Today in Physis 8: review I You learned a lot this semester, in priniple. Here s a laundrylist-like reminder of the first half of it: Generally useful things Eletrodynamis Eletromagneti plane wave propagation in a variety of media (linear, onduting, dispersive, guides) The Sream, by Edvard Munh (893). 6 April 4 Physis 8, Spring 4
Generally useful math fats Vetor and vetoralulus produt relation from the inside overs of the book Properties of the delta funtion Orthonormality of sines and osines iau ibu iu Ae + e = Ce A+ = C, a= b = π π π osmx os nxdx = sin mx sin nxdx = πδ osmx sin nxdx =, so os π t = π π os tdt = os xdx π = = sin t mn 6 April 4 Physis 8, Spring 4
Eletrodynamis (as opposed to statis or quasistatis) eyond magnetoquasistatis Displaement urrent, and Maxwell s repair of Ampère s Law The Maxwell equations Symmetry of the equations: magneti monopoles? gs units: Ε = ρ = E E = = J+ t t MKS units: ρ Ε = = ε E = = µ J+ µ ε t E t 6 April 4 Physis 8, Spring 4 3
Eletrodynamis (ontinued) The Maxwell equations in matter oundary onditions for eletrodynamis D = ρ = f D E = H = Jf + t t D H and E are ontinuous; is disontinuous by 4 πσ ; ( π ) is disontinuous by 4 K nˆ. ( D = ε E = µ H) In linear media, : ε E ε E = σ E above,above below,below µ,above,above above =,below E =,below,above below f,below = Kf nˆ µ f f 6 April 4 Physis 8, Spring 4 4
Eletrodynamis (ontinued) Potentials and fields Gauge transformations, espeially the Lorent gauge Energy onservation in eletrodynamis: Poynting s theorem dw meh. dt A E = V t = A V A + = t = ( ) d E a d ( + E ) dτ 8π dt ( umeh. + ue ) + S = t S= E, ue = E + 8π S V ( ) 6 April 4 Physis 8, Spring 4 5
Eletrodynamis (ontinued) Momentum onservation in eletrodynamis and the Maxwell stress tensor T = E E + E + dp meh. d = T da gedτ dt dt ( ) ij i j i j ij S ( gmeh. + ge ) T = t ge E LE = r ge = r E V ( ) δ 6 April 4 Physis 8, Spring 4 6
Waves Eletromagneti waves Waves on a string The simple solutions to the wave equation Sinusoidal waves µε E µε E, t t = = x f µ f = T t f = g( x± vt) = g( ). f xt, = Ae ( ), A = Ae ( ) ikx t iδ Polariation 6 April 4 Physis 8, Spring 4 7
Waves (ontinued) Refletion and transmission of waves on a string Impedane ik ( t f ) I = A Ie : f = fi + fr i( k t f ) R = A Re ik ( ) } t ft = A Te : f = ft (, ) ( +, ) f, (, ) f f t f t t ( + = =, t) v v Z Z A R = A I = A I v + v Z + Z v Z A T = A I = A I; Z = T v = Tµ v + v Z + Z 6 April 4 Physis 8, Spring 4 8
Plane eletromagneti waves in linear media Plane eletro-magneti waves Energy and momentum in plane eletro-magneti waves Radiation pressure Waves in linear media i( kr t E = E e ), = kˆ E E E u= =, S= kˆ = ukˆ E ˆ S u g = k = = kˆ = µε ˆ E S= E H = E µ u= + = E + 8π 8π µ εµ S εµ ε g = = E H = E ( E D H) ε 6 April 4 Physis 8, Spring 4 9
Plane eletromagneti waves in linear media (ontinued) The impedane of linear media Spaeloth oundary onditions for refletion and transmission of eletromagneti plane waves at interfaes Z = µ ε ε E, ε E, =,, = E E = = or ε,,,, µ µ ( ) E + E = ε E + = I R T I R T E + E = E + = ( ) Ix Rx Tx Ix Rx Tx µ µ E + E = E + = ( ) Iy Ry Ty Iy Ry Ty µ µ 6 April 4 Physis 8, Spring 4
Plane eletromagneti waves in linear media (ontinued) Snell s Law The Fresnel equations θi = θr sinθt ki v n = = = sinθi kt v n E I αβ E ki, kr, kt : E T =, E R = E I. + αβ + αβ E I α β E ki, kr, kt : E T =, E R = E I. α + β α + β osθt n α = sinθi osθi osθi n β = µ ε µ ε = Z Z ε ε = n n 6 April 4 Physis 8, Spring 4
