The electromagnetic wave and how it is made

What does the wave look like?

Throw a stone in a pond and the ripples spread. Shake a rope, and a pulse runs along it. In an electromagnetic wave what shakes is not a material. It is the electric field and the magnetic field themselves.

Take a plane wave that travels along the $z$ axis. The electric field points along $x$ and the magnetic field along $y$:

Here $k = 2\pi/\lambda$ is the wave number and $\omega = 2\pi f$ is the angular frequency. The speed of the wave is $c = \omega/k = f\lambda$.

The properties of the wave

• It is transverse. $\vec{E}$ and $\vec{B}$ are both at right angles to the direction of travel.
• The fields are at right angles to each other. The direction of travel is the direction of $\vec{E}\times\vec{B}$. In the picture, $\hat{x}\times\hat{y} = \hat{z}$.
• The fields are in phase. They are zero together and largest together, as the red and blue curves of the picture show.
• The strengths are tied together. In vacuum, at every point and instant, $E = cB$. So $E_0 = cB_0$.
• It needs no medium. Sunlight crosses empty space to reach us.
• It is not deflected by an electric or a magnetic field. The wave carries no charge.
• It carries energy and momentum, which we study in the third course.

Double the wavelength and look at the frequency. Then double the strength of $E$. Which readout changes, and which does not? The magnetic field doubles, and the frequency does not change.

A worked example. A plane wave in vacuum has $E_0 = 36$ V/m and a wavelength of $3.0$ m. It travels along $+z$, with $\vec{E}$ along $x$.

1. Frequency. $f = \dfrac{c}{\lambda} = \dfrac{3.00 \times 10^{8}\ \text{m/s}}{3.0\ \text{m}} = 100 \times 10^{6}\ \text{Hz}$.
2. Angular frequency and wave number. $\omega = 2\pi f = 6.28 \times 10^{8}\ \text{rad/s}$ and $k = \dfrac{2\pi}{\lambda} = 2.094\ \text{rad/m}$.
3. The magnetic field. We want $B_0$, so $B_0 = \dfrac{E_0}{c} = \dfrac{36\ \text{V/m}}{3.00 \times 10^{8}\ \text{m/s}} = 120\ \text{nT}$. It points along $+y$, since $\hat{x}\times\hat{y} = \hat{z}$.
4. The equations. $E_x = 36\sin(2.09\,z - 6.28 \times 10^{8}\,t)$ in V/m, and $B_y = 120 \times 10^{-9}\sin(\text{the same phase})$ in T.

In a material

The frequency is set by the source and does not change. The speed becomes $v = c/n$, and the wavelength becomes $\lambda = v/f = \lambda_0/n$. The ratio of the field strengths is $E_0/B_0 = v$, which is less than $c$.

A worked example. A radio wave of $100$ MHz enters a non-magnetic material with $\varepsilon_r = 4.0$ at that frequency. In air its wavelength is $3.0$ m.

1. The index. $n = \sqrt{\varepsilon_r} = \sqrt{4.0} = 2.0$.
2. The speed. $v = \dfrac{c}{n} = 1.50 \times 10^{8}\ \text{m/s}$.
3. The wavelength. $\lambda = \dfrac{v}{f} = \dfrac{1.50 \times 10^{8}\ \text{m/s}}{100 \times 10^{6}\ \text{Hz}} = 1.5\ \text{m}$, half of the wavelength in air.

How the wave is made

A charge at rest has a steady electric field. A charge moving at a steady speed is a steady current, and it has a steady magnetic field. Neither makes a wave. A wave needs fields that change and leave the source. That happens only when the charge is accelerated.

Think of a charge that swings up and down along an antenna. Its field cannot change everywhere at the same instant. The news of the change travels outwards at $c$. A kink in the field lines runs away from the charge. That kink is the wave. An oscillating charge makes a wave of the same frequency as its oscillation. The wave is strongest in the directions at right angles to the antenna, and there is none along its axis.

The power radiated by an oscillating charge grows very fast with frequency, in proportion to $\omega^4$ for the same amplitude. Doubling the frequency gives $16$ times the power. That is why the mains current at 50 Hz, with a wavelength of $6000$ km, hardly radiates, while a radio antenna at 100 MHz does. A good antenna is about half a wavelength long: for $100$ MHz that is $1.5$ m.

Hertz's experiment

Maxwell's theory predicted waves but nobody had made one. In 1887 and 1888 Heinrich Hertz did, in a laboratory.

• The transmitter. An induction coil charged two metal spheres until a spark jumped across a gap between them. The spark current swung to and fro very fast, and the spheres acted as a small antenna.
• The receiver. A loop of wire with a tiny gap stood some distance away. When the transmitter sparked, a small spark jumped across the gap of the loop, with no wire between them.
• The measurement. Hertz put a metal sheet behind the loop. The waves reflected from it and formed standing waves, with places of no field a half wavelength apart. Measuring that distance gave the wavelength $\lambda$. With the frequency of the transmitter known, he found $v = f\lambda$, and it matched the speed of light.
• The behaviour. He also showed reflection, refraction and polarisation of the waves, exactly as for light.

So the waves of Maxwell's theory were real, and light behaved as one of them.

Practise with new numbers

Try it in Python

The program takes a plane wave in vacuum with a given wavelength and peak electric field. It finds the frequency, the wave number and the peak magnetic field. Then it looks at one place at three moments, to show that $E$ and $cB$ rise and fall together.

Common mistakes

• Drawing $\vec{E}$ and $\vec{B}$ out of step. In vacuum they peak together. Out of step is for a standing wave or a wave in a conductor.
• Putting $\vec{B}$ along the direction of travel. A plane wave has no field along the way: both fields are at right angles to it.
• Mixing $E = cB$ with $E_0 = cB_0$. Both hold in vacuum, at every point. In a material, $E = vB$.
• Saying a charge moving steadily radiates. Only an accelerated charge does.

Remember these

• Plane wave: $E = E_0\sin(kz - \omega t)$, $B = B_0\sin(kz - \omega t)$. Transverse, $\vec{E}\perp\vec{B}$, in phase.
• Direction of travel is along $\vec{E}\times\vec{B}$. $E_0 = cB_0$ in vacuum.
• $c = f\lambda = \omega/k$. In a material, $v = c/n$, $f$ is unchanged and $\lambda = \lambda_0/n$.
• An accelerated charge radiates. Hertz made the waves in 1887 and found their speed to be the speed of light.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A plane wave travels along $+x$. Its electric field points along $+y$ at some instant. In which direction does its magnetic field point?

2. The peak magnetic field of a plane wave in vacuum is $24$ nT. What is the peak electric field?

3. A wave has a frequency of $150$ MHz. What is its wavelength in vacuum?

4. Does a charge moving in a straight line at a steady speed produce an electromagnetic wave? What about a charge going round a circle at a steady speed?

References

• OpenStax, University Physics Volume 2, Chapter 16 (Electromagnetic Waves): plane electromagnetic waves.
• NCERT Physics Class 12, Chapter 8 (Electromagnetic Waves): the nature of electromagnetic waves and Hertz's experiment.