# The Unapologetic Mathematician

## The Electromagnetic Wave Equations

Maxwell’s equations give us a collection of differential equations to describe the behavior of the electric and magnetic fields. Juggling them, we can come up with other differential equations that give us more insight into how these fields interact. And, in particular, we come up with a familiar equation that describes waves.

Specifically, let’s consider Maxwell’s equations in a vacuum, where there are no charges and no currents:

\displaystyle\begin{aligned}\nabla\cdot E&=0\\\nabla\times E&=-\frac{\partial B}{\partial t}\\\nabla\cdot B&=0\\\nabla\times B&=\epsilon_0\mu_0\frac{\partial E}{\partial t}\end{aligned}

Now let’s take the curl of both of the curl equations:

\displaystyle\begin{aligned}\nabla\times(\nabla\times E)&=-\frac{\partial}{\partial t}(\nabla\times B)\\&=-\frac{\partial}{\partial t}\left(\epsilon_0\mu_0\frac{\partial E}{\partial t}\right)\\&=-\epsilon_0\mu_0\frac{\partial^2 E}{\partial t^2}\\\nabla\times(\nabla\times B)&=\epsilon_0\mu_0\frac{\partial}{\partial t}(\nabla\times E)\\&=\epsilon_0\mu_0\frac{\partial}{\partial t}\left(-\frac{\partial B}{\partial t}\right)\\&=-\epsilon_0\mu_0\frac{\partial^2 B}{\partial t^2}\end{aligned}

We also have an identity for the double curl:

$\displaystyle\nabla\times(\nabla\times F)=\nabla(\nabla\cdot F)-\nabla^2F$

But for both of our fields we have $\nabla\cdot F=0$, meaning we can rewrite our equations as

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

which are the wave equations we were looking for.