Force on a Current-Carrying Conductor

The motor effect — a current-carrying conductor in a magnetic field experiences a force.

F = B I L sin θ

where θ is the angle between the current and the field.

The two extremes matter more than the formula

When the current is perpendicular to the field, sin θ = 1 and the force is maximum: F = BIL.

When the current is parallel to the field, sin θ = 0 and there is no force at all.

Fleming's left-hand rule gives the direction, for conventional current:

First finger — Field, from north to south.

seCond finger — Current, conventional, so positive to negative externally.

thuMb — Motion, the direction of the force.

Hold the three at right angles to each other. The left hand is for motors; the right hand is for generators, which is a distinction worth fixing early.

The force is perpendicular to both the current and the field — it is not along either of them. This is unlike every force met so far, and it is the source of most errors in this topic.

Defining the tesla — rearranging gives B = F/(IL), so one tesla is the flux density that produces a force of one newton on one metre of wire carrying one amp, placed at right angles to the field.

Example: this is the whole basis of the electric motor. A rectangular coil in a magnetic field has opposite currents in its two sides, so the forces on them act in opposite directions, producing a turning moment. A split-ring commutator reverses the current every half turn, so the torque always acts the same way round and the coil keeps rotating rather than settling.

Measuring B with a current balance — a stiff wire in a field is placed on a top-pan balance. Passing a current changes the reading by Δm, so the force is Δmg, and B = Δmg/(IL). Plotting the apparent change in mass against current gives a straight line whose gradient yields B, using all the readings rather than one.