Maxwell's equations have a remarkable consequence. In empty space, with no charges and no currents, a changing $\vec{E}$ makes $\vec{B}$ and a changing $\vec{B}$ makes $\vec{E}$, so a disturbance can carry itself along. Maxwell computed its speed from two numbers measured with charges and magnets on a lab bench, and found the speed of light. You met electromagnetic waves in Physics 2; here is where they come from.
Why it works
Take a wave moving along $x$, with $\vec{E}$ along $y$ and $\vec{B}$ along $z$. Apply Faraday's law to a thin rectangle in the $xy$-plane and the Ampère–Maxwell law to one in the $xz$-plane. In the limit of tiny rectangles they become, with partial derivatives,
$$\frac{\partial E}{\partial x} = -\frac{\partial B}{\partial t}, \qquad -\frac{\partial B}{\partial x} = \mu_0\varepsilon_0\frac{\partial E}{\partial t}.$$
Differentiate the first by $x$ and the second by $t$ and combine: $\dfrac{\partial^2E}{\partial x^2} = \mu_0\varepsilon_0\dfrac{\partial^2E}{\partial t^2}$, the wave equation with speed $1/\sqrt{\mu_0\varepsilon_0}$. A wave $E = E_0\sin(kx - \omega t)$ in the first equation forces $B = (E_0/c)\sin(kx - \omega t)$: in step with $E$, and smaller by $c$.
Key idea
In empty space electromagnetic waves travel at
$$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3.00 \times 10^{8}\ \text{m/s},$$
with $\vec{E}$ and $\vec{B}$ perpendicular to each other and to the direction of travel, in phase, and $E = cB$ at every point. The wave moves in the direction of $\vec{E}\times\vec{B}$.
Worked examples
Worked example: The speed of light from lab constants
Compute $1/\sqrt{\mu_0\varepsilon_0}$ with $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$ and $\varepsilon_0 = 8.854 \times 10^{-12}\ \text{F/m}$.
- $\mu_0\varepsilon_0 = (4\pi \times 10^{-7})(8.854 \times 10^{-12}) = 1.11 \times 10^{-17}\ \text{s²/m²}$.
- $c = 1/\sqrt{1.113 \times 10^{-17}} = 2.998 \times 10^{8}\ \text{m/s}$: the measured speed of light.
Answer: $c \approx 3.00 \times 10^{8}\ \text{m/s}$
Worked example: The magnetic field of a wave
A light wave has an electric field amplitude of $30\ \text{V/m}$. Find its magnetic field amplitude.
- $B_0 = E_0/c = 30/(3.00 \times 10^{8}) = 1.0 \times 10^{-7}\ \text{T} = 100\ \text{nT}$.
- Small in tesla, but it carries as much energy as the electric field, as you will see next.
Answer: $B_0 = 100\ \text{nT}$
Worked example: Finding directions
A wave travels along $+x$. At some instant $\vec{E}$ points along $+y$. Which way does $\vec{B}$ point?
- $\vec{E}\times\vec{B}$ must point along $+x$. Try $\vec{B}$ along $+z$: $\hat{\jmath}\times\hat{k} = \hat{\imath}$. That works.
- So $\vec{B}$ points along $+z$; half a cycle later both fields reverse together.
Answer: $+z$
Common mistake
"The electric and magnetic fields of light peak at different times, trading energy like an LC circuit." In a traveling wave they rise and fall together: $E = cB$ at every instant, as the step above showed. Each field is made by the other's change in space and time together, not by passing energy back and forth. Check: at a crest of $\vec{E}$, $\vec{B}$ is at a crest too.