Magnets & Electromagnetism

Electricity and magnetism are the same thing

Junior level — plain language, no maths

Magnets seem like their own kind of magic - a lump of metal that reaches out and grabs another one from across a gap. For centuries nobody connected them to electricity at all. Then in 1820 a Danish teacher called Hans Christian Ørsted was demonstrating an electric circuit to his class, and noticed that a compass needle sitting nearby twitched every time he switched the current on. That twitch changed physics forever.

What Ørsted had stumbled on is this: an electric current makes a magnetic field. Not a bit of one - a real one. Wind a wire into a coil, push current through it, and you have built a magnet as genuine as any lump of iron, with a north end and a south end. Switch the current off and the magnetism vanishes. Reverse the current and the north end becomes the south end. In the simulation above, turn up the current and watch a whole field of compass needles swing round to obey your coil instead of the Earth.

Then comes the beautiful part, found by Michael Faraday eleven years later: the street runs both ways. Move a magnet near a coil of wire and a current appears in the wire. Not because you connected a battery - there is no battery - but purely because the magnet moved. Push it in, current flows one way. Pull it out, current flows the other way. Hold it perfectly still inside the coil and, no matter how strong the magnet is, nothing at all happens. It is the changing that matters.

Nature also puts up a fight, and this is the part people find spooky. The current that appears always flows in whichever direction opposes what you are doing. Push the magnet in and the coil turns into a magnet that pushes back. Pull it out and the coil grabs at it. That is Lenz's law, and it is really just energy conservation wearing a costume: if the coil helped you instead of fighting you, you would get electricity for free. Every power station on the planet is someone forcing a magnet past a coil against exactly this resistance.

Fields, flux and Faraday's law

Student level — the core equations

A current-carrying wire is surrounded by a magnetic field that circles it, and the size of that field follows Ampère's law. Wind the wire into a long solenoid and the fields of the individual turns add along the axis, giving a nearly uniform interior field \(B = \mu_0 n I\), where \(n\) is turns per metre. Four thousand turns per metre at 3 A gives about 15 mT - roughly three hundred times the Earth's 50 µT, which is why a compass anywhere near it stops caring about north.

Outside the solenoid the field is a dipole, identical in shape to a bar magnet's, and it falls off as \(1/r^3\). That cube is worth pausing on. The radius at which the coil still beats the Earth's field scales only as \(I^{1/3}\): to double your reach you need eight times the current. Strong magnets are not far-reaching magnets.

Faraday's law handles the reverse direction. Define the magnetic flux through a coil of \(N\) turns and area \(A\) as \(\Phi = NBA\), and the induced electromotive force is \(\mathcal{E} = -\,d\Phi/dt\). Everything in that expression is a rate. A stationary magnet, however powerful, gives \(d\Phi/dt = 0\) and therefore nothing. Move it and the emf follows the gradient of the field, not the field itself - which is why, as a magnet drops through a coil, the induced voltage peaks twice with opposite signs and passes through exactly zero at the instant the magnet is centred and the field through the coil is strongest.

The minus sign is Lenz's law, and it is not decoration. The induced current circulates so that its own magnetic field opposes the change that produced it: an approaching north pole is met by an induced north pole, a departing one by an induced south. Flip the sign and you would have a coil that accelerated the magnet, extracting energy from nowhere. The mechanical work you do against that opposition is exactly the electrical energy you get out. That is a generator.

From Ørsted to Maxwell, and why magnetism is relativity

Scholar level — full mathematical depth

01Two laws, one field

Ampère's circuital law, \(\oint \vec B \cdot d\vec l = \mu_0 I_{\text{enc}}\), says a current threads a magnetic field around itself. Faraday's law, \(\oint \vec E \cdot d\vec l = -\,d\Phi_B/dt\), says a changing magnetic flux threads an electric field around itself. Written as a pair they already look suspiciously symmetric, and that symmetry is the seed of everything that follows. In differential form: \(\nabla \times \vec B = \mu_0 \vec J\) and \(\nabla \times \vec E = -\partial \vec B / \partial t\).

02The term Maxwell added

Ampère's law as stated is inconsistent: take the divergence of both sides and you get \(\nabla \cdot \vec J = 0\), which is false whenever charge accumulates - as it does on a charging capacitor. Maxwell repaired it with the displacement current, \(\nabla \times \vec B = \mu_0 \vec J + \mu_0 \varepsilon_0\,\partial \vec E/\partial t\). The repair was not cosmetic. With it, the equations in vacuum combine into a wave equation with speed \(c = 1/\sqrt{\mu_0 \varepsilon_0}\) - a number Maxwell could evaluate from purely electrical measurements, and which came out equal to the measured speed of light. Light was revealed to be an electromagnetic wave by an algebraic consistency fix.

03Lenz's sign, and why it cannot be otherwise

The minus sign in Faraday's law is fixed by energy conservation, not convention. Consider a coil of resistance \(R\): the induced current \(I = \mathcal{E}/R\) dissipates \(\mathcal{E}^2/R\), and that power must come from the agent moving the magnet. The retarding force on the magnet is therefore \(F = \mathcal{E}^2/(Rv)\), scaling as \(v\) for a given geometry - which is precisely why a magnet reaches terminal velocity in a copper tube, with eddy currents dissipating exactly the gravitational power input. Reverse the sign and the system becomes a runaway.

04Flux rule versus force law

Feynman flagged the flux rule \(\mathcal{E} = -d\Phi/dt\) as a rare case of a single formula covering two physically distinct mechanisms. When the circuit moves, the emf comes from the magnetic part of the Lorentz force, \(q\vec v \times \vec B\), acting on charges dragged through a static field. When the field changes and the circuit is still, the emf comes from a genuine induced electric field, \(\nabla \times \vec E = -\partial\vec B/\partial t\). Two different terms in the Lorentz force conspire to give the same answer - and there are contrived geometries where the flux rule fails while the force law does not.

05Magnetism as electrostatics seen from a train

The conspiracy is not a coincidence. Take a neutral current-carrying wire and a charge drifting alongside it. In the lab frame the wire is neutral and the force on the charge is purely magnetic. Boost into the charge's frame and the magnetic force vanishes - but the Lorentz contraction of the positive and negative charge densities differs, because they move at different velocities, so the wire is no longer neutral and there is an electric force instead. Same force, different names. Magnetism is what electrostatics looks like from a moving frame, and the factor \(1/c^2\) between them is why magnetic effects at everyday drift velocities of millimetres per second are as strong as they are: the enormous charge densities compensate.

06The missing monopole

Maxwell's equations are almost symmetric under swapping \(\vec E\) and \(\vec B\) - almost, because \(\nabla \cdot \vec B = 0\) has no magnetic-charge counterpart to \(\nabla \cdot \vec E = \rho/\varepsilon_0\). Cut a bar magnet in half and you get two magnets, never an isolated pole. Dirac showed in 1931 that the existence of even one monopole anywhere in the universe would force electric charge to be quantised, in units \(eg = n\hbar c/2\) - which would explain the otherwise unexplained fact that every electron carries exactly the same charge. None has ever been found, and the quantisation of charge remains unexplained.