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.