Maxwell and electromagnetic waves

Before Maxwell

By the 1850s electricity and magnetism were known to be linked, but the links were a collection of separate rules:

• Charges produce electric fields (Coulomb).
• Currents, which are moving charges, produce magnetic fields (Ørsted, Ampère).
• A changing magnetic field induces an emf, and so an electric field (Faraday).
• Magnetic field lines always form closed loops: there are no isolated north or south poles.

Faraday pictured these effects with field lines filling space. Maxwell took that picture seriously and wrote it in mathematics.

Unification of electricity and magnetism

Maxwell's work (1861–65) produced a set of equations, now called Maxwell's equations, that describe every classical electric and magnetic effect in one theory. You don't need the equations themselves for the HSC, but you do need what they say in words:

1. Gauss's law for electricity: electric field lines start on positive charges and end on negative charges.
2. Gauss's law for magnetism: magnetic field lines have no start or end; there are no magnetic monopoles.
3. Faraday's law: a changing magnetic field produces an electric field.
4. Ampère–Maxwell law: a magnetic field is produced by an electric current and by a changing electric field.

The part in italics was Maxwell's own addition. He noticed that Ampère's law gave contradictory answers for a charging capacitor, where current flows in the wires but not across the gap. He fixed it by adding a term, the displacement current, which says a changing electric field acts like a current. That one term is what makes electromagnetic waves possible. Electricity and magnetism were no longer two forces with some links; they were two aspects of a single electromagnetic field.

Predicting electromagnetic waves

Put Faraday's law and Maxwell's new term together. A changing magnetic field makes a changing electric field, and a changing electric field makes a changing magnetic field. Once started, the disturbance can keep regenerating itself, moving away from where it started, with no medium needed. Maxwell's equations predicted that:

• the electric and magnetic fields oscillate perpendicular to each other and to the direction of travel, so EM waves are transverse;
• the two fields oscillate in phase, reaching maximum together;
• the waves could have any frequency, so there should be a whole spectrum of them, not just visible light;
• they carry energy and momentum, and can be reflected, refracted and polarised.

Predicting the speed

The equations also gave the speed of the wave in a vacuum in terms of two constants that can be measured on a lab bench:

$$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$$

• $\varepsilon_0 = 8.854 \times 10^{-12}$ F/m is the permittivity of free space: it sets the strength of the electric force between charges (it appears in Coulomb's law).
• $\mu_0 \approx 4\pi \times 10^{-7}$ T m/A is the permeability of free space: it sets the strength of the magnetic field made by a current (you used it in Module 6).

Nothing in that calculation mentions light, yet the answer is the speed of light. The obvious conclusion, which Maxwell drew, is that light is an electromagnetic wave, and that it is only a small part of a much larger spectrum.

A modern footnote: since the 2019 redefinition of the SI units, $c$, $h$ and $e$ have exact defined values, and $\mu_0$ is now a measured quantity. It is still $4\pi \times 10^{-7}$ to better than one part in a billion, so the HSC value is fine.

How EM waves are produced

A charge at rest has an electric field but no magnetic field. A charge moving at constant velocity has both, but they just travel along with it. To send a wave out into space, a charge must accelerate: speed up, slow down or change direction. The kink this makes in its field lines moves away at $c$.

• In a radio antenna, electrons are driven up and down at the transmitter's frequency, so they radiate waves of the same frequency.
• In a hot object, charged particles jostle randomly, producing a broad spread of frequencies (7H).
• In an atom, an electron dropping between energy levels emits a photon of one particular frequency (7C).
• In a synchrotron, electrons are bent in a circle at almost the speed of light and give off intense X-rays and UV. The Australian Synchrotron in Melbourne bends 3 GeV electrons around a 216 m ring to make light for research on everything from viruses to bushfire-resistant materials.

How EM waves propagate

EM waves need no medium: the fields themselves are the wave. In a vacuum all frequencies travel at the same speed, $c = 3.00 \times 10^8$ m/s, so

$$c = f\lambda$$

In a material, the waves interact with the charges in the atoms and travel more slowly. The refractive index $n$ of the material compares the speeds:

$$n = \frac{c}{v}$$

When light enters glass the frequency stays the same, because it is set by the source, so the wavelength shrinks by the factor $n$.

Confirmation: Hertz

Maxwell died in 1879 without seeing his waves made on purpose. In 1887–88 Heinrich Hertz built a spark-gap transmitter: a spark jumping across a gap made charges surge back and forth at high frequency. A few metres away, a wire loop with its own tiny gap showed sparks too, so energy had crossed the room with no wires. Hertz showed his waves could be reflected, refracted, focused and polarised, measured their wavelength from standing waves, and found their speed was the speed of light. Within a decade Marconi was using them for radio.