# The Unapologetic Mathematician

## The Propagation Velocity of Electromagnetic Waves

Now we’ve derived the wave equation from Maxwell’s equations, and we have worked out the plane-wave solutions. But there’s more to Maxwell’s equations than just the wave equation. Still, let’s take some plane-waves and see what we get.

First and foremost, what’s the propagation velocity of our plane-wave solutions? Well, it’s $c$ for the generic wave equation

$\displaystyle\frac{\partial^2F}{\partial t^2}-c^2\nabla^2F=0$

while our electromagnetic wave equation is

\displaystyle\begin{aligned}\frac{\partial^2E}{\partial t^2}-\frac{1}{\epsilon_0\mu_0}\nabla^2E&=0\\\frac{\partial^2B}{\partial t^2}-\frac{1}{\epsilon_0\mu_0}\nabla^2B&=0\end{aligned}

so we find the propagation velocity of waves in both electric and magnetic fields is

$\displaystyle c=\frac{1}{\sqrt{\epsilon_0\mu_0}}$

Hm.

Conveniently, I already gave values for both $\epsilon_0$ and $\mu_0$:

\displaystyle\begin{aligned}\epsilon_0&=8.85418782\times10^{-12}\frac{\mathrm{F}}{\mathrm{m}}&=8.85418782\times10^{-12}\frac{\mathrm{s}^2\cdot\mathrm{C}^2}{\mathrm{m}^3\cdot\mathrm{kg}}\\\mu_0&=1.2566370614\times10^{-6}\frac{\mathrm{H}}{\mathrm{m}}&=1.2566370614\times10^{-6}\frac{\mathrm{m}\cdot\mathrm{kg}}{\mathrm{C}^2}\end{aligned}

Multiplying, we find:

$\displaystyle\epsilon_0\mu_0=8.85418782\times1.2566370614\times10^{-18}\frac{\mathrm{s}^2}{\mathrm{m}^2}=11.1265006\times10^{-18}\frac{\mathrm{s}^2}{\mathrm{m}^2}$

which means that

$\displaystyle c=\frac{1}{\sqrt{\epsilon_0\mu_0}}=0.299792457\times10^9\frac{\mathrm{m}}{\mathrm{s}}=299\,792\,457\frac{\mathrm{m}}{\mathrm{s}}$

And this is a number which should look very familiar: it’s the speed of light. In an 1864 paper, Maxwell himself noted:

The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.

Indeed, this supposition has been borne out in experiment after experiment over the last century and a half: light is an electromagnetic wave.

February 9, 2012