Torque on a Current Loop & the Magnetic Dipole Moment

In the previous lesson you learned that a straight wire carrying current I in a magnetic field B experiences a force F=IL×B. Now imagine taking that wire and bending it into a rectangular loop. Each of the four sides feels its own magnetic force, but the forces on opposite sides of the loop do not generally cancel — they point in different directions and act at different points, creating a torque that tends to rotate the loop.

This is a profoundly important situation. A current loop in a magnetic field is the simplest rotating machine imaginable, and understanding the torque it experiences opens the door to DC motors, galvanometers, and the very concept of a magnetic dipole. Before we can compute the torque, we need a compact way to describe the loop itself. That description is the magnetic dipole moment.

The magnitude of the magnetic dipole moment is simply m=IA, where I is the current and A is the area enclosed by the loop. The units are amperes times square meters (A·m²). The direction of m is perpendicular to the plane of the loop, found by curling the fingers of your right hand in the direction of the conventional current — your thumb points along m.

With this definition, a current loop is magnetically equivalent to a tiny bar magnet: m points from the south pole to the north pole. Every property of the loop — its tendency to align with a field, the field it produces far away, its energy in an external field — can be expressed compactly in terms of m rather than in terms of the detailed geometry of the wire.

When a current loop carrying dipole moment m is placed in a uniform external magnetic field B, the net force on the loop is actually zero — the forces on opposite sides cancel. But the forces act at different points, so there is a net torque:

The magnitude of the torque is τ=mB sinθ. When m and B are aligned (θ=0°), the torque is zero — the loop is in stable equilibrium, like a compass needle pointing north. When m and B are anti-aligned (θ=180°), the torque is also zero, but this is an unstable equilibrium. When m is perpendicular to B (θ=90°), the torque reaches its maximum value of mB.

The torque always acts to rotate the loop so that m aligns with B. This is exactly how a compass needle works, and it is the fundamental operating principle of every DC motor and analog galvanometer. In a DC motor, current is fed through a coil (the armature) sitting between the poles of a permanent magnet; the torque spins the coil, and a commutator reverses the current every half-turn so the torque always pushes in the same rotational direction.

The torque equation τ=m×B is one of the most practically important results in electromagnetism. Double the current and you double the torque. Double the number of turns and you effectively double the dipole moment. This direct proportionality between current and torque is what makes electromagnetic rotation so controllable — from the smallest micro-actuator in a phone's haptic engine to the traction motor in an electric vehicle, the same equation governs the conversion of electrical energy into rotational mechanical motion.