Faraday’s paradox is one of those beautiful knots in physics where motion, magnetism, and electricity seem to trip over each other.
It comes from Michael Faraday’s 1831 experiments with electromagnetic induction. The paradox is usually described with the Faraday disc (homopolar generator):
- Imagine a conducting copper disc rotating in a uniform magnetic field (say, field lines pointing along the axis of the disc).
- A wire touches the rim and another touches the axle.
- As the disc spins, a current flows – even though the magnetic field is uniform and there’s no obvious “cutting of flux lines” as Faraday’s own law seemed to require.
The paradox deepens when you change which part is rotating:
- Disc rotates, magnet fixed current flows.
- Magnet rotates, disc fixed no current flows.
- Both rotate together, same speed current still flows!
At first glance, this seemed to contradict Faraday’s own induction law, because the magnetic flux through the disc isn’t changing.
Resolution
It isn’t actually a contradiction once you dig into reference frames and Lorentz forces:
- Charges in the rotating disc move with the disc’s material.
- In their motion through the magnetic field, they feel a Lorentz force $\mathbf{F} = q(\mathbf{v} \times \mathbf{B})$.
- This force pushes them radially, creating a potential difference.
The key is that the disc’s motion relative to the magnetic field is what matters, not just the flux picture. The flux-cutting version of Faraday’s law fails in this setup, but the general Maxwell-Faraday equation and Lorentz force law explain it perfectly.
In short: Faraday’s paradox is that a current can be generated in a homopolar generator even when the magnetic flux doesn’t change – apparently contradicting Faraday’s induction law, but actually resolved by considering the motion of charges in the magnetic field.
Is the Magnetic Field Static for a Rotating Physical Magnet?
Excellent question – and it cuts right into the heart of why Faraday himself was so unsettled by his own disc experiment.
If you take a permanent magnet and spin it about its magnetic axis, you might expect that the magnetic field would “rotate” with the magnet. But from the point of view of classical electromagnetism, that isn’t what happens:
- A static magnetic field (from a permanent magnet or a DC current loop) is described by $\mathbf{B} = \nabla \times \mathbf{A}$.
- If the source distribution (the bound currents inside the magnet, or the actual current in a coil) rotates rigidly about the magnet’s symmetry axis, the field outside remains time-independent – it does not “spin” through space.
That’s because the field is axisymmetric: rotating the physical magnet around its magnetic axis doesn’t actually change the field configuration. The field lines are already symmetric about that axis.
So the answer is:
- Yes, the magnetic field is static (does not rotate in space) when the magnet spins about its own axis of symmetry.
- The field distribution only changes if you rotate the magnet about a different axis (tilting it, or flipping north/south). In that case the field in space does change with time, and you can induce emf in nearby conductors from that time-varying field.
This is exactly why, in Faraday’s paradox:
- When the magnet alone is rotated about its axis, no emf is induced in the stationary disc: the field remains static.
- When the disc rotates, the charges inside the conductor move through the static field, and the Lorentz force generates current.
- When both rotate together, the charges still move relative to the static field, so you still get current.
