For free electrons, equate the force to the electrostatics force to obtain
where is the mass on the electron, r is the position vector, is the charge on the electron, and E is the electric field. Now, the current density is given by
where is the number density of electrons, so the change in current density with time is given by
Now, a plane wave,
satisfies the wave equation
where k is the wavenumberand is the angular frequency, so
Plugging (3)-(6) into
where is the speed of light, gives
