8.2 Electromagnetic Waves and the Electromagnetic Spectrum

When you look at the Sun you see it as it was about eight minutes ago: light needs $\dfrac{1.5\times10^{11}\ \text{m}}{3.0\times10^{8}\ \text{m/s}}=500$ s to cross the gap. The radio commands sent to a spacecraft orbiting Mars take several minutes to arrive, so the engineers on Earth must plan every manoeuvre in advance. Sunlight, the radio link to the spacecraft, the infrared beam of a TV remote, the microwaves in an oven and the X-rays in a hospital scanner all travel at this same speed through empty space. They are all electromagnetic waves; what tells them apart is only their wavelength, which ranges from kilometres to less than the size of an atomic nucleus.

In Section 8.1 we found that Maxwell's equations allow a changing electric field to make a magnetic field, and a changing magnetic field to make an electric one. In this section we see what kind of charge motion launches such waves, describe the wave itself — its fields, its speed, the relation $E_0=cB_0$ — and work out the energy and momentum it carries. Finally we survey the whole electromagnetic spectrum: how each band is produced and detected, and what it is used for.

Sources of Electromagnetic Waves

A charge at rest has an electric field but no magnetic field. A charge moving with constant velocity — a steady current — has a magnetic field too, but its fields simply travel along with it; they do not break away. To launch a wave, the fields must keep changing, and that requires a charge whose velocity changes: an accelerated charge. Accelerated charges radiate electromagnetic waves.

The simplest source is a charge oscillating to and fro with frequency $\nu$. Its electric field changes with frequency $\nu$, which by the Ampère–Maxwell law makes a magnetic field changing at the same frequency, which by Faraday's law makes a changing electric field further out, and so on. The disturbance spreads outwards as an electromagnetic wave whose frequency equals the frequency of the oscillating charge. In a radio transmitter, an oscillating circuit drives charges up and down a metal rod, the antenna, millions of times a second; in a receiving antenna the arriving electric field pushes the electrons back and forth at the same frequency, giving a tiny alternating current that the receiver amplifies.

Visible light is radiated by charges oscillating far faster, about $10^{15}$ times a second — too fast for any circuit. Such light comes from electrons in atoms jumping between energy levels (Chapter 12) and from the jostling charges in hot bodies. X-rays come from fast electrons that are suddenly stopped in a metal target, and gamma rays from changes inside atomic nuclei.

The first experiments. In 1887 Heinrich Hertz drove sparks across a small gap between two metal spheres connected to an induction coil. Each spark set charges oscillating rapidly, and a separate ring of wire with its own tiny gap, several metres away, sparked in response: electromagnetic waves had crossed the room. Hertz showed that his waves could be reflected, refracted and made to interfere, just like light, and measured their speed. A few years later Jagadish Chandra Bose in Kolkata made waves between about $5$ mm and $25$ mm long, built receivers for them and studied their optics, and Guglielmo Marconi developed radio into a way of sending messages across oceans.

The Nature of Electromagnetic Waves

Far from its source a small piece of an electromagnetic wave looks like a plane wave: on every plane perpendicular to the direction of travel the fields are the same. For a sinusoidal wave travelling along the $+x$ direction, with $\mathbf E$ along $y$, Maxwell's equations allow only one kind of solution: $\mathbf B$ along $z$, oscillating in step with $\mathbf E$, with a fixed ratio of amplitudes and a fixed speed.

The ratio $E_0=cB_0$ means that the magnetic field of an electromagnetic wave is numerically tiny: a wave with an electric field of $300$ V/m — strong by everyday standards — has a magnetic field of only $1.0\ \mu$T. Yet, as we shall see, the two fields carry equal shares of the energy.

In a material. Inside a transparent medium of permittivity $\varepsilon=\varepsilon_r\varepsilon_0$ and permeability $\mu=\mu_r\mu_0$, the same reasoning gives a smaller speed.

Energy, Intensity and Momentum

An electromagnetic wave carries energy: sunlight warms the ground, and a microwave oven heats food. The energy is stored in the fields themselves. In Section 2.4 we found that an electric field has energy density $u_E=\tfrac12\varepsilon_0E^2$; a magnetic field has energy density $u_B=\dfrac{B^2}{2\mu_0}$ (you meet it with the energy of an inductor in Chapter 6). In a wave $B=E/c$ and $c^2=1/(\mu_0\varepsilon_0)$, so

The electric and magnetic fields carry equal shares of the energy, although $B$ looks so small. The total energy density at a point is $u=\varepsilon_0E^2$, and it oscillates because $E$ does. Averaged over a cycle, $\sin^2$ has the value $\tfrac12$, so $\langle u\rangle=\tfrac12\varepsilon_0E_0^2$.

Now imagine a surface of area $A$ facing the wave. In a time $\Delta t$, all the energy in a box of length $c\,\Delta t$ in front of it flows through it: energy $\langle u\rangle\,(Ac\,\Delta t)$. The energy per unit area per unit time is $\langle u\rangle c$.

Because $I\propto E_0^2$, doubling the field amplitude quadruples the intensity — the same rule as for every wave, where intensity goes as amplitude squared (Class 11 Physics, Chapter 14). Momentum delivery means that light pushes on whatever absorbs it. The push is minute — bright sunlight falling on your palm (about $0.01\ \text{m}^2$) pushes it with roughly $3\times10^{-8}$ N — but it is measurable, and in space, where nothing else acts, it can steer spacecraft and push the tails of comets away from the Sun.

The Electromagnetic Spectrum

Maxwell's equations put no limit on the frequency of an electromagnetic wave. The waves we know span more than twenty powers of ten in wavelength, and arranged in order they form the electromagnetic spectrum. The traditional bands are named after the way the waves were discovered or are produced; the boundaries between them are approximate, and neighbouring bands overlap (X-rays and gamma rays, for example, are told apart by their origin rather than their wavelength). All of them travel at $c$ in vacuum, obey $c=\nu\lambda$, and are emitted or absorbed in packets of energy $E=h\nu=hc/\lambda$ — the photons of Chapter 11. With $hc=1240$ eV nm, a photon of wavelength $\lambda$ (in nm) has energy $1240/\lambda$ eV.

A few features of the spectrum are worth knowing well:

Radio and microwaves pass easily through the atmosphere, clouds and walls, which makes them ideal for communication. Microwaves of $2.45$ GHz are absorbed well by water molecules, which is why a microwave oven heats food; radar uses the echo of a short microwave pulse to find the distance $d=ct/2$ of an aircraft or a rain cloud.

Infrared is the "heat" radiation of warm objects; your body radiates mostly at around $10\ \mu$m. The Earth's surface, warmed by sunlight, radiates infrared that carbon dioxide and water vapour absorb and partly send back down — the greenhouse effect that keeps the planet habitable, and that extra carbon dioxide strengthens.

Visible light is a narrow band — less than one factor of two in wavelength — to which our eyes happen to be sensitive, from red (about $700$ nm) through orange, yellow, green and blue to violet (about $400$ nm). The Sun's output peaks in this band.

Ultraviolet photons carry a few electronvolts or more, enough to break chemical bonds: they cause sunburn and can damage DNA, which is why UV lamps sterilise water. Most of the Sun's ultraviolet is absorbed by the ozone layer, about $15$–$35$ km up.

X-rays and gamma rays have photon energies of thousands to millions of electronvolts. They pass through soft tissue but are absorbed more strongly by bone and metal, which is what makes an X-ray image, and they ionise the atoms they hit — exposure must be kept as low as possible.

Exercises