The Motor Effect

In short: The motor effect is the force on a wire carrying a current in a magnetic field: $F = n I l B sin θ$ . The force is largest when the wire is at right angles to the field, and its direction comes from the right-hand palm rule.

Current makes a magnetic field

A wire carrying a current is surrounded by a magnetic field of circles around the wire.

Right-hand grip rule: grip the wire with your right thumb along the current. Your fingers curl the way the field goes.
The field is stronger close to the wire and weaker further away.

Put a current-carrying wire in an external magnetic field and it feels a force. This is the motor effect. The size of the force is:

$F = n I l B sin θ$

$n$ = number of wires (1 for a single wire)
$I$ = current (A)
$l$ = length of wire inside the field (m)
$B$ = magnetic field strength (T)
$θ$ = angle between the wire and the field

The force is biggest when the wire is at right angles to the field. A wire parallel to the field feels no force.

Which way is the force?

Use the right-hand palm rule:

Fingers point along the field $B$ (north to south).
Thumb points along the current $I$ (positive to negative, conventional current).
Palm pushes in the direction of the force $F$ .

The force is always at right angles to both the current and the field. Reverse the current or reverse the field and the force reverses. Reverse both and it stays the same.

Two parallel wires

Each wire sits in the magnetic field of the other, so each feels a force.

Currents in the same direction: wires attract.
Currents in opposite directions: wires repel.

The force per metre of wire is:

$\frac{F}{l} = \frac{μ _{0}}{2 π} \frac{I _{1} I _{2}}{r}$

$μ_{0} = 4 π \times 1 0^{- 7}$ N A⁻² (the magnetic constant), so $\frac{μ _{0}}{2 π} = 2 \times 1 0^{- 7}$
$r$ = distance between the wires (m)

By Newton's third law, the two wires push or pull on each other with equal and opposite forces, even if their currents are different.

What changes the force between wires?

Double one current: double the force.
Double the distance: half the force (not a quarter: it is $\frac{1}{r}$ , not $\frac{1}{r ^{2}}$ ).

The Motor Effect

Current makes a field

Circles round the wire
Right-hand grip rule

Force on a wire

F = nIlB sin θ
Largest at 90°
Zero when parallel

Direction

Right-hand palm rule
Fingers B, thumb I, palm F
Reverse I or B: force reverses

Parallel wires

Same direction: attract
Opposite: repel
F/l = μ₀I₁I₂ / 2πr

Newton's third law

Equal and opposite forces

Year 12Measuring fields and the ampere

Current balance. Place a horizontal wire on an electronic balance, inside a magnetic field. As the current rises, the balance reading changes by $I l B$ . A graph of force against current is a straight line through the origin with gradient $l B$ , so $B = \frac{gradient}{l}$ .

The ampere. The old definition of one ampere was the current in two long parallel wires 1 m apart that gives a force of $2 \times 1 0^{- 7}$ N per metre. It is built directly on the formula above.

Three wires. For a wire between two others, find the force from each neighbour separately (including direction), then add them as vectors.

Wire at an angle. If a wire crosses the field at an angle, only the part of it at right angles to $B$ counts. For a bent wire, use the straight-line distance across the field between the points where it enters and leaves.

Balancing forces. A rod hanging from two strings in a field, with the magnetic force upwards: $2 T + I l B = m g$ . If the force were downwards, $2 T = m g + I l B$ .

Force makes acceleration. A rod of mass $m$ across rails carrying current $I$ in a field $B$ starts to accelerate at $a = \frac{I l B}{m}$ (ignoring friction). Two wires on trolleys pushed apart by $F = \frac{μ _{0}}{2 π} \frac{I _{1} I _{2}}{r} l$ each accelerate at $\frac{F}{m}$ .

Between two wires. With equal currents in the same direction, the two fields at the midpoint point opposite ways and cancel, so the field there is zero. With opposite currents, they add.

AC in series. If two parallel wires are connected so the current flows out along one and back along the other, the currents are always opposite, even as the AC reverses. So the wires always repel, with a force that pulses.

Beams of charges. A beam of electrons is a current in the opposite direction to the electrons' motion. Electrons moving in the same direction as the current in a nearby wire act like an opposite current, so the beam is pushed away from the wire.

A balance that does not start at zero. If the force against current graph is straight but does not pass through the origin, the gradient still gives $l B$ . The intercept is a reading that has nothing to do with the current, such as the balance not being zeroed (a systematic error).

Quick summary

A current-carrying wire in a magnetic field feels a force: $F = n I l B sin θ$ .
The force is biggest when the wire is at 90° to the field, and zero when parallel.
Direction: right-hand palm rule (fingers $B$ , thumb $I$ , palm $F$ ).
Parallel wires attract if their currents are in the same direction and repel if opposite.
Force between wires: $\frac{F}{l} = \frac{μ _{0}}{2 π} \frac{I _{1} I _{2}}{r}$ . The forces are equal and opposite.

Quiz

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A wire carrying a current is placed in a magnetic field. What is this effect called?
1. A The generator effect
2. B The motor effect
3. C Induction
4. D The photoelectric effect
Show answer
B: The motor effect
A force on a current-carrying wire in a magnetic field is the motor effect.
A wire carries 2.0 A at right angles to a 0.50 T field. The length in the field is 0.40 m. The force is:
1. A 0.40 N
2. B 1.6 N
3. C 2.5 N
4. D 0.10 N
Show answer
A: 0.40 N
F = IlB = 2.0 × 0.40 × 0.50 = 0.40 N.
A wire lies parallel to a magnetic field. The force on it is:
1. A Largest
2. B Zero
3. C Half the maximum
4. D Upwards
Show answer
B: Zero
sin 0° = 0, so a wire parallel to the field feels no force.

The Motor Effect

Current makes a magnetic field

The motor effect

Which way is the force?

Two parallel wires

What changes the force between wires?

Quick summary

Quiz

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