Chapter 3. Electromagnetic Theory, Photons, and Light

Transcription

1 Chapte 3. lectomagnetic Theoy, Photons, and Light Hecht by YHL;100310; 3-1 Classical lectodynamics : negy tansfe by electomagnetic waves Quantum lectodynamics : negy tansfe by massless paticles(photons) 3.1 Basic Laws of lectomagnetic Theoy lectic chages, time vaying magnetic fields lectic field lectic cuents, time vaying electic fields Magnetic field The foce on a chage q F = q + qv B : Loentz foce A. Faaday s Induction Law A time-vaying magnetic flux though a loop Induced lectomotive Foce(emf) o voltage d emf = - F B dt z dl. B ds C zz A d db z dl = - B ds ds C dtzz Þ - A zz : Faaday s law A dt (Wong diection of dl in the book)

2 Hecht by YHL;100310; 3- B. Gauss s Law-lectic lectic flux : F = D ds = D ds Òòò A ^ Ò òò A The total electic flux fom a closed suface = The total enclosed chage Ò òò D ds = dv A òòò, V whee D = e, e = e eo, electic pemittivity Pemittivity of fee space Relative pemittivity, Dielectic constant A point chage at the oigin D is constant ove a sphee, F = q q Coulomb s law, = pe 4 o F = 4p. D C. Gauss s Law-Magnetic No isolated magnetic chage B ds = A zz 0 D. Ampee s Law A cuent penetating a closed loop induces magnetic field aound the loop Using cuent density J[ A / m ] Ñ ò H dl = J ds C òò : Ampee s law A whee B = mh with m = m m o, pemeability Pemeability of fee space Relative pemeability

3 Hecht by YHL;100310; 3-3 A capacito (1) Cuent though the suface A 1 ¹ 0 Magnetic field along cuve C ¹ 0 Cuent though the suface A = 0 Magnetic field along cuve C=0 (???) () Magnetic field aound the wie ¹ 0. B ¹ 0 between the plates even if thee is no cuent. (They ae indistinguishable) Between the capacito plates Q = e i = º e A A J D, Displacement cuent density Diffeentiate both sides The genealized Ampee s law æ D ö Ñ ò H dl = ç J + ds C òò : Maxwell A è ø

4 Hecht by YHL;100310; 3-4. Maxwell s quations In fee space : e = e, m = m, = = 0 o o J B B Ñ ò dl = - ds C òò Ñ = - A D D Ñ ò H dl = ds C òò Ñ H = A B ds òò = Ò 0 Ñ B = 0 A D ds òò = Ò 0 Ñ D = 0 A Diffeential fom (1) () (3) (4) 3. lectomagnetic Waves Thee basic popeties (1) Pependiculaity of the fields A time-vaying D geneates H that is pependicula to D [Fig. 3.1] B B [Fig. 3.5] Tansvese natue of and H () Intedependence of and H Time vaying and B egeneate each othe endlessly (3) Symmety of the equations The popagation diection will be symmetical to both and H Diffeential wave equation Fom (1) Ñ Ñ = - Ñ B e j e j Using () ÑeÑ j - Ñ = -eom o =0 Ñ = eom o In Catesian cood. by sepaation of vaiables x x x x e m x + + = o o, : y z Wave equation known long befoe Maxwell with v = 1/ eomo Maxwell calculated v = =» 3 10 m / s : e - - om 1 7 o p 10 Fizeau's measuement, c=315,300 Km / s Maxwell concluded that light is an electomagnetic wave.

