Electromagnetism

What is electromagnetism?

Electricity and magnetism feel like two different things. One shocks you off a doorknob, the other sticks a note to the fridge. They are really two faces of a single force.

Here is the link. A moving electric charge makes a magnetic field. That is how an electromagnet works, and how the motor in a fan or a drill spins. Run the trick backwards and a changing magnetic field pushes on charges and makes electricity flow. That is how a generator works, and how power stations light your house.

Now the surprise. When you wiggle a charge back and forth, it makes a changing electric field, which makes a changing magnetic field, which makes a changing electric field, and so on. That ripple of fields feeding each other travels outward all on its own, at the speed of light. The ripple is light.

Radio waves, the microwaves in your oven, the visible colours your eyes catch, and X-rays at the dentist are all exactly the same kind of ripple. The only thing that changes is how fast it wiggles.

How are electricity and magnetism connected?

Two simple rules start the story. Electric charges make electric fields, the invisible push that a charge feels near another charge. Moving charges, meaning electric currents, make magnetic fields, which is why a coil of wire carrying current behaves like a bar magnet.

In the 1830s Michael Faraday found the reverse effect. A changing magnetic field induces a voltage, and so drives a current, even with nothing moving but the field itself. Push a magnet through a coil and current flows. This is induction, and it runs every generator and transformer on the grid.

James Clerk Maxwell added the missing piece and made the picture symmetric. If a changing magnetic field makes an electric field, then a changing electric field ought to make a magnetic field. He wrote that term in. It was not obvious from experiments at the time, but without it the equations were inconsistent.

Put the two effects together and something remarkable falls out. A changing electric field makes a magnetic field, whose change makes an electric field, whose change makes a magnetic field. The pattern can sustain itself and roll forward as a wave, with the electric and magnetic fields at right angles to each other and to the direction it travels. Maxwell worked out how fast it must go:

\[ c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}. \]

Here \(\varepsilon_0\) and \(\mu_0\) are constants you can measure with simple bench experiments on charges and currents, no light involved. Plug them in and the speed comes out at about 300,000 kilometres per second. That was already the measured speed of light. The match was the clue: light is an electromagnetic wave.

What sets one kind of light apart from another is only its frequency, how many times per second the fields wiggle, or equally its wavelength, the distance between crests. Slow and long gives radio waves. Faster gives microwaves, then infrared heat, then the narrow band of visible colours from red to violet, then ultraviolet, X-rays, and gamma rays. One family, sorted by frequency.

Maxwell's equations and electromagnetic waves

The four equations

Classical electromagnetism is the whole of what Maxwell's four equations say. Gauss's law for the electric field: charge is the source of \(\mathbf{E}\), with field lines beginning and ending on charge. Gauss's law for magnetism: there are no magnetic charges, so \(\mathbf{B}\) lines never end, they only close on themselves. Faraday's law: a changing magnetic field curls up an electric field. The Ampère–Maxwell law: both a current and a changing electric field curl up a magnetic field. In SI form, with sources \(\rho\) and \(\mathbf{J}\),

\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}, \qquad \nabla \cdot \mathbf{B} = 0, \]

\[ \nabla \times \mathbf{E} = -\,\partial_t \mathbf{B}, \qquad \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0\, \partial_t \mathbf{E}. \]

The displacement current

The last term in the fourth equation, \(\mu_0 \varepsilon_0\, \partial_t \mathbf{E}\), is Maxwell's displacement current. It lets a changing electric field act as a source of magnetic field just as a real current does. Beyond making the equations mutually consistent, and saving charge conservation across a charging capacitor where no conduction current flows, it is the term that closes the loop between the two fields and permits waves.

From the curl to a wave

Take the curl of Faraday's law in empty space, where \(\rho = 0\) and \(\mathbf{J} = 0\). Using the identity \(\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}\) together with \(\nabla \cdot \mathbf{E} = 0\), and substituting the Ampère–Maxwell law for \(\nabla \times \mathbf{B}\), the two first-order equations collapse into one second-order equation,

\[ \nabla^2 \mathbf{E} = \mu_0 \varepsilon_0\, \partial_t^2 \mathbf{E}, \]

the wave equation, with propagation speed \(c = 1/\sqrt{\mu_0 \varepsilon_0}\). An identical equation holds for \(\mathbf{B}\). That Maxwell's measured constants delivered exactly the speed of light was the argument that light is electromagnetic.

Structure of the wave

For a plane wave both fields are transverse: \(\mathbf{E}\) and \(\mathbf{B}\) each lie perpendicular to the direction of travel and to each other, they oscillate in phase, and their magnitudes obey \(E = cB\). The orientation of \(\mathbf{E}\) defines the polarisation, which can be linear, circular, or elliptical. Because the fields are transverse, the polarisation state is a genuine extra degree of freedom, exploited by everything from sunglasses to LCD screens.

Energy and momentum

The fields are not bookkeeping devices; they carry energy and momentum. The energy flux is the Poynting vector \(\mathbf{S} = \tfrac{1}{\mu_0}\,\mathbf{E} \times \mathbf{B}\), pointing along the travel direction. Momentum density is \(\mathbf{S}/c^2\), so light exerts pressure when it lands on or reflects from a surface. Radiation pressure is faint in daylight but real, and it drives solar sails and shapes the tails of comets.

A door to relativity

Maxwell's equations single out one speed, \(c\), with no reference to who is watching. That refusal to pick a preferred frame is the hint Einstein followed. The equations are already Lorentz-invariant, and in the relativistic view the electric and magnetic fields are components of a single object, the electromagnetic field tensor. What one observer calls a pure electric field another, moving past, sees partly as magnetic. Electricity and magnetism are not merely linked. They are the same field seen from different frames.