Maxwell’s equations

Maxwell’s modification of Ampère’s law is one of four equations that together describe all of classical electromagnetism. These are known as Maxwell’s equations. We have now seen them all and can summarise them here:

Name Description Equation
Gauss’ Law Electric field lines start and end on charges $\oiint \vec{E} \cdot d\vec{A} = \frac{q_{\rm enc}}{\epsilon_0}$
Gauss’ Law for magnetism There are no magnetic charges, so magnetic field lines are closed loops $\oiint \vec{B} \cdot d\vec{A} = 0$
Faraday’s Law A changing magnetic field generates an electric field around loops $\oint \vec{E} \cdot d\vec{l} = - \frac{d\Phi_B}{dt}$
Ampère-Maxwell Law A current or a changing electric field generates a magnetic field $\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_{\rm enc} + \epsilon_0 \frac{d\Phi_E}{dt} \right)$

In your maths lectures you will learn about the divergence ($\nabla \cdot$) and curl ($\nabla \times$) operators, which allow us to write Maxwell’s four equations in a form without any integrals. However in the first year physics course, we use the integral form. The two are equivalent.

Together with the Lorentz force law from Lecture 9, \(\vec{F} = q (\vec{E} + \vec{v} \times \vec{B}),\) Maxwell’s equations describe all of electromagnetism (except for quantum effects).

  • Maxwell’s equations describe how fields are generated by charges and currents;
  • The Lorentz force law describes how charges and currents are affected by the fields.
  • Other equations (like the Biot-Savart law, or Coulomb’s law) are helpful, and historically came first – but today, they can all be derived from Maxwell’s equations.
  • From a modern perspective, these equations are part of a deeper structure known as a gauge theory. This makes explicit the connection to relativity, and describes the relationship between fields and charges in terms of symmetries. You will learn more about this in later courses.

Maxwell’s equation in a vacuum

In a vacuum, with no charges or currents, the equations simplify to:

\[\oiint \vec{E} \cdot d\vec{A} = 0, \quad \oiint \vec{B} \cdot d\vec{A} = 0,\]

and

\[\quad \oint \vec{E} \cdot d\vec{l} = - \frac{d\Phi_B}{dt}, \quad \oint \vec{B} \cdot d\vec{l} = \epsilon_0 \mu_0 \frac{d\Phi_E}{dt}.\]

Note how highly symmetric this is. It says there are no sources or sinks of electric or magnetic fields (first two equations) but that changing electric fields generate magnetic fields and changing magnetic fields generate electric fields (last two equations).

This is why electromagnetic waves can travel through a vacuum. The electric field generates the magnetic field, which generates the electric field, which generates the magnetic field, and so on. You can show starting from these equations that the speed at which these waves move is $c = 1/\sqrt{\epsilon_0 \mu_0}$, which is the speed of light, loosely speaking by substituting the vacuum Maxwell-Ampere law into the vacuum Faraday law; you then get a second order differential equation in time which describes a wave.

In the lecture, we will see roughly how this is done in YF 32.2. But it’s so much easier once you have the differential forms of Maxwell’s equations, so I will only show you this for completeness and the derivation won’t be examined.

The upshot is:

  • The electromagnetic field permits travelling waves, with $\vec{E}$ and $\vec{B}$ perpendicular to each other and to the direction of travel;
  • $\vec{E}$ and $\vec{B}$ are in phase with each other, i.e. a location of maximum electric field strength is also a location of maximum magnetic field strength;
  • The $\vec{E}$ and $\vec{B}$ field amplitudes are related by $\vert \vec{E}\vert = c \vert \vec{B} \vert$;
  • The waves travel at the speed of light $c$, where $c^{-2} = \mu_0 \epsilon_0 $ (because this combination appears in the vacuum Maxwell equations above).

It was this discovery that led Maxwell to realise that light is an electromagnetic wave. This really is a radical thought. People at the time could barely understand what he was saying. The famous physicist William Thomson (aka Lord Kelvin) said that Maxwell had “lapsed into mysticism”.1

Energy in electromagnetic waves

A key fact about electromagnetic waves in Maxwell’s theory is that they carry energy through a vacuum. This is because there is an energy density associated with the electric and magnetic fields.

