Maxwell's equations and the speed of light

Four laws, one story

By 1860 people knew several laws of electricity and magnetism, found one at a time by different people. Maxwell put them together, added the displacement current, and found that they form a complete set of four equations. The set also predicts a surprise: a wave.

Here are the four laws, first in words and then as equations. Each one connects a field to its source.

1. Gauss's law for electricity. Electric charge makes an electric field. The flux of $\vec{E}$ out of a closed surface is the enclosed charge divided by $\varepsilon_0$.

2. Gauss's law for magnetism. There are no isolated magnetic poles. Magnetic field lines have no beginning and no end, so the flux of $\vec{B}$ out of any closed surface is zero.

3. Faraday's law. A changing magnetic flux makes an electric field that goes round in a loop. This is the induced emf of a coil.

4. Ampere-Maxwell law. A current and a changing electric flux make a magnetic field that goes round in a loop.

One more law finishes the picture. The Lorentz force on a charge $q$ moving at velocity $\vec{v}$ is $\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$. It says what the fields do to a charge. The four equations say how charges and changing fields make the fields.

The symmetry that gives a wave

Look at laws 3 and 4 side by side.

• A changing magnetic field makes an electric field (Faraday).
• A changing electric field makes a magnetic field (Maxwell's displacement current).

Each change makes the other field. Picture a region of space with no charges and no currents. Let the electric field change. It makes a magnetic field. That field changes too, so it makes an electric field a little further on. The two fields keep making each other, and the disturbance moves outwards. It needs no wire and no material. That is an electromagnetic wave.

Raise the frequency and watch the magnetic field grow. The induced field is proportional to $\omega$, the rate at which $E$ changes. At the 50 Hz of the mains it is far too small to notice. At radio frequencies it is easy to detect.

The speed of the wave

Maxwell worked out how fast the disturbance moves. We follow his result without the full calculation. The speed depends only on the two constants that appear in the laws, $\mu_0$ (the magnetic constant) and $\varepsilon_0$ (the electric constant):

A worked example. Use $\mu_0 = 4\pi \times 10^{-7}\ \text{T m/A}$ and $\varepsilon_0 = 8.854 \times 10^{-12}\ \text{C}^2/(\text{N m}^2)$.

1. The product. $\mu_0\varepsilon_0 = 12.566 \times 10^{-7} \times 8.854 \times 10^{-12} = 1.113 \times 10^{-17}\ \text{s}^2/\text{m}^2$.
2. The speed. We want $c$, so $c = \dfrac{1}{\sqrt{1.113 \times 10^{-17}}}\ \text{m/s} = 2.998 \times 10^{8}\ \text{m/s}$.
3. Compare. Measurements of the speed of light give $3.00 \times 10^{8}$ m/s. Maxwell's number from electric and magnetic measurements is the same. Light had been found among the waves of electromagnetism. He concluded that light is itself an electromagnetic wave, a bold claim for 1865.

In a material the constants change to $\mu = \mu_r\mu_0$ and $\varepsilon = \varepsilon_r\varepsilon_0$. The speed becomes

For a non-magnetic material, $\mu_r = 1$ and $n = \sqrt{\varepsilon_r}$. Take care: $\varepsilon_r$ depends on the frequency. Water has $\varepsilon_r \approx 80$ for a steady field, but $n^2 = 1.77$ for visible light. Use the value that belongs to the frequency of the wave.

A second example. A radio wave enters a material with $\varepsilon_r = 4.0$ and $\mu_r = 1$ at its frequency. Then $v = \dfrac{3.00 \times 10^{8}\ \text{m/s}}{\sqrt{4.0}} = 1.50 \times 10^{8}\ \text{m/s}$, which is half of $c$.

What the laws do not need

Nothing in the four laws says "light". The same laws give radio waves, microwaves, X-rays and every other wave of the spectrum, all with the same speed in vacuum. Only the frequency differs. That is the subject of the next course.

Other books. Many books write the four laws in a different form, with $\nabla$ symbols, but the content is the same. Some books group Gauss's law for magnetism and Faraday's law as the "homogeneous" pair, and the other two as the pair with "sources". The integral form above is what Class 12 uses.

Practise with new numbers

Try it in Python

The program finds the speed of an electromagnetic wave from $\mu_0$ and $\varepsilon_0$. Then it lists the speed in three materials from their relative permittivity.

Common mistakes

• Thinking Faraday's law needs a coil. The induced electric field exists in empty space. A coil only shows it.
• Forgetting the minus sign in Faraday's law. It tells the direction: the induced field opposes the change of flux (Lenz's law).
• Believing magnetic charges exist. Gauss's law for magnetism says $\oint \vec{B}\cdot d\vec{A} = 0$. A bar magnet cut in two gives two magnets, each with two poles.
• Using $\varepsilon_r$ of a steady field for a fast wave. The relative permittivity depends on frequency. The value for light is not the value for a steady field.

Remember these

• Four laws: Gauss (electric), Gauss (magnetic), Faraday, Ampere-Maxwell. The Lorentz force completes them.
• A changing $B$ makes $E$, and a changing $E$ makes $B$. Together they make a wave that needs no medium.
• In vacuum, $c = 1/\sqrt{\mu_0\varepsilon_0} = 3.00 \times 10^{8}$ m/s.
• In a material, $v = c/\sqrt{\mu_r\varepsilon_r}$ and $n = \sqrt{\mu_r\varepsilon_r}$.

Check yourself

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

1. Which of Maxwell's four equations says that magnetic field lines form closed loops?

2. Name the two effects that make an electromagnetic wave keep going in empty space.

3. A non-magnetic material has $\varepsilon_r = 9.0$ at the frequency of a radio wave. What is the speed of the wave in it, and the refractive index?

4. Why was the agreement between $1/\sqrt{\mu_0\varepsilon_0}$ and the measured speed of light so important?

References

• OpenStax, University Physics Volume 2, Chapter 16 (Electromagnetic Waves): Maxwell's equations and electromagnetic waves.
• NCERT Physics Class 12, Chapter 8 (Electromagnetic Waves): Maxwell's equations and the speed $c = 1/\sqrt{\mu_0\varepsilon_0}$.