Maxwell’s Equations and Electromagnetic Waves

The Missing Term

Ampère's law as inherited was inconsistent. Apply it to a charging capacitor: a loop round the wire encloses current through one surface, and nothing at all through a surface bulging between the plates. Same loop, two answers.

Maxwell's fix was the displacement current, a term proportional to the rate of change of electric flux:

$$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}$$

Between the plates no charge flows, but $\mathbf{E}$ grows, and the new term supplies exactly the missing amount. The four equations then read:

The last two are now symmetric, and that symmetry has a consequence. In empty space, taking the curl of the third and substituting the fourth gives a wave equation:

$$\nabla^2\mathbf{E} = \mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}, \qquad c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$$

Putting in the measured constants gives $3.00 \times 10^8$ m/s, the speed of light, arrived at from electrical measurements that had nothing to do with optics. That is what identified light as an electromagnetic wave.

Plane waves are transverse, with $\mathbf{E} \perp \mathbf{B} \perp$ propagation, $E = cB$, and energy flow given by the Poynting vector $\mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0$.

Common pitfall: calling the displacement current a flow of charge. Nothing moves between the capacitor plates. It is a changing electric field that sources a magnetic field exactly as a real current would, the name is historical, and it misleads.