5 Hecht by YHL;100310; 3-5 A. Tansvese Waves A plane wave popagating along x-axis bx, y, z, tg bx, y, z, tg = = 0 y z ) ) ) = x, t Þ ( x, t) x + ( x, t) y + ( x, t) z ( ) Fom divegence eq. (4) Ñ = 0 x y z x x = 0 ) ) = ( x, t) y + ( x, t) z y z + + = 0 y z x = 0, x = const (not a taveling wave) = 0, = 0 x y z : Tansvese wave Assume = y $ x, t Fom (1) y x$ y$ z$ x y z 0 0 y b g B = - z$ y x$ B y$ B x y z$ Bz x = (5) = 0, = 0, to match $z Only and B exist : Tansvese wave A plane hamonic wave y y z (, ) = cos [ - w + e ] x t kx t o The B- field fom (5) y 1 Bz = -ò dt Þ o cos [ kx - w t + e] x c y = cbz : and B ae in phase B // Beam popagation diection

6 3.3 negy and Momentum A. The Poynting Vecto The electomagnetic wave tanspots enegy and momentum Hecht by YHL;100310; 3-6 The enegy density of an electic field : u e = 1 The enegy density of a magnetic field : u B B o = 1 mo Fom the elation = cb : u = u The total enegy density of an electomagnetic wave u = u + u B The powe density of an electomagnetic wave [W/m ] (enegy pe unit time pe unit aea) S = H B : Poynting vecto W / m A hamonic plane wave = o cos k - wt B = B cos k - wt o e j, e j e j e j S = c e B cos k - wt (1) o o o Time aveage of Hamonic Functions 1 t T + / ( ) º ò ( ) cos ( t) [ 1 sinc( T )cos( t) ] f t T T f t dt t -T / B. The Iadiance The time aveage of powe density Fom (1) and () I º S = c e B Þ H Þ c e T o o o o o o o 1 1 w = + n w» () T T >> 1/ n is moe efficient on a chage than B is called optical field The Invese Squae Law A point souce at the cente A spheical wave Fom the enegy consevation 4p I = 4p I, b g b g b g b g 1 1 = = const. 1 o 1 o lectic field ~ 1/, Iadiance ~ 1/ : Invese squae law

7 Hecht by YHL;100310; 3-7 C. Photons Light is absobed and emitted in photons engegy quanta Photons ae stable, chageless, massless. Photons exist only at the speed c. Photons can only be seen fom the esult of being ceated o annihilated. xpeiments on the quantum natue (1) Atoms emit localized light quanta in andom diections. A weak light souce suounded by the identical photodetectos at equal distances. Independent detections, one at a time. () Atoms ecoils when emitting photons. A beam of excited atoms Spontaneous emission of photons in andom diection Atoms ae kicked backwad The beam speads out. Radiation Pessue and Momentum Maxwell 1873 : Thee is a pessue in the diection nomal to the wave, which is equal to the enegy density. The electic field of a wave incident on a conducto Cuent on the conducto The wave s magnetic field geneates foces on the cuent. The adiation pessue,.p, by Maxwell.P S = u = u + ub Þ c S : Poynting vecto, c : speed of light powe foce speed foce = = aea speed aea speed aea xample The pessue of sola adiation on eath N / m (The atmospheic pessue: 10 5 N / m ) ~10 tons Viking spacecaft would have missed Mas by 15,000 Km without consideation of sola adiation pessue. Space ship diven by sola adiation pessue can be possible.

8 Hecht by YHL;100310; Radiation A stationay chage A constant -field, no B -field A unifomly moving chage and B fields but not coupled The souce of electomagnetic wave Nonunifomly moving chages Acceleated chages in a staight line Ciculating chages Oscillating chages : Linea acceleato : Cycloton : Radio antenna A. Linealy Acceleating Chages (a) A chage at est Unifom distibution of field lines (b) A chage with const. speed Nonunifom distibution of field lines Acceleating electon fom O 1 to O 4 Unifom acceleation Stationay at O fo t < 0. fom O 1 to O 4 Acceleation fom O to O 1. Linea move fom O 1 to O. The tansvese component stats to appea. (It changes in time in space and theefoe magnetic field is associated ) The adial component of the electic field : ~ 1 The tansvese component of the electic field : ~ 1 As, the tansvese component dominates. Radiation field

