An electromagnetic wave in a dielectric
In Eq. (9.17) we wrote out Maxwell's equations for the electric and magnetic fields in vacuum, including source terms – charge density ρ and current density J. Now we want to consider an electromagnetic field in an unbounded dielectric medium. The dielectric is a perfect insulator,
508 Electric fields in matter
we shall assume, so there is no free-charge current. That is, the last term on the right in Eqs. (10.76) through (10.79), the free-charge current density J, will be zero. No free charge is present either, but there could be a nonzero density of bound charge if div E is not zero. Let us agree to consider only fields with div E = 0. Then ρ, both bound and free, will be zero throughout the medium. No change is called for in the first induction equation, curl E = −∂B/∂t. For the second equation, we now take Eq. (10.77) without the free-charge-current term: curl B = μ0ϵ (∂E/∂t). The dielectric constant ϵ takes account of the bound-charge current as well as the vacuum displacement current. Our complete set of equations has become
curl E = −∂B/∂t, div E = 0;
curl B = μ0ϵ∂E/∂t, div B = 0. (10.80)
These differ from Eq. (9.18) only in the replacement of ϵ0 with ϵ in the second induction equation.
As we did in Section 9.4, let us construct a wavelike electromagnetic field that can be made to satisfy Maxwell's equations. This time we give our trial wave function a slightly more general form:
E = z^ E₀ sin(ky−ωt),
B = x^ B₀ sin(ky−ωt). (10.81)
The angle (ky−ωt) is called the phase of the wave. For a point that moves in the positive y direction with speed ω/k, the phase ky−ωt remains constant. In other words, ω/k is the phase velocity of this wave. This term is used when it is necessary to distinguish between two velocities, phase velocity and group velocity. There is no difference in the case we are considering, so we shall call ω/k simply the wave velocity, the same as v in our discussion in Section 9.4. At any fixed location, such as y = y₀, the fields oscillate in time with angular frequency ω. At any instant of time, such as t = t₀, the phase differs by 2π at planes one wavelength λ apart, where λ = 2π/k.
The divergence equations in Eq. (10.80) are quickly seen to be satisfied by the wave in Eq. (10.81). For the curl equations, the space and time derivatives we need are those listed in Eq. (9.24) with small alterations:
curl E = x^ E₀ k cos(ky−ωt), ∂E/∂t = −z^ E₀ ω cos(ky−ωt),
curl B = −z^ B₀ k cos(ky−ωt), ∂B/∂t = −x^ B₀ ω cos(ky−ωt).
(10.82)