# 1 Maxwell's greatest triumph

This course presents Maxwell's greatest triumph – the prediction that electromagnetic waves can propagate vast distances through empty space and the realisation that light is itself an electromagnetic wave. Visible light has a very narrow range of wavelengths, but this tells us more about the sensitivity of our eyes than about the nature of electromagnetic radiation. A few years after Maxwell's death other types of electromagnetic radiation, including radio waves, X-rays and gamma rays, were discovered. Compared to light, radio waves have very long wavelengths, while X-rays and gamma rays have very short wavelengths. Different parts of the electromagnetic spectrum are used in different ways (Figure 1). Radio waves are used for broadcast radio and television, satellite communications and mobile phones. Gamma rays are used to treat cancer and X-rays are used in medical diagnosis. Yet all these waves have the same underlying description in terms of electric and magnetic fields.

Maxwell was in a position to predict the existence of electromagnetic waves because, by the mid-1860s, he had developed a comprehensive theory of electromagnetism. You may already have met some or all of Maxwell's four equations: let's take a brief look at *Gauss's law*, the *no-monopole law* and *Faraday's law* first. These three laws can be expressed in terms of volume, surface and line integrals or in terms of partial derivatives. In this course we will make use of the differential versions of Maxwell's equations.

## SAQ 1

Write down the *differential* versions of Gauss's law, the no-monopole law and Faraday's law. Are these laws true under all circumstances?

### Answer

The three laws are:

where **E** and **B** are the electric and magnetic fields, *ρ* is the charge density and *ε*_{0} is the permittivity of free space. All three laws have general validity: they apply to time-varying situations as well as static or steady-state ones.

The differential version of *Ampère's law* is

where **J** is the current density and *μ*_{0} is the permeability of free space. However, Ampère's law has a different status: it requires steady currents and is not valid for currents that vary in time. This means that Ampère's law is not general enough to count as one of Maxwell's four laws of electromagnetism.

Fortunately, Ampère's law can be rescued. Maxwell realised that an extra term, , can be added to the right-hand side of Ampère's law. This term makes no difference in static situations, but it extends the validity of the law to general, time-varying situations. The extended equation is called the *Ampère–Maxwell law* and takes the form

Our first task is to justify this law. To achieve this, we will make use of a basic principle of electromagnetism – the conservation of charge. Section 2 will show that the law of conservation of charge leads to a relationship between current density and charge density known as the *equation of continuity*. This relationship will be used in Section 3 to help justify the Ampère–Maxwell law. Then, with all four of Maxwell's equations in place, we will be in a position to demonstrate that electromagnetic waves are a direct consequence of the laws of electromagnetism.