9 Hecht by YHL;100310; 3-9 Radiation by a linealy acceleating chage B. Synchoton Radiation Chages ae acceleated in a cicula path by a magnetic field and adiate in a naow cone. The fequency of the emission ~ The fequency aound the obit Tunable fom IR to X-ay The highe speed Fo v The shote backwad lobe, the longe fowad lobe.» c The polaization in the plane of motion. [Fig. 3.31] Synchoton adiation fom the Cab Nebula Due to ciculating chages tapped by magnetic field C. lectic Dipole Radiation An oscillating dipole The dipole moment p t po t b g = cos w : po : One plus and one minus chages vibate to and fo along a staight line = qd, (chage X max. sepaation)

10 Hecht by YHL;100310; 3-10 In the nea egion In the closed loop egion In the fa egion (adiation zone) The iadiance p Ibqg = 3p c 4 o w 3 eo sin q : has the fom of a static electic field : No specific wavelength ( is composed of 5 tems) : A fixed wavelength. and B ae tansvese, mutually pependicula, and in phase. p k k t o sin q cosb - w g = 4pe o : The highe the feq. w, the stonge the adiation

11 Hecht by YHL;100310; 3-11 D. The mission of Light fom Atoms The souce of emission and absoption of light The bound chages (paticulaly, the oute electons of atoms) Valence electons Light is emitted duing eadjustments of the oute chages. An atom is in gound state vey electon in the lowest possible enegy state. An atom is in excited state One o moe electons into a highe enegy level At low tempeatues Atoms in thei gound states At highe tempeatues Moe atoms in excited state due to atomic collisions (glow dischage, flame, spak and so on) nough enegy to an atom (By collision with anothe atom, electon, photon) xcitation of the atom The absobed enegy = The enegy diffeence between the initial and final states -8-9 The lifetime of the excited atoms : 10 ~ 10 seconds Relaxation to the gound state by emitting light o by eleasing themal enegy D = hn, esonance feq. Inteatomic collision The emission specta of single atoms o low pessue gases Shap lines The emission specta of solids o liquids Fequency bands

12 3.5 Light in Matte In a homogeneous, isotopic dielectic (nonconducting mateial) e e, m m v = 1/ em o o Hecht by YHL;100310; 3-1 The absolute index of efaction c em n º = Þ e m v o o e m n = e, Maxwell s elation m» 1, except fo feomagnetic mateials - ( m = fo diamond) The index of efaction depends on fequency Dispesion Scatteing and Absoption Inteaction of the incident light with atoms (1) Dissipative absoption at esonance feq. Photon s enegy = xcitation enegy of the atom Absoption of the photon by the atom Tansfe of the absobed enegy to themal enegy by collisions befoe emitting photons () Nonesonant scatteing (Loentz Model) Incident photon enegy < xcitation enegy of atom Oscillation of electon cloud at the same feq. as the incident light. No excitation, the atom is still in the gound state. (Oscillating dipole) Radiation at the same feq. with the same enegy as the incident light. (lastic scatteing) (3) Spontaneous emission xcitation Spontaneous emission xcitation -» 10 8 s An atom can emit 10 8 photons pe second The satuation, constantly emitting and eexcited, at I» 10 W / m An atom behaves as a souce of spheical electomagnetic waves.

