Physics 218, Spring February 2004

Transcription

1 Physis 8 Spring February 004 Today in Physis 8: dispersion in onduting dia Semilassial theory of ondutivity Condutivity and dispersion in tals and in very dilute ondutors : group veloity plasma frequeny Aurora in the ionosphere over northern Canada seen from the International Spae Station Photo by Don Pettit 0 February 004 Physis 8 Spring 004 Semilassial theory of ondutivity On mirosopi sales ondutors differ from dieletris mostly in that many of the eletrons are free: there are no springs and shok absorbers binding them to their host atoms or moleules Thus the net fore on an eletron at x = 0 interating with i t a wave of light and its eletri field E = E0 e is dx d x F = qe γ 0 = a= dt dt and the equation of motion is d x dx q it + γ 0 = Ee 0 dt dt 0 February 004 Physis 8 Spring 004 Semilassial theory of ondutivity for whih the solution is obtained easily in the manner employed last leture: q E 0 it x = x0e x0 = + iγ0 The damping is provided by our old friend Ohmi resistane and if the onduting dium is isotropi we will need to refer to only one damping onstant γ 0 rather than a whole host of γ j In turn γ 0 will be larger or smaller if the rate at whih eletrons suffer ollisions is larger or smaller as we ll see in a mont 0 February 004 Physis 8 Spring () University of Rohester

2 Physis 8 Spring February 004 Semilassial theory of ondutivity ow under the influene of the passing wave the eletron moves with speed v= dx dt Suppose that the dium around the eletron onsists of atoms per unit volu eah of whih ontributes f 0 free eletrons Then the mobile harge density is ρ mobile = f0 q whene the urrent density is q Ee it 0 dx J = ρmobilev = = ( i) dt + i γ 0 = E γ0 i 0 February 004 Physis 8 Spring Semilassial theory of ondutivity But J = σ E so we have obtained here another expression for the ondutivity: σ = γ0 i There are two useful limiting ases to distinguish: tals (liquid or solid) for whih is large and the rate of ollisions of ondution eletrons and ions is also large so that γ 0 is large onduting gases for whih and γ 0 are muh smaller 0 February 004 Physis 8 Spring Condutivity in tals In tals γ 0 and the real part of the ondutivity is muh larger than the imaginary part: i artanγ f 0 0q σ = e γ0 + γ 0 Compare this with the expression we derived in PHY 7: nq σ = t = t where n was the number of eletrons per unit volu and t was the average ti between ollisions Evidently γ 0 is the average rate of eletron ollisions: γ 0 = t 0 February 004 Physis 8 Spring () University of Rohester

3 Physis 8 Spring February 004 Flashbak: ollisions drift veloity and ondutivity (PHY 7 8 ovember 00) Consider a hunk of tal with mobile harges in it and an eletri field E present If it has been a ti t i sine partile i last suffered a ollision and if it left that ollision at speed v i then the montum of this partile is pi = mvi + qeti So a snapshot of the tal would reveal an average value of arrier montum given by p = mv = i ( i i) p = mv + qet i= i i= The starting speeds v i are endowed by ollisions with the fixed ions; their energy in turn os from the thermal energy (heat) of the dium 0 February 004 Physis 8 Spring Flashbak: ollisions drift veloity and ondutivity Thermal motions in a solid or liquid are (essentially) random in magnitude and diretion so if is large mvi = 0 ; thus i= qe qe v = ti = t m m i= Drift veloity nq t and J = ρv = nqv = E σe m where n is the number density of arriers Clearly J should be linear in E if the arrier veloities are mostly thermal and if ollisions take plae 0 February 004 Physis 8 Spring Condutivity in very dilute ondutors In the limit of small damping γ 0 the ondutivity is purely imaginary: σ i This is what you get in gases under two onditions that are worth distinguishing: ionized gases in whih ollisions (lose interations of pairs of harges) still provide most of the resistane plasmas ionized gases in whih ollisions are so infrequent that the motions of eletrons are olletive: the fields of all the harges in an eletron s neighborhood are important in determining its motion 0 February 004 Physis 8 Spring () University of Rohester 3

