j j ( j ) + j resonance: ω ω j for some j ω j E = far from resonance: n Nq f j 1+ mε ω ω j j Cauchy s formula: 1 1 ω (1 + ) j j j ω ω ω ω n B 1 + A(1 + ) λ Physics BYU 54
Physics BYU 55
Wave guides EM waves confined to the interior of a hollow pipe of (perfect) conductor Boundary conditions ˆn nˆ E= at boundary (wall surface) nˆ B = the dispersion relation for free waves ω = ck is no longer valid! and EM waves are no longer purely transverse! Physics BYU 56
assume z propagation direction and k propagation wave number s= ω t kz in z - direction F e is (real part) (stationary waves in x-y) Physics BYU 57
(stationary in x-y, traveling in z direction) Physics BYU 58
xy, Transverse directions k x, k y are quantized (normal modes) due to B.C.! x a y b a b mπ nπ k = k = k = k a b mn, =,1,, x y z D Alembertian becomes: 1 ( ) ( ) c x y ω k + Physics BYU 59
ω + k f( xy, ) = c (no z dependence) where = + x y two independent solutions: TE - modes TM - modes f( xy, ) B = E z z E B z z ( xy, ) TE ( xy, ) TM = but Bz = but Ez Physics BYU 6
X Rectangular wave guides X or x and a a 1π mπ = a kx = m =,1,... kx a nπ ky = n =,1,... b except m = n = f( x, y) = X( xy ) ( y) Physics BYU 61
ω Dispersion relation: m π a n π b = c k + + ck + ω mn ω mn there is a cut-off frequency ω c ω ω = ω c mn the lowest cut-off frequency is for TE mode ω 1 = cπ / a Physics BYU 6
B Given: E Maxwell (plane waves): ik wave eq.: using, t z z iω k determine B E k E= k B= ω k E= ωb k B= E c ω k = k E c E E = k k E = k ( k E ) k ( k E ) k E = ke and ω B = k E z z Physics BYU 63
we get E = kk E z ωk B z ω / c k for plane waves! for Fxyzt (,,, ) = f( xye, ) ω i( t kz) we replace k i E = i k E z + ω B z ω / c k and similarly: B = i k Bz ω Ez / c ω / c k Physics BYU 64
rectangular case: f( x, y) = X( xy ) ( y) X( x) and Y( y) sin or cos functions For TE modes Neumann B.C. so if a E z = nˆ B = nˆ ( ) = B z mπx mπy Bz( xy, ) = cos cos B a b b cπ lowest frequency is ω = a > Physics BYU 65
for: B = cos( k x)cos( k y)cos( ωt kz) B while B = B(, xyzt,,) z z x y z TE mode kkx Bx = sin( k )cos( )cos( ) xx kyy ωt kz B ω / c k kky By = cos( k )sin( )cos( ) xx k yy ωt kz B ω / c k mπ nπ and: kx = k y = a b Physics BYU 66
mπ nπ plane wave k' = eˆ ˆ ˆ 1+ e + ke3 standing waves a k cosθ = k ' b vg ωmn = ccosθ = 1 ω Physics BYU 67
b a Coaxial transmission line Non dispersive transverse waves k ω c Ez = Bz = E = B = statics-like -d solutions: sˆ A φˆ E = A B = s c s A E( s, ϕ, z, t) = cos( kz ωt) sˆ and s TEM modes A B = cos( kz ωt) φˆ cs Physics BYU 68 = and