The energy per unit volume comes from adding the electric and magnetic energy densities:

\[u = \frac{1}{2} \epsilon_0 \vert \vec{E} \vert^2 + \frac{1}{2 \mu_0} \vert \vec{B} \vert^2 = \frac{\vert \vec{B} \vert^2}{\mu_0}\]

from substituting $\vert \vec{E} \vert = c \vert \vec{B} \vert$.

Since the wave moves at the speed of light, the energy flows along at the speed of light and the energy flux $S$ (i.e. energy passing a unit area per unit time) is given by

\[S \equiv uc = \frac{c \vert \vec{B} \vert^2}{\mu_0} = \frac{\vert \vec{E} \vert \vert \vec{B} \vert}{\mu_0},\]

again using $\vert \vec{E} \vert = c \vert \vec{B} \vert$.

In fact this energy flow has a direction – along the direction the wave propagates. So we may write it as a vector $\vec{S}$, and then

\[\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}\,,\]

because the wave propagation direction is perpendicular to both $\vec{E}$ and $\vec{B}$.

$\vec{S}$ is known as the Poynting vector.

Experimental evidence for electromagnetic waves

Maxwell was only proved right via 1887 experiments by Heinrich Hertz

  • Hertz used an inductor to generate a very high oscillating voltage which sparked across a gap in the circuit to make visible sparks
  • He positioned another wire loop with a gap at different distances away, and showed that sparks also appeared there — despite there being no power source within that second loop.
  • By careful experimentation with a metal reflecting plate, he showed that this wasn’t just induction, but rather an electromagnetic wave with the properties (including speed) predicted by Maxwell.
  • In retrospect, Hertz had built the first ever radio transmitter and receiver. But Hertz made no mention of any possible use for electromagnetic waves in his writing – this was pure discovery science.2

Generating electromagnetic waves of different wavelengths

More generally, electromagnetic waves are generated by oscillating charges. The faster the charge oscillates, the longer the wavelength (according to the standard relationship for waves, $\lambda = c/f$ where $f$ is the frequency and $\lambda$ is the wavelength).

You can try moving the charge below at different oscillation frequencies to see how the wavelength changes. (Note that there are actually two charges, a positive and a negative, initially stacked on top of each other to keep the overall charge neutral. This is just like a radio antenna or other typical source of electromagnetic waves, where there is no overall charge.)

Thermal emission is a special case where electrons in a material are moving essentially randomly, so they generate electromagnetic waves with a range of wavelengths, dependent on the temperature. This is why hot objects glow.

Wavelength range Frequency range Name Example natural source Example artificial source
$>1$ cm $< 30$ GHz Radio waves Planetary magnetospheres Electric currents in antennae
$1$ mm – $1$ cm $30 - 300$ GHz Microwaves Molecular rotation Microwave oven magnetron; radar systems
$1\,\mu$m – $1$ mm $300$ GHz – $300$ THz Infrared Thermal emission from warm objects Infrared lamps
$400$ nm – $700$ nm $430$ THz – $750$ THz Visible light Thermal emission from stars like the Sun LEDs, lasers
$10$ nm – $400$ nm $750$ THz – $30$ PHz Ultraviolet Electron transitions in atoms; thermal emission from very hot stars UV lamps (e.g., using mercury vapour)
$0.01$ nm – $10$ nm $30$ PHz – $30$ EHz X-rays Electron transitions in atoms Electron deceleration (e.g., X‑ray tubes)
$<0.01$ nm $> 30$ EHz Gamma rays Radioactive decay; catastrophic astrophysical explosions Nuclear reactors, particle accelerators, positron annihilations (in PET scans)

Final thought

In this class, I hope you will learn not merely results, or formulae applicable to cases that may possibly occur in our practice afterwards, but the principles on which those formulae depend, and without which the formulae are mere mental rubbish.

— James Clerk Maxwell, 1860, inaugural lecture at Kings College London3

Footnotes

  1. In retrospect, Thomson wasn’t a good judge of theories; he also argued strongly against natural evolution. For more about the reaction to Maxwell’s theory see Faraday, Maxwell and the Electromagnetic Field by Nancy Forbes and Basil Mahon (2019), chapter 13. Sadly it’s not available as an e-book but there’s a copy in the library. ↩

  2. The more colourful quotes about Hertz proudly saying his research was of “no use whatsoever” as given on wikipedia and elsewhere don’t have reliable primary sources. But certainly his writing makes no reference to any possible practical applications. ↩

  3. A lovely sentiment, but apparently Maxwell was a terrible lecturer. ↩