## The Electromagnetic Wave Equation

We can use Maxwell's equations to derive the wave equation that electromagnetic waves obey.

We start from the equation

Substitute into this the equationrearranged to giveand the equationWe obtain

We take the curl of both sides of this equation to obtain

Assumingandto be constant allows us to write

Now use the vector identityto obtain

Free space has no charge soto give

A similar procedure can be followed to obtain

These are both wave equations, with the wave speed give by

In a vacuumandThey are given the symbolsand respectively. In other materialsThis means that the speed of electromagnetic radiation is less in these materials.