Homopolar Motor

The whole motor: a wire, a battery, a magnet

Stand a AA battery upright on a neodymium magnet. Bend a short piece of copper wire so its top touches the battery's positive terminal and its bottom legs brush the edge of the magnet. Let go — the wire spins, sometimes hundreds of times a minute, with no gears, no coils, no electronics. That is a homopolar motor, and it is the single simplest electric motor that exists.

The trick is geometry. Current enters at the top terminal, flows down and outward through the wire to the magnet's rim, then back up through the magnet and battery body. That outward leg of current sits inside the magnet's vertical field. A current carrying conductor in a magnetic field feels a sideways push — the same force that runs every loudspeaker and electric car. Here the push is always tangential, so instead of jerking once and stopping, the wire keeps getting shoved around the circle, turn after turn.

"Homopolar" means single polarity: the field and the current geometry never flip. That one word explains why this motor needs none of the switching hardware that every other DC motor relies on.

How it works — the force is always sideways

The force on a straight current-carrying wire in a magnetic field is:

F = I L × B

where I is the current, L is a vector along the wire in the direction of current flow with length equal to the wire segment, and B is the magnetic field. The cross product means the force is perpendicular to both the current and the field.

Now set up the homopolar geometry. The magnet's field B points straight up (axial, along the spin axis). The useful part of the current flows radially outward (or inward), away from the axis. The cross product of a radial vector with a vertical vector is azimuthal — it points around the circle:

radial (I) × axial (B)  =  azimuthal (F)
r̂ × ẑ  =  −θ̂   (and ẑ × r̂ = +θ̂)

Either way the result lies along θ̂ — purely azimuthal. The sign just tells you which way around the circle the wire turns, and that flips if you reverse the current or flip the magnet.

Because the geometry of "radial current, axial field" looks identical no matter how far the wire has rotated, the tangential force is the same at every angle. There is never a point in the rotation where the torque wants to reverse — so unlike a coil motor, there is nothing to commutate. The wire simply accelerates until something balances the torque.

The governing physics and torque

To get torque, multiply the tangential force by the lever arm (the radius at which it acts). For a wire carrying current I that crosses the field over a radial span from r₁ to r₂ in a uniform axial field B, the torque about the axis is:

τ = ∫ r · (I B) dr  =  I B · (r₂² − r₁²) / 2

For a Faraday disc of radius R carrying its current from axle to rim, this is simply:

τ = ½ I B R²

As the conductor spins, it sweeps through the field and generates a motional (back) EMF by the same Lorentz physics, now acting on the moving charges:

ε_back = ∫ (v × B) · dL  =  ½ B ω R²

where ω is the angular speed. The current that actually flows is set by Ohm's law with this back-EMF subtracted:

I = (V_battery − ε_back) / R_total

At startup ω = 0, so back-EMF is zero and current (hence torque) is maximum. As it speeds up, back-EMF rises, current falls, and torque drops. The motor settles at the steady-state speed where the remaining torque just overcomes friction. This is the same speed-regulation mechanism every motor uses — it just happens here without any switching.

Key conditions to make it actually spin

Three things must all be true, or the wire just sits there:

• The current must have a radial component crossing the field. A purely axial current (straight up the axis) is parallel to B, and L × B = 0 — no force. The wire's geometry must carry current sideways through the field.
• There must be a low-resistance closed loop with a sliding contact. The spinning wire has to keep touching the magnet (or a mercury pool, as Faraday used) so current keeps flowing as it rotates. That contact point is both essential and the motor's main source of friction.
• The field must be strong and the loop nearly shorted. Toy builds use neodymium magnets (B ≈ 0.3–0.5 T at the surface) and a bare copper loop of well under an ohm, so currents of several amps flow. Weak fridge magnets (B ≈ 0.01 T) barely move the wire.

