Wave Phenomena Physics 15c

Transcription

1 Wave Phenomena Phyi 15 Leture 18 EM Wave in Matter (H&L Setion 9.7)

2 What We Did Lat Time! Reviewed refletion and refration! Total internal refletion i more ubtle than it look! Imaginary wave extend a few λ beyond the urfae! Studied how to reate EM wave! Aelerated harge radiate EM wave! Power given by Larmor formula! Proportional to (aeleration)! Intability of atom " QM! Polarization parallel to the aeleration! Explain Brewter angle µ q inθ r ET a( t ) 4π r µ qa P 6π a

3 Oillating Charge! Now we onider an oillating harge x i t x e ω! Tranvere omponent of E at (r,θ) i! What thi ign? i t a x ω xe ω! a point oppoite to x # Thi i where the minu ame from! But E T point oppoite to a! It more appropriate to write!! θ µ q inθ µ qω x r inθ ET a( t ) e 4π r 4π r E T point the ame diretion a x E T + r iω t q µ qω x inθ e 4π r E T r iω t

4 Power! Trivial to alulate power denity! NB: Mut take the real part before alulating a! S µ q ω x in θ 16π r 4 o r ( t ) { ω } Time average! Eay enough to alulate total power of radiation! Integrate over a phere at ditane r 4 π π µ q ω x π 3 P dφ S r inθdθ in θdθ 16π 4 µ q ω x 1π S 4 µ q ω x in θ 3π r Inreae with frequeny a ω 4

5 Rayleigh Sattering! Sunlight paing through the air make air moleule oillate! Inoming light ha a broad petrum! All frequenie are more or le equal E q F qe! Moleule radiate power aording to 4 µ q ω x P 1π! More power i aborbed and re-emitted at higher frequenie! Thi i why the ky look blue

6 Sunrie/Sunet! Air atter blue light! Sunlight loe blue a it travere atmophere! It turn red

7 Polarization! Sunlight ontain two polarization Viewed from right! Only one aue radiation that reahe the oberver! Sattered light (what you ee in the blue ky) i polarized! Photographer ue polarizing filter to deepen the olor of the ky

8 Goal for Today! We know an aelerated harge radiate EM wave! Matter i made of harged partile! EM wave paing through matter aelerate them! They radiate EM wave in return! What happen in the end?! It hould explain how EM wave behave in matter! In partiular, why doe it travel lower?! We put together many harge and aelerate them! And try to figure out what happen to the EM field around

9 Sheet of Charge! Imagine an infinite array of harge making a heet! All harge are oillating together a x i t x e ω E? z y x! What kind of EM radiation would they make at a ditane z from the heet?! Suppoe that there are n harge of +q per unit area

10 Sheet of Charge! Conider a little piee! Charge of thi piee i q n dxdy! E due to thi piee i dx dy z r y dx dy x µ qnω x inθ iω t Edxdy T e dxdy 4π r! Conider a imilar piee at x! y-z omponent anel r x θ x E r x We only need x omponent µ qnω x in θ iω t Edxdy x Edxdy T inθ e dxdy 4π r r

11 Radiation From the Sheet! Integrating E x over dxdy i pain r x + y + z r + + µ qnω x + + in θ iω t x Edxdy e dxdy 4π r! Calulation i diffiult, but irrelevant! We know that! E i parallel to the x axi! E doe not depend on x or y inθ y + r z Solution mut be plane wave in ±z diretion! Atual olution i Proportional to veloity in oppoite diretion E (, t z) x iω t iµ qnωx e µ qn z vt ( ) z

12 Wall of Charge! Imagine the heet ha a thikne dz! Denity of harge i n per unit volume " n dz per unit area! Plane EM wave are arriving from z! Suppoe E in move the harge a x qe k in Fore Spring ontant! Thi i impliti iω qe iωt! Veloity i v e k! We an alulate radiation from the heet z i t Ein Ee ω dz z

13 Radiation From the Wall! Charge in the wall radiate µ qn dz iωµ q n E z v t dze e z rad ( > ) ( ) k! Add thi to the inoming wave iωµ q n Ein + Erad 1+ dz Ee k! For mall dz, we an ue e α! E jut after the wall (z dz) i z iω t 1+ α +" z iω t dz z iωµ q n 1 dz µ q n dz iω t iω + dz k k iωt in + rad z dz ( ) E E e E e E e e E in

14 Thin " Thik Wall! Paing the thin wall hange the E field Ez i t E e ω! Add more wall Ez i t E e ω z Ndz 1 µ iω + iωt Ez dz Ee e E E e e z Ndz Ee q n k dz 1 µ q n iω + Ndz iωt k 1 µ q n iω t + z k Multiply the ame fator N time! Solution look jut like plane wave Propagation of plane EM wave through a thik wall of harged partile

15 Phae Veloity Ezt (, ) Ee 1 µ q n iω t z + k! We an alulate the phae veloity by ( ) 1 µ q n t C t + k z ont. C z 1 + µ! Differentiate: qn! That i n 1+ Index of refration!! We found k! p i maller than OK with Relativity! p i ontant No diperion q n dz p 1 µ q n µ q n q n dt + k 1+ k 1+ ε k ε k

16 Quik Summary! We tarted from aelerated harged partile! Eah of them radiate EM wave! Matter i denely populated by thoe! Inoming EM wave make them oillate! Colletive radiation from a thin heet make plane wave! Phae of the EM wave i hifted lightly by adding it! Aumulating the phae hift over finite thikne make plane wave again, but with modified phae veloity! We end up with a lower peed of light! We found origin of refration!