13 Hecht by YHL;100310; 3-13 A. Dispesion Macoscopic view: A matte esponses to the electic o magnetic field via e o m Micoscopic view: The atom inteacts with electic field via electic dipole Applied electic field Distoted chage distibution Intenal field (xtenal) (lectic dipole moment) The electic polaization P D = e + P Þ e o : Dipole moment pe unit volume The oigin of polaization (1) Oientational polaization : Alignment of pola molecules in the electic field Pemanent dipole moment fom unequal shaing of valence electons (wate molecule) Pola molecules Rotation by Reduced esponse (heavy) Highe feq. xample Wate e» 80 fom 0 Hz to Hz, e falls off quickly above Hz () lectonic polaization : Distotion of electon cloud in nonpola molecules. lectons Good esponse to optical feq. (light) ( Hz ) Ÿ (3) Ionic o atomic polaization : Shift of the positive and negative ions in ionic cystals NaCl, The eq. of motion Restoing foce of the electon fo small x : F = - kx Fom Newton s second law i t d x dx qoe m ox m m dt - w = w + + g : w o = k / m dt Damping effect (negy loss duing oscillation due to Mass X Acceleation inteaction between neighboing atoms) Restoing foce Diving foce fom the incident wave The solution é ù q m x ( t ) = ê ú oe ê ( wo - w ) - igw ú ë û / -iwt (1) Without the diving foce (no incident wave) The oscillato vibates at the esonance feq. w o () Fo w = wo : t (3) Fo w? wo : x t b g and xbtg ae in phase b g is 180 o b g out of phase with t

14 Hecht by YHL;100310; 3-14 The electic polaization, The pemittivity q No / m P = qxon Þ, w - w - i gw ( o ) ( ) e = e P t q N / m o + t Þ e + w - w - i gw o ( ) ( o ) The dispesion elation n Nq æ 1 ö = 1 + eom ç wo - w - igw è ø > 1 fo w < w o < 1 fo w > w o A molecule with many oscillatos with diffeent esonant fequencies Nq æ f ö j n = 1 + å, eom ç j woj - w - i g jw è ø f j : oscillato stength, tansition pobability w oj : chaacteistic feq. Fo w Fo w Fo w << w oj : w can be neglected in the eq. constant n w oj : n gadually inceases with fequency (nomal dispesion)» w oj : The damping tem becomes dominant (stong absoption) dn < 0 (anomalous dispesion) dw Shaded egions: absoption bands Shaded egion: visible band (Note ise in UV and fall in IR) A mateial opaque nea the esonance fequency can be tanspaent at othe fequencies.

15 3.6 The lectomagnetic-photon Spectum In 1867, Maxwell published his theoy. The fequency band known at that time : IR-VIS-UV Hecht by YHL;100310; 3-15 A. Radiofequency Waves xpeiments by Hetz in 1887 Oscillatoy dischage (Oscillating electic dipole) Spak is seen Coppe point Bass knob Tansmitte Receive Hetz s expeiments Fequency in his expeiments : Focusing, eflecting, efacting of adiation. Measuement of polaization. Intefeence fo a standing wave and measuement of the wavelength. : A few Hetz ~ 10 9 Hz (Radiofequency) B. Micowaves ~10 Hz Molecules : lectonic, otational, vibational enegy levels Pola molecules Wate molecules : Stong otational esonances (Micowave oven.45 GHz) C. Infaed Hz ~10 Hz Nea IR Intemediate IR Fa IR xteme IR : 780 ~ 3,000 nm : 3,000 ~ 6,000 nm : 6,000 ~ 15,000 nm : 15,000 nm ~ 1.0 mm Any mateial can adiate and absob IR via themal agitation of its molecules. Molecules have both vibational and otational esonances in the IR xample ~ 1 of the electomagnetic enegy fom the sun is IR Moe IR than light fom light bulbs Human body : 3,000 nm peak nea 10,000 nm exteme IR

16 Hecht by YHL;100310; 3-16 D. Light Hz ~ Hz Souces of light Reaagnement of valence electons in geneal Themal adiation by incandescent mateials (hot, glowing metal filaments) Randomly acceleated electons Boad emission spectum Collisions Gas dischage Atoms of a gas xcitation Radiation lectic dischage, Seies of feq. lines. Ultaviolet ~10 Hz F. X-ays ~10 Hz Its wavelengths ae mostly smalle than an atom. G. Gamma Rays The highest enegy, shotest wavelength electomagnetic adiations Difficult to obseve wavelike popeties