Numbers — a real desktop build

Quantity	Typical value	Note
Battery voltage V	1.5 V (AA cell)	Acts as a near-short-circuit source
Loop resistance R	~0.3 Ω (mostly internal)	Bare wire adds almost nothing
Current I	~3–5 A	I = V / R, limited by internal resistance
Magnet field B	0.3–0.5 T (neodymium)	At the magnet face
Wire radius r	~0.01 m (1 cm)	Lever arm for the torque
Tangential force F = I·B·L	~0.01–0.03 N	For a ~1–2 cm crossing length
Steady speed	~few hundred to ~2000 RPM	Set by friction + back-EMF
Run time on one cell	A few minutes	High current drains the AA fast

The forces are tiny — hundredths of a newton — but the wire is light and nearly frictionless on its pivot, so even a small steady tangential push spins it up quickly. The limiting factor is almost never torque; it is the battery overheating and dying from the huge short-circuit current.

Homopolar motor vs ordinary DC motor

Property	Homopolar motor	Ordinary brushed DC motor
Conductor geometry	Single radial conductor in axial field	Multi-turn coil in a transverse field
Force direction	Always azimuthal — same at every angle	Reverses every half turn as coil flips
Commutator / brushes	None needed (single polarity)	Required to reverse coil current each half turn
Output voltage if run as generator	Pure DC, single polarity (Faraday disc)	AC, mechanically rectified to DC
Voltage vs current character	Very low voltage, very high current	Higher voltage, more modest current
Practical efficiency	Low — near short circuit, contact losses	High — 70–95% in good designs
Torque ripple	None — perfectly smooth	Cogging / ripple from commutation
Typical use today	Demonstrations; high-current pulsed power	Tools, toys, appliances, vehicles

Where it shows up — from Faraday's mercury to railguns

• Faraday's 1821 rotation. The very first device to convert electricity into continuous mechanical motion: a current-carrying wire dipping into mercury around a magnet, rotating endlessly. Every motor since descends from it.
• The Faraday disc (1831). Run the same geometry as a generator — spin a copper disc in an axial field and tap axle-to-rim — and you get the first DC generator, producing steady single-polarity voltage with no commutator.
• Homopolar generators for pulsed power. Industrial homopolar machines spin massive conducting rotors to store kinetic energy, then dump enormous low-voltage, high-current pulses — used to drive railguns, fusion experiments, and resistance welders. Devices like Australia's old ANU machine reached the megajoule and megawatt-pulse scale.
• Teaching the Lorentz force. Because it has just three parts and one equation, the homopolar motor is the canonical classroom demo for F = I L × B and the right-hand rule.
• Concept seed for maglev and rail propulsion. The same "current-crossing-field-makes-force" idea, scaled and arranged differently, underlies linear motors and electromagnetic launch.

Common misconceptions and edge cases

• "The current spins the magnet." No — the force acts on the moving charges in the conductor, so the part free to move (usually the wire) is what spins. Build it so the magnet is free and the wire is fixed, and the magnet spins instead. Either way it is F = I L × B on the conductor.
• "It needs alternating current." The opposite. A homopolar motor runs on pure DC and would not work cleanly on AC — reversing the current 50–60 times a second would just reverse the torque that often and produce no net spin.
• "Stronger battery voltage means faster spin." Up to a point, but the bottleneck is current, not voltage, and the loop is nearly a dead short. A higher-voltage source mostly means more heat and a faster-dying battery, not proportionally more speed.
• "There is no back-EMF because there are no coils." There is. Any conductor moving through B has a motional EMF (v × B) that opposes the supply. It is what sets the steady-state speed — without it the wire would (ideally) accelerate forever.
• "Flipping the magnet over does nothing." It reverses B, which reverses the cross product, which reverses the spin direction. Flipping the battery does the same. Flip both and it spins the original way.
• The Faraday-paradox edge case. If you make the magnet itself the rotor and ask "does the field rotate with it?", you reach the classic homopolar/Faraday paradox — the field is axially symmetric, so rotating the magnet does not change the flux geometry the way intuition expects. What matters is the relative motion of the conductor through B, not whether the magnet turns.