Plane eletromagneti waves in linear media (ontinued) Total internal refletion Polariation on refletion Interferene in layers of linear media Transmission and refletion in stratified linear media, viewed as a boundary-value problem n θic > arsin n n tan θ I = β =. n dn osθt λ m = m =,,, m ( ) 6 April 4 Physis 8, Spring 4
Plane eletromagneti waves in linear media (ontinued) Matrix formulation of the fields at the interfaes in stratified linear media Y Y, TE = osθt = osθt µ Z, TM = ε ε µ osθ T E, osδ isin δ/ Y E, E, = M H iy, sinδ osδ H, H, E, E, p+ = MM Mp H, H, p+ M = M M M = p m m m m. 6 April 4 Physis 8, Spring 4 3
Plane eletromagneti waves in linear media (ontinued) Charateristi matrix formulation of refleted and transmitted fields and intensity Examples: Single interfae Plane-parallel dieletri in vauum Multiple quarterwave staks m Y + m Y Y m m Y r = m Y + m Y Y + m + m Y Y t = m Y + m Y Y + m + m Y p+ p+ p+ p+ p+ p+ S ρ = = S R, I, S Y τ = = S Y τ + ρ = Tp, +, p+ I, r t 6 April 4 Physis 8, Spring 4 4
Plane eletromagneti waves in ondutors Eletromagneti waves in ondutors Attenuation of the waves, and an eletroni analogy Penetration of waves into ondutors: skin depth d E εµ E σµ E = + t t ε ρε τ = =. πσ π σ = = + κ µε ε. 6 April 4 Physis 8, Spring 4 5
Plane eletromagneti waves in ondutors (ontinued) Good and bad ondutors Relative phase of E and of waves in ondutors σ ε ε good, σ bad. k πµσ πµσ = ( + i), k = κ = good, k πσ µ σ = k+ iκ, k µε κ, κ = = Z bad. ε iφ ( k iκ ) ke + = E = E S k = ˆ E e 8 πµ κ 6 April 4 Physis 8, Spring 4 6
Plane eletromagneti waves in ondutors (ontinued) Refletion from onduting surfaes The harateristi matrix of a onduting layer E β T = E I, E R = E I + β + β µ k µ πσ β = γ ( + i) γ = ε µ εµ I I R I Y k, good ( γ) ( γ) = δ = µ + γ = + + γ k and k d, πµ σ ( + i) good = πσ µ µε + i bad ε 6 April 4 Physis 8, Spring 4 7
Plane eletromagneti waves in dispersive media Motion of bound eletrons in matter, and the frequeny dependene of the dieletri onstant Dispersion relations Ordinary and anomalous dispersion d x dt ε = + dx + γ + = dt Nq m x me M ( ) ( ) e j= j j Dilute gas: I = I e, q E e α it ( ) j ( ) j ( ) j + j M Nq f γ α κ =, m e j= γ f j iγ M π Nq f j n k +, me j= γ n = n+ i α. j j + j 6 April 4 Physis 8, Spring 4 8
Plane eletromagneti waves in dispersive media (ontinued) Nf q Semilassial theory σ = of ondutivity me γ i Condutivity and Nfq Nfq σ metals, σ i dispersion in metals meγ me and in very dilute p ondutors k =, p Light propagation in very dilute v = = = > k n ondutors: group ( ) p veloity, plasma d frequeny vg = = p < dk d = = <, always, in nondispersive media. k dk n 6 April 4 Physis 8, Spring 4 9 Nfq m e gases.
6 April 4 Physis 8, Spring 4 Guided waves Metalli waveguides Light propagation in hollow ondutive waveguides. x y x y E i E k x y k E i E k y x k E i k x y k E i k y x k = + = = = + TE waves TM waves E = =
Guided waves (ontinued) The TE modes of retangular metal waveguides m x n y π π = os os, a b mn, =,,,... (but not both ) E i i,, x = E y = y x k k ik, ik. x = y = x y k k 6 April 4 Physis 8, Spring 4
Guided waves (ontinued) Waveguide modes, e.g. TE: S i mπ mπx mπx nπ y sin os os xˆ k a a a b nπ mπx nπy nπy os sin os ˆ + y b a b b = 8π + ( k ) 6 April 4 Physis 8, Spring 4 k nπ mπx nπ y os sin b a b mπ mπx nπ y + sin os ˆ. b a b
Guided waves (ontinued) Dispersion and ut-off in waveguides Massive photons? The real reason there are no TEM modes in hollow onduting waveguides TEM modes in oaxial waveguides k v m π n π = a b mn = = = > k mn d vg = = mn < dk 6 April 4 Physis 8, Spring 4 3