9 Wave solution of Maxwells equations.

Transcription

1 9 Wave solution of Maxwells equations. Contents 9.1 Wave solution: Plane waves 9.2 Scalar Spherical waves 9.3 Cylindrical waves 9.4 Momentum and energy of the electromagnetic field Keywords: Plane, cylindrical and spherical waves. Poynting vector Ref: J. D. Jackson: Classical Electrodynamics; A. Sommerfeld: Electrodynamics. 9.1 Wave solution: Plane waves For finding solutions of wave equation (3.14), let us assume E = E(z,t) = E x (z,t)i+e y (z,t)j+e z (z,t)k, (9.1) wherei,jandkareunitvectorsalongx,y andz directionsrespectively. With the above assumption, (9.1), Maxwell equation (3.1) becomes, E = E z z = 0. (9.2)

2 2 9 Wave solution of Maxwells equations. The above condition makes the z-component of the wave equation as 1 c 2 2 E z t 2 = 0. (9.3) Since we are looking for periodic solutions, equation (9.3) shows that the longitudinal polarisation, E z = 0. At this point we make another simplification and choose E y = 0, which then trivially satisfies the y-component of the wave equation. Now we are left with the x-component of the wave equation which reads Substitution of a trial periodic solution, 1 c 2 2 E x t 2 2 E x z 2 = 0. (9.4) E x (z,t) = Acos(kz ωt+α), (9.5) in (9.4) demands, ω 2 = c 2 k 2. (9.6) The relation (9.6) is known as the dispersion relation for light. It shows that the angular frequency, ω, of the electromagnetic wave is linearly proportional to the wave number, k = 2π/λ. So we see that in this simplistic assumption we have a sinusoidal travelling wave in the z direction whose amplitude is in the x-direction. Now we turn our attention to the components of the Maxwell

3 9.2 Scalar Spherical waves 3 equation (3.2). With the above solution, (9.5), we have ( E) x = y E z z E y = 0 B x = 0, (9.7) ( E) y = z E x x E z = Aksin(kz ωt+α) = t B y B y = A c cos(kz ωt+α) = E x c, (9.8) ( E) z = x E y y E x = 0 B z = 0. (9.9) The above shows that the magnetic field is polarised in the y-direction. So for the complete wave the electric field is confined in the x-z plane whereas the magnetic field is confined in the y-z plane and the wave is travelling in the z-direction. Problem 1: Plot a snapshot of the above wave in x-y-z space for t = 0, α = 0, k = π, showing the E and B fields. Problem 2: Plot the same wave in x-y-t space for x = 0.5,y = 1 and z = Scalar Spherical waves An ideal point source creates a spherical wave. Let us see how a spherical wave satisfies wave equation. First we separate the temporal part of the wave, like it was done in (9.5) and substitute in the wave equation, ( 2 + ω2 c2)ψ = 0 (9.10) We now assume that the wave is dependent only on r, the magnitude of the radial co-ordinate, so that it is spherically symmetric. The above choice

4 4 9 Wave solution of Maxwells equations. simplifies the equation (9.10) to, 2 (rψ) r 2 + ω2 c2(rψ) = 0 (9.11) Which gives the spherical wave as, ψ = A 0 r exp(±iω(r c t)), (9.12) where A 0 is the source strength. Notice that the r in the amplitude would reduce the intensity as 1/r 2 (intensity being proportional to the square of the amplitude). Problem 3: Complete the intermediate steps of the above solution. 9.3 Cylindrical waves In case of cylindrical wave there is a symmetry about any line, usually this line is taken as the z-axis in the literature. They may be produced by a very long source of of negligible width. So one works with the cylindrical co-ordinates (ρ, φ and z) in this case. Now again separating the temporal part and assuming that the dependence is only on ρ, we have The solution can be written as, 2 ψ ρ + 1 ψ 2 ρ ρ + ω2 c2ψ = 0 (9.13) ψ = B 0 J 0 ( ω c ρ)e iωt, (9.14)

5 9.4 Momentum and energy of the electromagnetic field 5 where J 0 is cylindrical Bessel function of zeroth order and B 0 is the strength of the source. There is another solution to the above equation but it is infinite at the origin. For large distance from the source the solution may be approximated by ψ = B 0 ρ cos(ω( ρ c t) α), (9.15) where α is some phase. Problem 4: Fill the intermediate steps of the cylindrical wave solution, and show that asymptotically the solution reduces to (9.15). 9.4 Momentum and energy of the electromagnetic field To create an electromagnetic field one needs energy, and the if the field is dynamic like the case of electromagnetic waves it would also have momentum. The conservation of energy is often called Poynting s theorem. For a single charge q moving in an electromagnetic field (E, B) the instantaneous force on the charge is the Lorentz force, F = q(e+v B), where v is its instantaneous velocity. Since the magnetic force is always perpendicular to the velocity there is no work done by the magnetic field. So the rate of work done is given by qv E. The quantity qv is nothing but the current density j so the rate of workdoneisequaltoj E. Tofindthetotalrateofworkdoneforacontinuous

6 6 9 Wave solution of Maxwells equations. charge and current distribution one has to integrate over a finite volume. j E d 3 x (9.16) If we now eliminate j using the fourth Maxwell equation (3.4) we have, [ j E d 3 x = E ( H) E D ] d 3 x (9.17) t Using the vector identity, (E H) = H ( E) E ( H), (9.18) and Faraday s law, we have, j E d 3 x = [ (E H)+E D B +H t t Now for LIH we assume that total energy density as ] d 3 x. (9.19) u = 1 2 (E D+B H) = 1 2 (ǫe E+µ 1 B B) = 1 2 (ǫe2 +µ 1 B 2 ). (9.20) The (9.19) now becomes j E d 3 x = [ (E H)+ u ] d 3 x. (9.21) t This equation can be written in a differential continuity equation since the volume is arbitrary, u t + S = j E, (9.22) where the vector S = E H is known as Poynting vector. It represents the energy flow of an electromagnetic field or the momentum density of the

7 9.4 Momentum and energy of the electromagnetic field 7 field. The theorem in either (integral or differential) form says that the rate of change of energy of an electromagnetic field in a certain volume plus the energy flowing out of the volume is equal to the negative of the rate of total work done by the field on the sources within the volume. Problem 5: Evaluate the Poynting vector and energy density for a plane electromagnetic wave.