1. The electric field of a plane electromagnetic wave propagating in free space is described by

Determine the corresponding magnetic field and the time average Poynting vector for the wave.

The wave is propagating in +x direction so that the propagation vector is . Using Faraday’s law, we have,  which gives magnetic field directed along the z direction,

The Poynting vector is given by

The time average of the Poynting vector is .

2. Show that an s –polarized wave cannot be totally transmitted to another medium.

Considering Fresnel equations for s polarization,

For total transmission , so that . From Snell’s law, we have, . Squaring and adding, we get , which is not correct.

3. An electromagnetic wave   given by

is incident on the surface of a perfect conductor and is totally reflected.  The  incident and the reflected waves combine and form a pattern. What is the average Poynting vector?

The incident wave is

Since the tangential component of the electric field must vanish at the interface (z=0), the reflected wave us given by

The corresponding magnetic fields are given by

The individual Poynting vectors can be calculated and on adding it will turn out that the average Poynting vector is zero. The waves of the type obtained by superposition of the two waves have the structure,

where the space and time parts are separated. These are known as “standing waves” and they do not  transport energy.