Why a wire carrying current jumps in a magnet, and how that becomes a motor

How to use this simulation

1. Start in “Straight wire” mode with the defaults: B = 0.5 T, I = 4 A, L = 0.1 m and θ = 90°. The red arrow is the force, F = 0.200 N. The dashed line is where the wire sat before the field pushed it.
2. Drag the angle slider towards 0°. The arrow shrinks, and at 0° the wire lies along the field and the force is 0.000 N. Try 30° too: the force is 0.100 N, exactly half the maximum.
3. Tick “Reverse the current”. The charges turn round and the force arrow flips to the other side of the page. Tick “Reverse the field” as well and it flips back. Two reversals cancel.
4. Compare the small hand diagram with the scene. First finger for the field, second finger for the current, thumb for the force: that is Fleming’s left-hand rule.
5. Switch to “DC motor”. A coil sits between the poles, seen end-on. Each side feels an equal and opposite force, so the coil turns. Watch the dots and crosses (current out of and into the page) swap each half turn as the split ring changes contact.
6. Untick the commutator. The coil swings a little, overshoots, and settles standing upright. Without the reversal, the push turns into a pull halfway round. Tick it again and the spin continues.

What the motor effect is

A moving charge feels a sideways push when it travels through a magnetic field. A current is a stream of moving charges, so a wire carrying current in a magnetic field feels the same push, added up over all its charges. The wire gets shoved, and because the charges cannot leave the metal, the wire itself moves. That is the motor effect.

Three things have to be there together: a current, a magnetic field, and a wire that is not lined up with the field. Take away any one and nothing happens. A compass needle near a wire is the same story told in reverse: the wire’s own field pushes the needle.

The force is not along the current and not along the field. It is at right angles to both, which is why people find it hard to picture. The hand rule below is the trick that makes it easy.

The formula: F = BIL sin θ

For a straight piece of wire of length L inside a uniform field B, carrying a current I, the force is the product of all three, times a factor for the angle.

The angle θ is between the wire and the field lines. When the wire is across the field (θ = 90°) the sine is 1 and the force is at its largest, BIL. When the wire runs along the field (θ = 0° or 180°) the sine is 0 and there is no force at all. In between, only the part of the field that is across the wire counts.

Fleming’s left-hand rule

Hold your left hand with the thumb, first finger and second finger at right angles to one another, like three axes in a corner of a room. Point the first finger along the magnetic field (north to south). Point the second finger along the conventional current. The thumb then points along the force.

Remember it by the first letters: Thumb for Thrust (force), First finger for Field, seCond finger for Current. Left hand for motors, because a motor is a case where current and field produce motion.

In the simulation the force is always perpendicular to the page, so the force arrow is drawn slanted, as if it were coming out of or going into the screen. Up and to the right means into the page, down and to the left means out of it. Reverse the current and the arrow flips, because the thumb has to turn round when one finger does.

Worked examples with the formula

Four short problems. Try each one on the sliders afterwards.

From a push to a spin: the coil

One wire only gets pushed once, then it runs out of magnet. To keep going we need something that turns. Bend the wire into a rectangular loop, put it between the poles on an axle, and run current round it.

The two long sides carry current in opposite directions (up one side, down the other). By the left-hand rule, the force on one side is up and on the other is down. A pair of equal and opposite forces with different lines of action is a couple. It does not move the loop sideways. It turns it.

The short sides at the ends are parallel to the field in the flat position, so they feel nothing useful, and their pushes cancel in any case.

The torque formula: τ = NBIA sin θ

For a coil of N turns and area A (length L times width W), each of the two long sides feels N B I L. The turning effect is that force times the distance across the coil, which comes out as the compact result below.

Here θ is the angle between the coil’s normal (the line sticking out of its face) and the field. When the coil lies flat along the field, the normal is at 90° and the torque is at its largest. When the coil faces the field, the normal is along it, θ = 0°, and the forces point along the coil’s own plane. They pull the loop straight and the torque is zero.

The commutator: why the motor does not get stuck

There is a problem. As the coil rotates past the vertical position, the side that was going up is now on the other side of the axle, and the left-hand rule now makes it push back down. Without any change, the coil would swing past, slow, turn round, and settle. Switch the commutator off in the simulation and watch exactly that.

The fix is the split-ring commutator. It is a ring cut into two halves, one joined to each end of the coil. Two fixed carbon brushes press on the halves and bring current in. As the coil goes through the vertical position, each half slides from one brush to the other, so the current in the coil is reversed at exactly the right moment. The side that is now on the left always has current in the direction that pushes it up.

The graph on the right of the simulation shows the effect. Without a commutator the torque follows sin θ and goes negative in the second half-turn. With one it follows |sin θ|, always positive. The torque still drops to zero twice per turn. The coil’s momentum carries it through those two dead spots.

• Brushes are fixed and carry the supply current. The split ring turns with the coil.
• Reversal happens when the coil plane is perpendicular to the field, where the torque is already zero.
• The average torque over a full turn is 0.637 of the maximum, the factor 2/π. For the simulated motor that is 0.204 N·m.

Making a real motor stronger and smoother

Four things raise the turning power: more turns, a stronger field (a bigger magnet or an electromagnet), a larger current, and a larger coil area. Each of them appears in the torque formula, and each has a slider in the simulation.

To make the spin smooth rather than pulsing, real motors use many coils set at different angles, with a commutator of many segments, so that when one coil is at its dead spot another is at full torque. They also wind the coils on a soft iron core, which concentrates the field.

To reverse a DC motor you can reverse the current or reverse the field. If you reverse both it spins the same way again. Try the two toggles together in the simulation.

Common mistakes

These are the places marks are usually lost.

• Using the angle to the field’s normal in F = BIL sin θ. For a wire, θ is the angle between the wire and the field.
• Saying the force is along the field or along the wire. It is always at right angles to both.
• Using the right hand. Left hand is for the motor effect.
• Forgetting that conventional current points from + to −, and that the second finger follows it.
• Calling the commutator a switch for the supply. It reverses the current in the coil, not the supply.
• Writing τ = BIL for a coil. A coil has N turns and an area, so τ = NBIA sin θ.

Exam corner

Typical short questions ask you to state Fleming’s left-hand rule, to give the direction of the force in a sketch, to write F = BIL sin θ and say when it is zero. Longer ones ask you to explain, with a diagram, how a DC motor works, naming the coil, magnet, split ring and brushes, and saying why the ring is split.

Numericals mostly use F = BIL sin θ and τ = NBIA sin θ. Write the formula, substitute with units, and give the final answer in newtons or newton metres. If a direction is asked for, draw the three fingers.

Revision list

Say these out loud before you close this page.

• A current in a magnetic field feels a force at right angles to both.
• F = BIL sin θ, with θ the angle between wire and field. Zero along the field, maximum across it.
• Left hand: thumb force, first finger field, second finger current.
• Reverse the current or the field and the force reverses. Reverse both and it stays.
• A coil feels a couple, so it turns: τ = NBIA sin θ.
• The split-ring commutator reverses the current each half turn, so the torque keeps one sign.
• Raising N, B, I or A raises the torque in proportion.

Frequently asked questions