Maxwell's Equations and Electromagnetic Waves in Vacuum: Identical to Elastic String Waves, Study notes of Physics

Download Maxwell's Equations and Electromagnetic Waves in Vacuum: Identical to Elastic String Waves and more Study notes Physics in PDF only on Docsity! BK 3 1 Background, III Electromagnetic waves We now turn to a second example of a field theory—electromagnetism. You are probably familiar with the summary of electric and magnetic phenomena known as Maxwell’s Equations. In their integral form these are: ! 0 ! E • d ! A =" Q ! B • d ! A = 0" ! E •" d ! l = # d dt ! B " • d ! A 1 µ 0 ! B " • d ! l = I +! 0 d dt ! E • d ! A " Note that we are using ! E to represent electric field because E is going to be reserved for energy. The two integrals— ! ! !" —in the first line are over closed surfaces, while in the second line are over closed loops. The ordinary integral signs in the second line are over open surfaces defined by the loops on the left hand sides. The quantities Q and I are charge and current, respectively, and µ0 and ε0 are fundamental constants that describe the magnetic and electric properties of free space. James Clerk Maxwell’s sole contribution to these equations is the second term on the right hand side of the very last equation—a term he introduced based on symmetry, something like “if time-changing magnetic fields can make electric fields, why not the other way around?” We are going to restrict our discussion of electromagnetism as follows: (a) everything takes place in a vacuum—that is, Q = 0, I = 0; (b) ! E points in the y-direction and ! B points in the z- direction, as shown in the figure to the right; (c) ! E and ! B have planar symmetry, meaning that their components only depend spatially on x not on y or z: i.e., E = E(x, t) and B = B(x, t). (Please note the arrows have been erased indicating that these are components, not vectors.) It is possible to rewrite Maxwell’s Equations in integral form as partial differential equations by restricting the integrals to infinitesimal loops and infinitesimal areas. Without going through the pretty straightforward details, the results—for the geometry described above—can be summarized as follows: the first line of Maxwell’s integral equations give no information, both becoming 0 = 0 (it’s not that the fields are zero, it’s that the sums of the dot products over the surfaces are); on the other hand, the second line becomes, respectively, !E !x = " !B !t , 1 µ0 !B !x = "#0 !E !t If we differentiate the first equation with respect to x and the second with respect to t and eliminate the mixed second partials of B we see that the electric field satisfies the wave equation, provided v W 2 = 1 µ 0 ! 0 . Alternatively, differentiating the first with respect to t and so forth shows BK 3 2 that the magnetic field also satisfies the wave equation (with the same wavespeed). Thus, Maxwell’s first order differential equations have waves encoded in them. (They are like the components of a “Dirac” form of a second order partial differential equation. As with the string example, we could write the Maxwell Equations as one first order equation, the solution of which would be a large array containing both electric and magnetic fields. In fuller geometric generality, where the electric and magnetic fields are both vectors, the resulting solution would have six entries.) The first order equations of motion for transverse waves on a string are !" !t = !v !x and !v !t = v W 2 !" !x = # µ !" !x . If we make the identifications E !"v, B !# , $ 0 ! µ , 1/µ 0 !% (careful: don’t confuse µ, the mass density in the string, with µ0, the magnetic permeability of free space), we see that the two Maxwell partial differential equations are formally identical to the two partial differential equations of an elastic string. Transverse elastic waves on a string and electromagnetic waves (in the restricted geometry defined in the figure) are mathematically identical. The associated wave speed is v W = ! µ " 1/µ 0 # 0 = c , the speed of light. Please note that without Maxwell’s guess, the right hand side of the second partial differential equation would be zero and there would be no electromagnetic waves! That is, no light, TV, radio, microwaves, X-rays, …! The associated energy density for electromagnetism can be written down from the correspondences: 1 2 (µv2 + !s2)" 1 2 (# 0 E 2 + 1 µ 0 B 2) . Then there’s the gauge freedom thing. Remember, lurking behind the physically relevant v and σ fields for the string is another “potential” field, q, whose derivatives yield v and σ. The same is true for ! E and ! B . There are actually two hiding potential fields for electromagnetism, one a scalar called the “electric potential,” φ , the other a vector called the “electromagnetic vector potential,” ! A . For the restricted geometry in our example here we can set φ = 0 and E = ! "A "t B= "A "x Because the physical fields are derivatives of A, A can take on infinitely many values and this ambiguity is called electromagnetic gauge freedom. Phonons and photons The previous notes offered a few cryptic comments about “quantum field theory.” Although QFT is a very complicated mathematical business, its essence can be understood with what we have so far plus a little quantum mechanics. Imagine a string of length L or, equally well, an uncharged capacitor—an electromagnetic “cavity”—whose plates are separated by L. The boundaries in these two situations force the associated elastic or electromagnetic waves to “stand,” i.e., to have fixed nodes that are separated by certain allowed distances related to L. For example, if the boundaries require some of the fields to vanish at x = 0 and L then the allowed wavelengths are 2L/l , where l can be 1, 2, 3, … . Because the wave speed is determined by τ and µ for a string or by ε0 and µ0 for electromagnetism, wavelength times frequency is constant