17 Corretion! There wa a mall omiion in the above diuion! Thin heet radiate EM wave in both ide! I ignored the bakward-going wave! Bakward wave ompliate analyi! But the onluion hange little! We till find plane wave with modified phae veloity! Index of refration turn out n + true 1 qn ε k intead of n 1+ qn approx. ε k! For material with mall n, n true n approx.

18 Maxwell Equation! Now we go bak to Maxwell equation E ρ ε B E t B 1 E B + µ J t! Movement of the harge in matter " Current q! We aumed E q E x v k k t! Uual trik with BAC-CAB rule give u 1 E µ qn E E + t k t J qn v Simple wave equation

19 Plane Wave Solution 1 µ qn +! Wave equation redue to! Diperion relation i! We found the ame olution E i( kx ωt) E E E k e t 1 µ qn k E ω + k k ω qn 1 + εk 1+! We ued the hort-ut by truting Maxwell J term! It an be made even impler p ω k qn ε k n 1+ E qn ε k Same n true

20 Maxwell Equation! Take the equation! We are auming q J qn v v k! We ould define ε E B ε µ + µ J t E t εµ E t qn ε + B k E µ qn E B εµ + t k t qn E ε + µ k t! We aborbed the J term into the matter permittivity ε! Now it eay to get ε n 1+ ε qn ε k

21 Quik Summary Again! We took three approahe to get the ame anwer! Miroopi: what doe eah eletron do?! Gave u a glimpe of really why light goe lower! Integration wa too tough! Current denity in vauum! Repreent eletron olletive movement by J! Wave equation i eay enough to olve! Permittivity of matter! Aborb J by replaing ε with ε! Solution i totally trivial! They repreent different level of abtration

22 Realiti Example! In the previou analyi, we ued a impliti model for the movement of the harge x qe k! Let try omething more realiti! We diu plama firt! I know it ound unfamiliar! But it a lot impler tuff than mot matter! Then we talk about more ordinary inulator! I ll take air for an example

23 Plama! Imagine a pae filled with free eletron! We neutralize the eletron ga by adding poitive ion! The ion are heavy Ignore their movement! Suh a mixture i alled a plama! You an make them by heating tuff up really hot! It rather eay to analyze! Eah eletron equation of motion i i t q m!! x qe qee ω x E e mω! Current denity iq n J qn v E mω e iωt iω t

24 Plama Frequeny! Wave Equation i 1 E J µ t t E + ω qn ω ω p! Diperion relation i k 1 1 εmω ω qn! ω p i alled the plama frequeny ε m! Plama i a diperive medium:! ω p i a ut-off frequeny iq n J qn v E mω ω µ qn k E E+ E m 1 ω p! For ω < ω p, k beome imaginary " Wave diappear p ω k ω e iωt

25 Phae and Group Veloitie ω ω( k) ωp + k ω k p 1 ω ω p ω p k g dω 1 ω p ω dk ω Slope p Slope g ω p! p i greater than, but g remain le than! I hope you remember thi from Leture #1

26 Ionophere! Sunlight ionize air in the upper atmophere! Ionophere ha free eletron denity n ~ 1 1 /m 3! Varie with unpot, eaon, day/night, latitude, et qn 7 ω p rad/ ε m ω p 6 ν p 9 1 Hz π! Viible light (ν ~ 1 14 Hz) pae through eaily! Shortwave radio (ν ~ 5 MHz) i refleted! You an hear BBC, Deuthe Welle, Voie of Ruia

27 Inulator! Unlike in plama or in metal, eletron in inulator are bound to the moleule! Binding fore i imilar to a pring! Ret of the moleule muh heavier " Ignore the movement! Equation of motion qe m kx i t m!! x qee ω k x! Fored oillation We know the olution i t e ω x x mω x qe k x qe q k mω E x m( ω ω )! Current denity i iω q n k iωt J qnv E e ω m( ω ω ) m M

28 Diperion Relation! Wave Equation i 1 E J µ t t E +! Diperion relation i k ω! It diperive again ω p k 1 + ρω ( ω ω ) ω µ qn ω k E E E m ω ω qn ω ω ρ εm( ω ω ) ω ω n 1 + ρω ( ω ω ) iω q n m( ω ω ) J qn v E ρ Idential to the LC tranmiion line diued in Leture #1 e qn ε k iωt

29 Example: Air! Air i a mixture of N, O, H O, Ar, et! Many reonane exit in UV and horter wavelength! Thing are impler in viible light p 1 + ρω ( ω ω ) + ρ 1! At STP, n 1.3 " ρ.6 n 1+ ρ 1+ ρ! ρ i proportional to denity n! Uing the ideal ga formula PV nmolrt P(atm) n + Index of refration of air T (K) 1 for low-freq. EM wave 1 ρ qn ε k

30 Mirage! In a hot day, air temperature i higher near the ground 73 P(atm) n 1+.3 T (K) 1! Light travel fater! Refration make light bend upward! You may ee the ground reflet light a if there i a path of water low fat ool hot T

31 Summary! Studied miroopi origin of refration in matter! Started from EM radiation due to aelerated harge! Uniform ditribution of uh harge make EM wave appear to low down! Three level of deribing EM wave in matter! Point harge in vauum! Current denity in vauum! Permittivity (and permeability) of matter! Analyzed EM wave in plama and in inulator n ω ω plama 1 p! Plama refletive for ω < ω p n 1 + ρω ( ω ω ) inulator ω p qn ε m