Electromagnetism Spring 2018, NYU

Transcription

1 Eectromagnetism Spring 08, NYU March 6, 08 Time-dependent fieds We now consider the two phenomena missing from the static fied case: Faraday s Law of induction and Maxwe s dispacement current. Faraday observed that a transient current was induced in a test circuit if: the steady current fowing in a nearby circuit was turned on off, The adjacent circuit with stationary current was moved reative to the test circuit, 3 a permanent magnet was moved coser further from the test circuit. Faraday interpreted the transient current in the test circuit in each of these cases due to the charge in the magnetic fied inked by the test circuit: the charging magnetic fux induces an eectric fied around the test circuit, whose circuation is the eectromotive fce ξ, which in turn causes a current fow as dictated by Ohm s aw. This can be summarized as: ξ = k df dt where k is a constant, and the minus sign refects enz s aw: the direction of the induced current driven by the eectromotive fce EMF ξ produces a magnetic fied, which tends to oppose the change in magnetic fux produced it. We now write ξ = E d where E is the eectric fied induced at test circuit, and F = B ds S any surface that has as boundary 3 Then: S ξ = E d = E de = d S dt S B ds B = k d S 4 where the second equation is due to stokes aw and the ast equation is fixing circuit. Maxwe argued that this reation shoud be vaid everywhere in space irrespective if one put a test circuit not, then we have the differentia fmat Faraday s induction: E = k B that is, in any point in space where the magnetic fied changes with time, a rotationa eectric fied is generated. To determine k, et s consider what happens under Gaiean transfmations good f ow veocities compared to c. 5

2 We start from the Lentz fce: F = q E + v c B, where v is the veocity of charge q in system S. Gaiean covariance means that the fm of the Lentz fce is the same in another system S that moves at veocity u unifmy with respect to S F = q E + v c B v = v u 6 where we assumed as known experimentay, that q is invariant. Then F = q E v u + B c = q E uc B + q v c B = qe + q v c B 7 where the ast equation is due to Gaiean invariance crect as we wi see ater to Ov /c. Setting the terms that depend on v to be equa since v is arbitrary { B = B + Ou/c E = E + u c B + Ou /c 8 Let s now go back to Faraday s Law and see how Gaiean invariance fixes k /c. Consider the test circuit moving with respect to the ab frames with veocity u as above, the test circuit is at rest in S E d = k d B ds dt 9 Looking at it in S, the circuit is moving so d dt = + u, : d B ds = B dt S S d S + u B ds 0 S But B u = B u u B + u B B u = u B 0 = u B d dt S B ds = S S [ E ku B] d = k B d S u B d S B d S But now we can interpret this as Faraday f a fixed circuit in S cresponding to instantaneousy, which woud read E d B = k S d S 3 and Gaiean invariance means that E = E + ku B 4 and from transfmation derived befe we must have k = c.

3 Let s now consider the remaining Maxwe s equations. We have E = 4πρ B = 0 E = B c B = 4π c j 5 However as they stand, these equations have an inconsistency f time-dependent fieds: since B = 0 j = 0, but j + ρ = 0, so in der to aow f ρ 0, we must modify B eqn. It was Maxwe who first proposed how to do this: keeping B = 0 no monopoes to be consistent with Gauss s aw: j + ρ = j + E = j + E = 0 6 4π 4π j E 4π 4π c j 4π c j + E c me generay f materia media, using again j free + ρ free = 0 { D { = 4πρ free B = 0 E = B c H = 4π c c j free + D c 7 8 D The extra term c of EM waves and radiation, as we sha see. In the absence of sources and materia media we have simpy that Maxwe added he caed it dispacement current. It is crucia in his prediction B = c E 9 B Comparing to Faraday s aw E = c, we see that if Lentz + Faraday gave us E = E + u c B, from B = E c one woud expect B = B u c E. In fact, as we noted, using the crect transfmation to the Lentz fce to Ou /c, we can derive precisey this. Therefe, the eading der transfmations of the EM fieds are: { B = B u c E E = E + u c B 0 Let s consider what happens with the description of EM fieds by potentias φ, A in the time-dependent case. Since B = 0 sti, we can define again B = A But now since E is no onger potentia we have E = c B = A c 3

4 Now the potentia vect is instead: E + c A = 0 3 Then we redefine the scaar potentia through: E + c A E = φ c φ 4 Let s see what the remaining Maxwe equations impy f the potentias. F this purpose, assume no media so D = E, H = B, so we have: E = φ A = 4πρ 6 c φ + c which generaizes Poisson. The remaining Maxwe equation says : A B = A = A A 4π = c j + c 5 A = 4πρ 7 φ c A 8 A A c = A + φ = 4π c c j 9 which again generaizes Poisson in Couomb gauge f A in the static case. These two fied equations f φ, A can now be decouped by using the Lentz gauge: which eads to: A + c φ = 0 30 A c A A = 4π c j 3 φ φ c φ = 4πρ 3 that is, the potentias φ and Cartesian components of A satisfy the inhomogeneous wave equation in the Lentz gauge. One can easiy verify that in a inear and isotropic medium characterized by ɛ and µ, these equations get simpy modified by the repacement: c ɛµ c 33 4

5 In this case the wave speed is just v c ɛµ = c n, where n = ɛµ is the refraction index. Ceary, in absence of sources φ, A satisfy the homogeneous wave equation, which ed Maxwe to the prediction verified 5 years ater by Hertz that EM waves propagate in vacuum with speed c. Athough the resut above f φ, A is in the Lentz gauge, physica resuts shoud be independent of gauge choice. So et s derive the wave equation f E, B which are gauge independent. From: { E = B c E = 0 B = E c B 34 = 0 We have E B Then, indeed { = E = B E = c B = E c B = c E = 35 B c E E c = E = 0 B B c = B = 0 Let s consider gauge transfmation, i.e. transfmations of φ, A that give rise to the same E, B: φ φ, A A such that { E = φ A c = φ c A c B = A = A F A is simpe: A = A + Λ, where Λ is an arbitrary scaar fied. This yieds: E = φ c A + Λ = φ A c φ c φ = φ c 38 = φ 39 Λ Λ 40 Now suppose we find φ, A so that they satisfy Lentz gauge: Λ c Λ A + c φ = 0 4 = Λ = A + φ c In other wds, if A, φ aready satisfied Lentz Gauge, Λ satisfies the wave equation. Therefe once in Lentz gauge we can aways add soutions of wave equation to φ, A and we wi stay in Lentz gauge. Thus Lentz gauge does not uniquey specify φ, A. But it has the advantage of being reativistic invariant. 4 5