Electromagnetic Wave Equation

The wave equation for a plane electric wave is

with the same form applying to the magnetic field wave in a plane perpendicular the electric field. The wave equation for electromagnetic waves arises from Maxwell's equations. The form of a plane wave solution for the electric field is

and that for the magnetic field

To be consistent with Maxwell's equations, these solutions must be related by

Energy in Electromagnetic Waves

Electromagnetic waves carry energy as they travel through empty space. There is an energy density associated with both the electric and magnetic fields. The rate of energy transport per unit area is described by the vector

which is called the Poynting vector. This expression is a vector product, and since the magnetic field is perpendicular to the electric field, the magnitude can be written

A condition of the wave solution for a plane wave is so that the average intensity for a plane wave can be written