The wave equation

In lih materials in regions with  r_f = 0   and  j_f = 0  both E  and B  satisfy the wave equation.

Sinusoidal plane wave solutions    exist.  We have

,

   (SI units),              (Gaussian units).

In regions with  r_f=0  and   j_f=s_cE  both E and B satisfy the damped wave equation.

The skin depth is defined as the distance it takes to reduce the amplitude by a factor of .

Plane waves at boundaries

are the boundary conditions the electromagnetic fields have to satisfy on any boundary.   For waves propagating across a dielectric-dielectric boundary or from a dielectric into a conductor we use From these boundary conditions we derive the law or reflection, q_i = q_r, and the law of refraction, (Snell’s law), n₁sinq_i = n₂sinq_t,  for a wave propagating from a medium with index of refraction n₁ into a medium with index of refraction n₂.  Here q_i, q_r, and q_t are the angles the incident, reflected, and transmitted wave vectors make with the normal to the boundary.

The plane of incidence is the plane containing the wave vector k_i and the normal to the boundary.  A linear polarized wave has p-polarization if E lies in the plane of incidence, and it has s-polarization if E is perpendicular to the plane of incidence.  The reflection coefficients for s and p polarization are , and the transmission coefficients for s and p polarization are .

Here       and        .

The reflectance is    and the transmittance is   

We have R+T=1  (energy conservation).

For conductors  are complex numbers.

Polarization by reflection

When q_i = q_B, the Brewster angle, then only light with s-polarization is reflected.  We have

Total internal reflection

If n₂<n₁ , then for  we have total internal reflection.  The wave vector k_t then has a component parallel to the interface which is real and a component perpendicular to the interface which is imaginary.

Reflection by a good conductor

For good conductors The wave does not penetrate into the conductor.  We then use the boundary conditions