5.4.1 Twelve lectures, four equations

This section ends the electrodynamics part of the course, and it is a good place to look back (Figure 5.4.2). We began with a force between two charges at rest, Coulomb's law of Section 3.1, and we end with a wave that needs no charges at all, the plane wave of Home exercise 5.1.1. Each step on the way removed one assumption. First we assumed that the charges were still: that gave the electric field, Gauss's law, the potential and the behaviour of material media and conductors, all of Chapter 3. Then we allowed charges to move but required the currents to be steady: that gave the Lorentz force, the Biot–Savart law and Ampère's law, the vector potential, inductance, magnetic energy and magnetic circuits, all of Chapter 4. Finally, in Sections 5.1 to 5.4, we dropped the assumption that the fields do not change in time, and the two halves became one subject.

What survives all of it is remarkably compact: the pair of curl equations and the pair of divergence equations (5.1.1), together with $D=\epsilon E$, $B=\mu H$ and $J=\sigma E$ for the material, and the Lorentz force $F=q(E+v\times B)$ for what the fields do to matter. Home exercise 5.1.1 shows that this set has a solution in empty space which is a travelling wave, and that the wave is possible only because of the displacement current: without $\partial D/\partial t$ the magnetic field of the wave would have nothing to sustain it. Exercise 5.1.2 derives the wave equation itself and computes the speed $1/\sqrt{{\mu }_0{\epsilon }_0}$ from the two constants of this book, ${\epsilon }_0=10^{-9}/(36\pi )F/m$ and ${\mu }_0=4\pi \times 10^{-7}H/m$: the product ${\mu }_0{\epsilon }_0$ is exactly $10^{-16}/9$ and the speed exactly $3\times 10^{8}m/s$. Coulomb's constant comes from the force between charges and ${\mu }_0$ from the force between currents, two table-top experiments, and together they give the speed of light. This was Maxwell's evidence, in 1865, that light is an electromagnetic wave. Section 5.5 derives the wave equation in general, and Section 5.6 treats the time-harmonic waves of engineering practice.

Change ${\epsilon }_r$ and $f$ and watch $v$, $\lambda$ and $\eta$. In vacuum $\omega /k$ is exactly $3\times 10^8$ m/s.