Electromagnetic waves propagate much differently in conductors than they do in dielectrics or in vacuum. If the resistivity of the conductor is sufficiently low (that is, if it is a sufficiently good conductor), the oscillating electric field of the wave gives rise to an oscillating conduction current that is much larger than the displacement current. In this case, the wave equation for an electric field:

Del(ρ/ε₀) - (Del)²E = -dμ₀J/dt

Explanation:

From Maxwell's fourth equation

Curl B = μ₀J + μ₀ε₀dE/dt (1) where the second term is the displacement current.

If the oscillation conduction current in the conductor is much larger than the displacement current then, the displacement current goes to zero. So we have

Curl B = μ₀J  (2)(since μ₀ε₀dE/dt = 0)

From maxwell's third equation

Curl E = -dB/dt  (3)  

taking curl of the above from the left

Curl(Curl E) = Curl(-dB/dt)

Curl(Curl E) = (-d(CurlB)/dt)  (4)

Substituting for Curl B into (4), we have

Curl(Curl E) = -dμ₀J/dt

Del(DivE) - (Del)²E = -dμ₀J/dt    (5)

From Maxwell's first equation,

DivE = ρ/ε₀

Substituting this into (5), we have

Del(ρ/ε₀) - (Del)²E = -dμ₀J/dt

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