Torque on a Current Loop in a Magnetic Field Explained

A rectangular loop carrying current in a uniform magnetic field experiences forces on its sides that do not cancel as a turning effect. The two sides perpendicular to the field are pushed in opposite directions, forming a couple that rotates the loop. This torque is the working principle of the electric motor and the moving-coil galvanometer.

It matters because nearly every rotating electric machine begins with this single loop. Understanding how torque depends on turns, area, current and angle tells you how to make a motor stronger and why it needs a commutator to keep turning.

Key Concepts

Torque τ: turning effect in newton metre (N m). Loop area A: length × breadth in m². Number of turns N: multiplies torque because each turn carries current. Angle θ: between the normal to the loop plane and the field B; the plane's orientation decides sin or cos in the formula. Couple: two equal, opposite, parallel forces whose net force is zero but torque is not.

Formulas

τ = N I A B sinθ where τ in N m, I in ampere (A), A in m², B in tesla (T) and θ is angle between the loop's normal (area vector) and B. Maximum torque τ = NIAB when the plane of the loop is parallel to the field (normal perpendicular, θ = 90°); zero when the normal aligns with B. Also, τ = m × B with magnetic moment m = NIA. Assumption: field uniform across the loop. Limit: radial-field designs in meters keep τ proportional to I by making B effectively always perpendicular.

Worked Example

A coil of N = 100 turns, area A = 0.01 m², carries I = 0.5 A in B = 0.2 T with its plane parallel to the field (θ = 90°).

1. τ = N I A B sinθ = 100 × 0.5 × 0.01 × 0.2 × 1.
2. τ = 0.10 N m.

The coil experiences a torque of 0.10 N m. A spring opposing this in a galvanometer would twist until spring torque balances it, giving deflection proportional to current.

Common Misconceptions

• Net force on the loop is what turns it: In a uniform field net force is zero; pure torque from a couple does the turning.
• Torque is maximum when the loop faces the field: When the normal is along B the torque is zero; it is maximum when the plane is parallel to B.
• More turns change only resistance: Turns multiply torque (and magnetic moment) directly through N.

Real-World Applications

• DC motors, where a commutator reverses current each half-turn to sustain rotation.
• Moving-coil galvanometers and analog meters with controlling springs.
• Electric vehicle drive motors and servo actuators built from many such coils.

A current loop in a field is a magnetic dipole acted on by torque τ = NIAB sinθ — the simple physics at the heart of motors, meters and electromagnetic actuators.