Torque on Current Loop in Uniform B

Torque tends to align the magnetic moment with B. θ is the angle between m and B. Maximum when m ⊥ B, zero when aligned. Basis of galvanometer and electric motor operation.

Setup

A rectangular loop of sides $a$ and $b$ carries current $I$ in a uniform field $\vec{B}$. The normal to the loop makes angle $\theta$ with $\vec{B}$.

Forces on Each Side

Using $\vec{F} = I\vec{L} \times \vec{B}$:

• Sides parallel to $\vec{B}$: $F = 0$ (since $\vec{L} \parallel \vec{B}$).
• Two sides of length $a$ perpendicular to $\vec{B}$: each experiences force $F = BIa$, in opposite directions.

These two forces form a couple — net force is zero but net torque is non-zero.

Torque

The moment arm between the two forces is $b\sin\theta$:

$$\tau = F \cdot b\sin\theta = BIa \cdot b\sin\theta = BIA\sin\theta$$

For $N$ turns:

$$\boxed{\tau = NBIA\sin\theta = mB\sin\theta}$$

In vector form:

$$\boxed{\vec{\tau} = \vec{m} \times \vec{B}}$$

Equilibria

• $\theta = 0°$: $\tau = 0$, $\vec{m} \parallel \vec{B}$ — stable (minimum PE).
• $\theta = 90°$: $\tau = mB$ — maximum torque.
• $\theta = 180°$: $\tau = 0$, $\vec{m}$ antiparallel to $\vec{B}$ — unstable (maximum PE).