5.1.1 Starting points
We begin by making some simplifying assumptions about the electric field. This is legitimate because we are not looking for the most general solution to Maxwell's equations, but only for special solutions that exhibit wave-like behaviour. We will ultimately check that our solutions for the fields satisfy all of Maxwell's equations, and hence obtain retrospective support for our initial assumptions.
If you drop a pebble in a pond, waves spread out in all directions on the surface. Many electromagnetic waves spread out radially like this, but we will consider a disturbance that propagates in a fixed direction, like the parallel beam from a searchlight. We will take the direction of propagation to be the z-axis. For simplicity, we assume that the electric field depends only on z and t, and does not depend on x or y at all. At any given instant, the surfaces on which the electric field has a constant value are planes perpendicular to the z-axis. These planes are infinite in extent, corresponding to an infinitely wide beam. Disturbances of this type are called plane waves. We will also assume that the electric field oscillates along a fixed direction. Disturbances of this type are called linearly polarised waves.
With these assumptions, the electric field takes the form
where f (z, t) is some (as yet unspecified) function of z and t and u = uxex + uy ey + uzez is a fixed unit vector. This electric field, and any associated magnetic field, must satisfy all four of Maxwell's equations in empty space. We will now show that this can be achieved provided that certain conditions are met. So our confirmation that electromagnetic waves can exist will also predict some of their properties.