Question

From Maxwell’s equations of electromagnetism in free space show that electromagnetic plane waves are transverse. For a plane electromagnetic wave travelling along the z-axis in free space, show that

Answer 1

We consider the case of plane waves travelling in the z-direction (along  k , the direction of propagation of the wave) so that the wave fronts are planes parallel to the xy-plane. If the vibrations are to be represented by variations of vector E and vector H , we see that in any wave front they must be constant over the whole plane at any instant, and their partial derivatives with respect to x and y must vanish. None of the components of E and H depends on either of the transverse coordinates x and y. Now,

which says that E_z is independent of z. That E_z is also independent of t can be seen by considering Maxwell’s Eqn. (11.4) in free space:

Thus, E_z is a constant. For simplicity, we take this constant to be zero. Similarly, we can show that Hz is a constant and we again take H_z to be zero. Thus, we conclude that apart from the nonwave like constant fields, the electromagnetic plane waves are transverse waves. Thus, the electric and magnetic fields are perpendicular to the direction of propagation k (z-direction):

(i) We take the x-component of Maxwell’s equation (11.3) and the y-component of Maxwell’s equation (11.4):

Thus, E_y and H_x are coupled. (ii) Similarly if we take y-component of Eqn. (11.3) and the x-component of Eqn. (11.4), we shall find that E_x and H_y are coupled: