For free electrons, equate the force to the electrostatics force to obtain

(1)

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

(2)

where is the number density of electrons, so the change in current density with time is given by

(3)

Now, a plane wave,

(4)

satisfies the wave equation

(5)

where k is the wavenumberand is the angular frequency, so

(6)

Plugging (3)-(6) into

(7)

where is the speed of light, gives

	(8)

and

(9)