Physis 8 Spring 4 6 April 4 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 + Be Ce A + B C a b π π osmx os nxdx sin mx sin nxdx πδmn π osmx sin nxdx so π os t os tdt π π os xdx sin t π 6 April 4 Physis 8 Spring 4 Eletrodynamis (as opposed to statis or quasistatis) Beyond magnetoquasistatis Displaement urrent gs units: Ε ρ B and Maxwell s repair B π E B J+ of Ampère s Law The Maxwell MKS units: equations ρ Ε B ε Symmetry of the B equations: magneti E B J+ t monopoles? 4 E t t E µ µ ε t 6 April 4 Physis 8 Spring 4 3 () University of Rohester
Physis 8 Spring 4 6 April 4 The Maxwell equations in matter Boundary onditions for eletrodynamis Eletrodynamis (ontinued) D ρ f B B D E H Jf + t t B and E are ontinuous; D is disontinuous by 4 πσ f ; H is disontinuous by ( 4 π ) Kf nˆ In linear media ( D ε E B µ H) : εabove E above εbelow E below σ f B above B below Eabove Ebelow B above B below K f nˆ µ above µ below 6 April 4 Physis 8 Spring 4 4 Eletrodynamis (ontinued) Potentials and fields Gauge transformations espeially the Lorentz gauge Energy onservation in eletrodynamis: Poynting s theorem A E V t B A V A + t dwmeh ( ) d dt E B a S d ( B E ) dτ 8π dt + V ( umeh + ueb ) + S t S E B ueb E + B 8π ( ) 6 April 4 Physis 8 Spring 4 5 Eletrodynamis (ontinued) Momentum onservation in eletrodynamis and the Maxwell stress tensor ( ) Tij EiEj + Bi Bj E + B δij dp meh d d EBdτ dt T a dt g S V ( gmeh + geb ) T t geb E B LEB r geb r ( E B) 6 April 4 Physis 8 Spring 4 6 () University of Rohester
Physis 8 Spring 4 6 April 4 Waves Eletromagneti waves Waves on a string The simple solutions to the wave equation Sinusoidal waves µε E µε B E B t t f µ f T x t f g( x± vt) g( z) ikx ( t) i f ( xt ) Ae A Ae δ Polarization 6 April 4 Physis 8 Spring 4 7 Refletion and transmission of waves on a string Impedane Waves (ontinued) ikz ( t f ) I A Ie z : f fi + fr i( kz t f ) R A Re ikz ( t f ) T A Te } z : f ft + f ( t) f ( t) f f + ( t) ( t) z z v v Z Z A R A I A I v + v Z + Z v Z A T A I A I; v + v Z + Z Z T v Tµ 6 April 4 Physis 8 Spring 4 8 Plane eletromagneti waves in linear media ( ) i kr t Plane eletro-magneti E E e B kˆ E waves E B E u S kˆ ukˆ Energy and momentum in plane eletro-magneti E ˆ S u g k kˆ waves Radiation pressure B µε zˆ E Waves in linear media S E H E B µ u ( E D+ B H) εe + B 8π 8π µ εµ S εµ ε g E H E B 6 April 4 Physis 8 Spring 4 9 () University of Rohester 3
Physis 8 Spring 4 6 April 4 Plane eletromagneti waves in linear media (ontinued) The impedane of linear media Spaeloth Boundary onditions for refletion and transmission of eletromagneti plane waves at interfaes µ Z ε ε E ε E B B E E B B µ µ or ε( E Iz + E Rz ) εe Tz B Iz + B Rz B Tz E Ix + E Rx E Tx ( B Ix + B Rx ) B Tx µ µ E Iy + E Ry E Ty ( B Iy + B Ry ) B Ty µ µ 6 April 4 Physis 8 Spring 4 Plane eletromagneti waves in linear media (ontinued) Snell s Law θ θ The Fresnel equations I R 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 Polarization 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 θib β n dn osθt λ m ( m ) m 6 April 4 Physis 8 Spring 4 () University of Rohester 4
Physis 8 Spring 4 6 April 4 Plane eletromagneti waves in linear media (ontinued) Matrix formulation of the fields at the interfaes in stratified linear media ε Y TE osθt osθt µ Z ε Y TM µ osθt 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 MMMp 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 my + myyp+ m myp+ r my + myyp+ + m + myp+ Y t my + myyp+ + m + myp+ SR ρ r SI STp + Yp+ τ t SI Y τ + ρ 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 E εµ E σµ E + z t t ε ρε τ πσ π σ d + κ µε ε 6 April 4 Physis 8 Spring 4 5 () University of Rohester 5
Physis 8 Spring 4 6 April 4 Plane eletromagneti waves in ondutors (ontinued) Good and bad ondutors Relative phase of E and B of waves in ondutors ε ε σ 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 i ) ke B + κ E E k ˆ E z e κ S z 8 πµ 6 April 4 Physis 8 Spring 4 6 Plane eletromagneti waves in ondutors (ontinued) Refletion from E β T E I E R E I onduting + β + β surfaes k β γ ( + i) γ The harateristi matrix of a onduting layer µ µ πσ ε µ good εµ IR ( γ) + γ I I + ( γ) + γ k Y µ and δ k d πµ σ ( + i) good k 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 dx q it + γ + x Ee dt dt me M Nq f j ε + me j j iγ j ( ) α z Dilute gas: I( z) Ie M π Nq f j( j ) n k + m e j ( j ) + γ j M Nq f jγ j α κ m e j ( j ) + γ j n n+ i α 6 April 4 Physis 8 Spring 4 8 () University of Rohester 6
Physis 8 Spring 4 6 April 4 Plane eletromagneti waves in dispersive media (ontinued) Semilassial theory Nfq σ of ondutivity me γ i Condutivity and Nfq Nfq σ metals σ i gases dispersion in metals meγ me and in very dilute p Nfq ondutors k p m e 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 Guided waves Metalli waveguides Light propagation in hollow ondutive waveguides E TE waves B TM waves i E B E x k + x y k i E B E y k y x k i B E B x k x y k i B E B y k + y x k 6 April 4 Physis 8 Spring 4 Guided waves (ontinued) The TE modes of retangular metal waveguides m x n y B π π B os os a b mn (but not both ) i B i B E x E y y x k k ik B ik B B x B y x y k k 6 April 4 Physis 8 Spring 4 () University of Rohester 7
Physis 8 Spring 4 6 April 4 Guided waves (ontinued) Waveguide modes eg TE: B i mπ mπx mπx nπ y S sin os os ˆ 8π x k a a a b nπ mπx nπy nπy os sin os ˆ + y b a b b + k nπ mπx nπ y os sin b a b ( k ) mπ mπx nπ y + sin os ˆ b z a b 6 April 4 Physis 8 Spring 4 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 m π n π k a b mn v > k mn d vg mn < dk 6 April 4 Physis 8 Spring 4 3 () University of Rohester 8