Electromagnetic Wave Equation

Consider the differential form of the four Maxwell's equations in a vacuum, which mutually relate the electric and magnetic fields.

Separate equations for the electric and magnetic fields can be found by applying the curl to both sides of the third and fourth equations, using geometrical identities, the first and second equations, the rule of partial derivatives, and the third and fourth equations.

Simplification reveals that the electric and magnetic fields follow the same equation. It implies that each Cartesian component satisfies the wave equation. So, the electromagnetic fields are three-dimensional waves.

They propagate in a vacuum with a speed determined by natural constants.

Assume the wave traverses in the z-direction. The general solutions for the fields are written.

To these solutions, the first and second Maxwell's equations are applied. Simplification reveals that the fields' z components are zero. So, electromagnetic waves are transverse waves.

Maxwell's third equation applied to the solutions gives relations between their x and y components, which can be clubbed in vector form. So, the traveling electric and magnetic fields are mutually perpendicular.