Motor effect and Fleming's rule

Finding the direction with Fleming’s left-hand rule

For a conductor at right angles to the magnetic field, the force acts perpendicular to both the current and the field. There are therefore three directions to keep track of.

Hold your left thumb, first finger and second finger mutually at right angles:

• First finger → magnetic field, from the magnet’s north pole towards its south pole across the gap.
• Second finger → conventional current, from positive towards negative through the wire.
• Thumb → force on the conductor.

Rotate your whole hand until two fingers match the two known directions. The remaining finger then gives the unknown direction. The thumb shows the force, whether or not the conductor is free to move.

For example, suppose the field points from left to right and the current flows out of the page towards you. Point your first finger right and your second finger towards you: your thumb points upwards, so the force on the wire is upwards. The force on the magnet is downwards.

Reversing the current reverses the wire’s magnetic field and therefore reverses the force. Reversing the magnet’s field also reverses the force. Reversing both leaves the force in its original direction.

Calculating the force

When the current-carrying conductor is at right angles to the magnetic field, use:

$$F = BIl$$

Here:

• $F$ is the force on the conductor, in newtons (N).
• $B$ is the magnetic flux density, in tesla (T).
• $I$ is the current, in amperes (A).
• $l$ is the length of conductor within the magnetic field, in metres (m).

Magnetic flux density measures how strong the magnetic field is. One tesla is equivalent to one newton per ampere metre, written $1\text{ T} = 1\text{ N}/(\text{A}\,\text{m})$.

For example, a wire carries a current of $3.0\text{ A}$, with $0.10\text{ m}$ of wire at right angles to a field of magnetic flux density $0.50\text{ T}$:

$$F = 0.50 \times 3.0 \times 0.10 = 0.15\text{ N}$$

This is the size of the force on the wire. The magnet experiences a force of the same size in the opposite direction; Fleming’s rule determines the direction on the wire.

What changes the force?

The equation shows that increasing the magnetic flux density, current or length of wire in the field increases the force. Doubling any one of these, while keeping the others unchanged, doubles the force.

Orientation matters too. The force is greatest when the conductor is perpendicular to the field. A conductor parallel to the field experiences no motor-effect force. The equation $F = BIl$ applies to the perpendicular case, not to every possible angle.

If another quantity is unknown, rearrange the equation:

$$B = \frac{F}{Il}, \qquad I = \frac{F}{Bl}, \qquad l = \frac{F}{BI}$$

Reverse current or field → reverse force. Reverse both → force unchanged.

Force calculation

For a conductor at right angles to the field:

$$F = BIl$$

$F$: N; $B$: T or N/(A m); $I$: A; $l$: m of conductor within the field.

Force increases with $B$, $I$ and $l$. It is greatest at right angles and zero when the conductor is parallel to the field.