Electromagnetic Waves

Fields that keep each other going

Faraday's law says that a changing magnetic field makes an electric field. The Ampere-Maxwell law says that a changing electric field makes a magnetic field. Suppose an electric field somewhere starts to change. It makes a changing magnetic field around it. That changing magnetic field makes a changing electric field a little further out, and so on. The disturbance spreads outward on its own, with no charge and no medium needed.

Such a travelling pair of fields is an electromagnetic wave. What starts one? A charge at rest has only a steady electric field. A charge moving at a steady velocity carries its fields along with it, and sends nothing away. A charge that accelerates changes its fields suddenly, and that change travels outward as a wave. So accelerated charges radiate.

The antenna

An antenna is a conductor in which charges are driven up and down. Join a straight rod to an alternating source, and charge surges to its top, then to its bottom, \(f\) times each second. The charges accelerate all the time, so the rod radiates. Its fields change \(f\) times each second, so the wave it sends out has the frequency \(f\) of the source.

In the figure above the crests always move out at the same speed, \(c = 3\times10^8\,\text{m/s}\). One crest leaves in each period \(1/f\), so the crests are \(\lambda = c/f\) apart. At \(100\,\text{MHz}\) that is \(3\,\text{m}\), and at \(200\,\text{MHz}\) it is \(1.5\,\text{m}\). An antenna works best when its length is about half a wavelength, so an antenna for \(100\,\text{MHz}\) is about \(1.5\,\text{m}\) long.

The plane wave

Far from the antenna, a small part of the spreading wave is nearly flat. It is a plane wave. Let it travel along \(x\). Its electric field points along \(y\) and its magnetic field along \(z\). Both are travelling waves of the form met in the chapter Introduction to Waves.

\[ \begin{gathered} E_y = E_0\sin(kx - \omega t) \\ B_z = B_0\sin(kx - \omega t) \end{gathered} \]

Here \(k = 2\pi/\lambda\) and \(\omega = 2\pi f\), and the wave travels at \(\omega/k = c\).

Watch the figure above. The two fields rise and fall together, in phase, at every point. They are at right angles to each other. Both are at right angles to the direction of travel, so the wave is a transverse wave. The wave moves along \(\vec E\times\vec B\). Here \(\hat j\times\hat k = \hat i\), so it moves along \(+x\).

Why \(E_0 = cB_0\)

The two amplitudes are not free. Faraday's law ties them together. Take a thin rectangle in the plane of \(\vec E\), of width \(dx\) along the travel. The electric field is a little different on its two long sides. The difference gives a circulation of \(\vec E\), which must equal minus the rate of change of the magnetic flux through the rectangle. For the plane wave this reads \(\partial E/\partial x = -\partial B/\partial t\).

So an electric field of \(300\,\text{V/m}\) goes with a magnetic field of only \(300/(3\times10^8) = 10^{-6}\,\text{T}\), one microtesla. In tesla the magnetic field of light is tiny. The force \(qvB\) it exerts on a charge is \(v/c\) times the electric force \(qE\). For slow charges the electric field does almost all the pushing.

The speed of light from two constants

Maxwell's equations give the speed of the wave from the two constants of electricity and magnetism. It is \(1/\sqrt{\mu_0\epsilon_0}\). Here \(\mu_0\epsilon_0\) \(= 4\pi\times10^{-7}\times8.85\times10^{-12}\) \(= 1.11\times10^{-17}\,\text{s}^2/\text{m}^2\). So the speed is \(1/\sqrt{1.11\times10^{-17}}\) \(= 3.0\times10^8\,\text{m/s}\), the speed of light.

Every electromagnetic wave in vacuum, of any frequency, travels at this one speed. Radio waves, light and X-rays differ only in frequency and wavelength. In a material the constants become \(\mu = \mu_r\mu_0\) and \(\epsilon = \epsilon_r\epsilon_0\), so the wave slows to \(v = c/\sqrt{\mu_r\epsilon_r}\). The refractive index of the chapter Refraction, Total Internal Reflection and Prisms is then \(n = c/v = \sqrt{\mu_r\epsilon_r}\).

For glass, \(\mu_r = 1\) and \(\epsilon_r = 2.25\) at the frequency of light. So \(v = 3\times10^8/\sqrt{2.25} = 2\times10^8\,\text{m/s}\) and \(n = 1.5\). Maxwell's constants explain the bending of light in a lens.