investigate and analyse theoretically and practically the force on a current-carrying conductor in a magnetic field, $F = n B I L$, and apply this to the operation of a simple DC motor including the role of the split-ring commutator

What this dot point is asking

VCAA wants you to apply $F = n B I L$ to a current-carrying conductor in a magnetic field, find the direction with the right-hand slap rule, and use these ideas to explain how a simple DC motor produces continuous rotation, including the role of the split-ring commutator.

The answer

Force on a straight current-carrying conductor

A straight wire of length $L$ carrying current $I$, with $n$ turns or wires, in a uniform field $B$ perpendicular to the current, feels a force:

(For a single wire, $n = 1$.) If the current is at angle $\theta$ to the field, replace $B$ with $B \sin\theta$. If $I$ is parallel to $B$, the force is zero.

The direction is given by the right-hand slap rule:

• Fingers point in the direction of conventional current $I$.
• Curl them toward the field $B$.
• The thumb points in the direction of the force $F$.

The force is perpendicular to both $I$ and $B$.

Torque on a current loop

Consider a rectangular coil of $n$ turns, width $w$ and length $L$, carrying current $I$ in a uniform field $B$ parallel to the plane of the coil. The forces on the two long sides are equal in magnitude ($F = n B I L$) but opposite in direction (one up, one down). These forces form a couple that rotates the coil about its axis.

The torque is $\tau = n B I L w$ when the coil is parallel to the field, falling to zero when the coil is perpendicular to the field (in the vertical position). The torque varies as $\cos\alpha$ where $\alpha$ is the angle between the coil plane and the field.

The simple DC motor

A simple DC motor consists of:

1. A rectangular coil of $n$ turns mounted on an axle.
2. A pair of permanent magnets producing a roughly uniform field across the coil.
3. A split-ring commutator attached to the axle, with two carbon brushes that touch the commutator and connect it to the external DC supply.

When current flows through the coil:

• The force on one side of the coil is up; the force on the other side is down (right-hand slap rule).
• These forces produce a torque that rotates the coil.
• As the coil passes the vertical position, the two halves of the split-ring commutator swap brush contacts. This reverses the current direction in the coil.
• After the reversal, the force on each side again pushes the coil in the same rotational direction.

The coil continues to rotate continuously in one direction. Without the commutator, the torque would reverse every half turn and the coil would oscillate rather than rotate.

Why a real motor uses many turns and an iron core

A real motor has hundreds of turns ($n$ large) to multiply the torque without needing huge currents. A laminated iron core inside the coil concentrates the field $B$, increasing torque further. Multiple coils at different angles smooth out the torque so the rotation is uniform rather than pulsing.

Worked example with numbers

A DC motor coil has 50 turns, each side 8.0 cm long. The coil sits in a 0.20 T field. Current is 1.5 A. Find the maximum force on one side and the maximum torque if the coil is 6.0 cm wide.

Force on one long side:

$F = n B I L = 50 \times 0.20 \times 1.5 \times 0.08 = 1.2$ N.

Maximum torque (coil parallel to field):

$\tau = F \times w = 1.2 \times 0.06 = 0.072$ N m.

(Equivalent to $\tau = n B I L w = 50 \times 0.20 \times 1.5 \times 0.08 \times 0.06$.)

Try it: DC motor torque calculator - enter turns, field, current and coil dimensions and get the maximum torque.

Common traps

Forgetting the $n$ in $F = nBIL$. VCE coils have many turns; missing $n$ gives an answer too small by a factor of 50 or 100.

Calculating the force on the short sides as if it adds to the torque. Forces on the short sides are parallel to the axis, do no useful work, and try to stretch the coil rather than rotate it. Only the long sides contribute torque.

Confusing split-ring with slip ring. A split-ring commutator reverses the current direction in the coil every half-turn (used in DC motors and DC generators). Slip rings keep the connection continuous (used in AC generators).

Stating the motor stops at the vertical position. The torque is momentarily zero at the vertical position, but the coil's inertia carries it past, and after the commutator swaps the contacts the torque resumes in the same rotational direction.

Using the right-hand rule for electrons. Conventional current flows from + to -, opposite to the electron drift. Apply the right-hand rule using conventional current.

In one sentence

A current-carrying conductor in a magnetic field feels $F = n B I L$ perpendicular to both $I$ and $B$ (right-hand slap rule), and in a DC motor the two opposite forces on a coil produce a torque while the split-ring commutator reverses the coil current every half-turn so the rotation continues in one direction.