31.5 Advanced Topic: Electromagnetic Waves

NOTE   This optional section goes through the mathematics of showing that Maxwell’s equations predict electromagnetic waves. The key results of this section are summarized at the beginning of Section 31.6, so this section may be omitted with no loss of continuity.

A photograph of a large radar installations used to track radars, rockets and missiles.

Large radar installations like this one are used to track rockets and missiles.

Maxwell developed his four equations as a mathematical summary of what was known about electricity and magnetism in the mid-19th century: the properties of static electric and magnetic fields plus Faraday’s discovery of electromagnetic induction. Maxwell introduced the idea of displacement current—that a changing electric flux creates a magnetic field—on purely theoretical grounds; there was no experimental evidence at the time. But this new concept was the key to Maxwell’s success because it soon allowed him to make the remarkable and totally unexpected prediction of electromagnetic waves—self-sustaining oscillations of the electric and magnetic fields that propagate through space without the need for charges or currents.

Our goals in this section are to show that Maxwell’s equations lead to a wave equation for the electric and magnetic fields and to discover that all electromagnetic waves, regardless of frequency, travel through vacuum at the same speed, a speed we now call the speed of light. A completely general derivation of the wave equation is too mathematically advanced for this textbook, so we will make a small number of assumptions—but assumptions that will seem quite reasonable after our study of induced fields.

To begin, we will assume that electric and magnetic fields can exist independently of charges and currents in a source-free region of space. This is a very important assumption because it makes the statement that fields are real entities; they’re not just cute pictures that tell us about charges and currents. The source-free Maxwell’s equations, with no charges or currents, are

∮E⃗⋅dA⃗=0∮E⃗⋅ds⃗=−dΦmdt∮B⃗⋅dA⃗=0∮B⃗⋅ds⃗=ϵ0μ0dΦedt(31.22)

Any electromagnetic wave traveling in empty space must be consistent with these equations.