4 Physis 8 Spring February 004 Let s start with the expression for the omplex wavenumber that we got diretly from the wave equation for the fields in onduting dia (leture 3 February): 4 k πσµ = µε + i Assu that the gas is suffiiently rarefied to take µ = ε = ; then 4π k = + i i 4π p = = = k m 0 February 004 Physis 8 Spring where we have defined the (angular) plasma frequeny: p 4π m Dilute ondutors provide us with the simplest example of dispersion (whih you ll rember is a variation of refrative index with frequeny) We shall now onsider two new features of dispersion using this example Wave speed With k = n the phase veloity of light in a dilute ondutor is v= = = > (!!!) k n p ( ) 0 February 004 Physis 8 Spring 004 ow we experts in the speial theory of relativity all know that ause-and-effet relationships an hange no faster than by propagation at the speed of light in vauum But any hange of this wave suh as would onstitute a signal would have to be due to a hange in the wave s amplitude or energy and suh hanges turn out not to propagate at the phase veloity To see this onsider broadasting a plane wave for whih we have sohow modulated the amplitude sinusoidally at so angular frequeny mod : 0 February 004 Physis 8 Spring 004 () University of Rohester 4

5 Physis 8 Spring February 004 ikz ( t) ik ( ) ( ) ( modz modt) ikz ( t E z t = E ) 0 z t e = E0mode e The peaks and troughs of the wave itself move of ourse at v = k but the peaks and troughs of the modulation move so as to keep kmodz modt = onstant ; kmoddz moddt = 0 That is dz mod vg = dt kmod We don t know a priori what k mod is But we do know as a funtion of k and an write mod = = ( k ) ( k) kmod = k k 0 February 004 Physis 8 Spring And sine mod we an express as a Taylor series about and neglet all but the first two terms: vg = = + ( k k) + k k k k dk Group veloity dk For our dilute ondutor dk d vg = = = p dk d = ( p ) = ( p ) 0 February 004 Physis 8 Spring vg = ( p ) = p < (!!!!) Thus intensity hanges and other signals travel at a speed smaller than even though the waves that make up the signal travel faster than Relativity still works ote that for nondispersive dia = = < always k dk n 0 February 004 Physis 8 Spring () University of Rohester 5

6 Physis 8 Spring February 004 Opaity for < p For angular frequenies less than the plasma frequeny < 0 so the square root of this quantity is purely imaginary: and waves are strongly attenuated in this dium: the dium is opaque p i k = p = p z ikz ( t ) p it E = E 0e = E 0 e e ; 0 February 004 Physis 8 Spring This strong attenuation is a manifestation of a very large resistive part to the dium s impedane Sine vauum has only a non-resistive impedane the mismath between vauum and plasma would result in strong refletion by the plasma of light at this frequeny ( < p ) A good example of a plasma is the ionosphere: the partially-ionized upper reahes ( km) of Earth s atmosphere Here the free eletron density is of order 5-3 f 0 = 0 m for a plasma frequeny of p 4π 6 ν p = 3 0 Hz 3 MHz π = π m = = e 0 February 004 Physis 8 Spring Thus the ionosphere is a good refletor for AM radio frequenies (500 khz 6 MHz) but a poor refletor for FM (87 MHz 0 MHz) This explains several phenona familiar to those who still listen to the radio: AM radio stations are long-ranged and FM stations aren t That s beause an AM signal from over the horizon an reflet off the ionosphere and get to your radio; the FM signal just keeps going off into spae Intense solar ativity ruins AM radio reeption Solar flares bombard Earth with lots of ionized gas severly disrupting the ionosphere and hanging the plasma frequeny all over the plae Thus the refletion and AM reeption are inonsistent 0 February 004 Physis 8 Spring () University of Rohester 6

7 Physis 8 Spring February 004 You an reeive very distant AM stations at night that you an t get during the day The ionization is produed by sunlight At night the eletrons and ions reombine This reombination takes plae muh faster the denser the gas is The gas is denser lower in the atmosphere so reombination proeeds from the bottom up the refleting surfae retreats to higher altitude at night Thus a single refletion an reah you from farther away When the sunlight os bak the edge of the ionosphere returns to the lower altitude Too bad nobody listens to the radio any more 0 February 004 Physis 8 Spring () University of Rohester 7