Electromagnetic Waves

An electromagnetic wave is simply a ripple of electric and magnetic field that keeps itself going through empty space, with each field's change creating the other. The name just says what it's made of: an electric part and a magnetic part, traveling together as a wave. Radio, Wi-Fi, microwaves, visible light and X-rays are all this one kind of wave. The headline result, $c = 1/\sqrt{\mu_0\varepsilon_0}$, has just two pieces: $\mu_0$ (the constant from magnetic forces between currents) and $\varepsilon_0$ (the constant from electric forces between charges), and out comes $c$ (the speed of light). The derivation has a lot of partial derivatives, but it's really just Faraday's law and the Ampère–Maxwell law fed into each other. Bear with it.

The scene. Push charge up and down a radio antenna. The changing current makes a changing $\mathbf{B}$ nearby. By Faraday's law that makes a curling $\mathbf{E}$ a little farther out. That $\mathbf{E}$ is changing too, so by the Ampère–Maxwell law it makes $\mathbf{B}$ farther out still. The disturbance feeds itself and leaves the antenna at a definite speed. So how fast does it go?

Deriving the wave equation. In empty space, take fields that depend only on $x$ and $t$, with $\mathbf{E} = E_y(x, t)\,\hat{\mathbf{y}}$ and $\mathbf{B} = B_z(x, t)\,\hat{\mathbf{z}}$. Only one component of each curl survives:

Differentiate the first in $x$ and the second in $t$. The mixed partials of $B_z$ are equal (Clairaut's theorem), so

That's the wave equation $u_{tt} = c^2u_{xx}$ with $c^2 = 1/(\mu_0\varepsilon_0)$, and the same steps in the other order give it for $B_z$. In full 3D, take the curl of Faraday's law and use $\nabla\times(\nabla\times\mathbf{E}) = \nabla(\nabla\cdot\mathbf{E}) - \nabla^2\mathbf{E}$ with $\nabla\cdot\mathbf{E} = 0$: then $-\nabla^2\mathbf{E} = -\mu_0\varepsilon_0\,\partial^2\mathbf{E}/\partial t^2$.

The number. $c = 1/\sqrt{(4\pi\times10^{-7})(8.85\times10^{-12})} = 3.00\times10^8$ m/s. Think about how surprising that is. Nothing in $\mu_0$ or $\varepsilon_0$ knows about light: one comes from forces between currents, the other from forces between charges, both measured with wires and plates on a lab bench. When this calculation first matched the measured speed of light, it showed that light is an electromagnetic wave.

Plane waves. By d'Alembert's solution, any shape $E_y = f(x - ct)$ travels along $+x$ unchanged. A sinusoid $E_y = E_0\cos(kx - \omega t)$ works when $\omega = ck$, and Faraday's law then fixes $\mathbf{B}$: $\partial B_z/\partial t = -\partial E_y/\partial x = kE_0\sin(kx - \omega t)$, so

Three facts fall right out of that: $B_0 = E_0/c$; $\mathbf{E}$ and $\mathbf{B}$ peak together (they're in phase); and the wave moves along $\mathbf{E}\times\mathbf{B}$, here $\hat{\mathbf{y}}\times\hat{\mathbf{z}} = \hat{\mathbf{x}}$.

Transverse. Gauss's law in empty space, $\nabla\cdot\mathbf{E} = 0$, says a field that depends only on $x$ can't have an oscillating $E_x$: the fields wiggle perpendicular to the motion, never along it. The same goes for $\mathbf{B}$. Faraday's law then turns an $\mathbf{E}$ along $y$ into a $\mathbf{B}$ along $z$, so the two fields are perpendicular to each other as well as to the direction of travel. The direction $\mathbf{E}$ points is the wave's polarization.

The spectrum. The equation doesn't care about frequency. AM radio ($\lambda \approx 300$ m), Wi-Fi ($12$ cm), visible light ($400$ to $700$ nm) and X-rays ($\approx 0.1$ nm) are all the same wave, with $c = f\lambda$.

In matter. In an insulator with dielectric constant $\kappa$, $\varepsilon_0$ becomes $\kappa\varepsilon_0$ and the speed drops to $c/\sqrt\kappa$. Solid polyethylene ($\kappa = 2.25$) inside a coaxial cable slows signals to $c/1.5$, two thirds of $c$, a number cable data sheets print as the velocity factor.

A common mistake. Since $B_0 = E_0/c$ comes out as a tiny number in teslas, it's tempting to think the magnetic part of the wave hardly matters. It does. It carries exactly half the energy, as the next lesson shows; teslas are just a big unit. Two smaller slips: mixing up $\omega$ and $f$, or $k$ and $\lambda$, and getting $\mathbf{B}$ backward. Always check that $\mathbf{E}\times\mathbf{B}$ points the way the wave travels.

Every wireless link is engineered with these relations. Antenna sizes are set by the wavelength (a quarter-wave antenna for 2.4 GHz Wi-Fi is about 3 cm) and radar measures distance by timing echoes at $c$. A GPS receiver turns signal travel times into position, where an error of 1 ns is 30 cm. Coaxial cables carry signals at $c/\sqrt\kappa$ and optical fibers at $c/n$. The same equation covers light, so optics, lasers and solar panels all start here.