Know that a current-carrying coil in a magnetic field may experience a turning effect and that the turning effect is increased by increasing: (a) the number of turns on the coil (b) the current (c) the strength of the magnetic field

4.5.5 The d.c. motor

Learning Objective

Know that a current‑carrying coil in a magnetic field may experience a turning effect and that the turning effect is increased by increasing:

1. the number of turns on the coil,
2. the current,
3. the strength of the magnetic field.

Principle of the Turning Effect

When a straight conductor of length \$l\$ carrying a current \$I\$ is placed in a uniform magnetic field \$B\$, it experiences a force

\$\mathbf{F}=I\,\mathbf{l}\times\mathbf{B}\$

For a rectangular coil of \$N\$ turns, each side of length \$l\$, the forces on opposite sides are opposite and produce a couple (torque) that tends to rotate the coil.

Torque on a rectangular coil

The magnitude of the torque \$\tau\$ produced by a single turn is

\$\tau = B I A \sin\theta\$

where \$A = l \times w\$ is the area of the coil and \$\theta\$ is the angle between the normal to the coil and the magnetic field. For \$N\$ turns the torque is multiplied by \$N\$:

\$\tau = N B I A \sin\theta\$

Factors that increase the turning effect

Construction of a simple d.c. motor

A basic d.c. motor consists of:

• A rectangular coil (armature) mounted on a shaft so it can rotate.
• Two permanent magnets that create a uniform magnetic field across the coil.
• A split-ring commutator that reverses the direction of current in the coil each half‑turn, ensuring continuous rotation.

Why the commutator is needed

Without a commutator the torque would reverse every half‑turn because the direction of the force on each side of the coil would change when the coil passes through the position where the magnetic field is perpendicular to the plane of the coil. The split‑ring commutator swaps the connections to the power source, keeping the torque in the same rotational direction.

Key points to remember

• The turning effect (torque) on a coil is given by \$\tau = N B I A \sin\theta\$.
• Increasing any of \$N\$, \$B\$, or \$I\$ increases the torque proportionally.
• The commutator ensures that the direction of current in the coil is reversed each half‑turn, producing continuous rotation.
• In practical motors the magnetic field is often produced by strong permanent magnets or electromagnets, and the coil may have many turns of fine wire to maximise torque.

Typical exam question

Question: A rectangular coil of \$20\$ turns has dimensions \$5\ \text{cm} \times 10\ \text{cm}\$ and carries a current of \$2\ \text{A}\$ in a uniform magnetic field of \$0.3\ \text{T}\$. Calculate the maximum torque produced by the coil.

Solution:

1. Calculate the area: \$A = 0.05\ \text{m} \times 0.10\ \text{m} = 5.0\times10^{-3}\ \text{m}^2\$.
2. Maximum torque occurs when \$\sin\theta = 1\$.
3. Use \$\tau = N B I A\$:

   \$\tau = 20 \times 0.3\ \text{T} \times 2\ \text{A} \times 5.0\times10^{-3}\ \text{m}^2 = 0.06\ \text{N·m}.\$

Thus the maximum torque is \$0.06\ \text{N·m}\$.