Q.A non-conducting sphere has a mass of 100 g and radius 20 cm. A flat compact coil of wire with 5 turns is wrapped tightly around it, with each turn concentric with the sphere. This sphere is placed on an inclined plane such that the plane of the coil is parallel to the inclined plane. A uniform magnetic field of 0.5 T exists in the region, in the vertically upward direction. Compute the current $I$ required to keep the sphere at rest in equilibrium.

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. …