Q.Derive the expression for the torque on a current-carrying coil in a magnetic field.

Torque on a Current Loop

The Intuition First

Imagine a compass needle in the Earth's magnetic field. The needle always turns until it points north. Why? Because the needle itself is a tiny magnet, and the field exerts a twist — a torque — that tries to align it.

A current-carrying loop behaves exactly like that tiny magnet. It has a magnetic moment $\vec{m}$, which is like its own internal compass arrow. When you place this loop in an external magnetic field $\vec{B}$, the field pulls on one side of the loop and pushes on the other, creating a turning effect.

The loop doesn't feel a net force (if the field is uniform), but it does feel a torque. That torque always tries to rotate the loop so that its magnetic moment points along the field — just like a compass needle.

The Key Players

The magnetic moment of a planar current loop is:

where $I$ is the current, $A$ is the area of the loop, and $\hat{n}$ is a unit vector perpendicular to the plane of the loop (direction given by the right-hand rule: curl your fingers along the current, your thumb points along $\vec{m}$).

The external field $\vec{B}$ is uniform — same magnitude and direction everywhere in the region of the loop.

The Torque: Two Equivalent Forms

The torque on the loop is:

where $\theta$ is the angle between $\vec{m}$ and $\vec{B}$. The torque is maximum when $\vec{m}$ is perpendicular to $\vec{B}$ ($\theta = 90^\circ$), and zero when they are parallel or antiparallel ($\theta = 0^\circ$ or $180^\circ$).

The vector form captures both magnitude and direction:

The cross product tells you: the torque is perpendicular to both $\vec{m}$ and $\vec{B}$, and its direction is given by the right-hand rule. This torque always rotates $\vec{m}$ toward $\vec{B}$.

Why It Happens (The Physics)

Consider a rectangular loop of sides $a$ and $b$, carrying current $I$, placed in a uniform field $\vec{B}$. Let the plane of the loop make an angle $\theta$ with the field.

The two sides of length $a$ are perpendicular to $\vec{B}$. On each of these sides, the magnetic force is $F = I a B$, but the forces on opposite sides are in opposite directions. These two forces form a couple — equal and opposite, not along the same line — which produces a torque.

The lever arm for each force is $(b/2) \sin\theta$, so the net torque is:

Since $m = I A$, we get $\tau = m B \sin\theta$.

The Stable Equilibrium

When $\vec{m}$ is aligned with $\vec{B}$ ($\theta = 0$), the torque is zero. This is a stable equilibrium — if you nudge the loop slightly, the torque brings it back. …