A circular coil having N turns and radius r carries a current I. It is held in the XZ plane in a magnetic field $$B\hat{i}$$. The torque on the coil due to the magnetic field is:

We recall the general result for a current-carrying coil in a uniform magnetic field. A planar coil with $$N$$ turns, carrying current $$I$$, and having vector area $$\vec A$$ possesses a magnetic dipole moment $$\vec\mu$$ defined by the formula

$$\vec\mu = N\,I\,\vec A.$$

Here the vector $$\vec A$$ has magnitude equal to the geometrical area of the coil and its direction is perpendicular to the plane of the coil, obtained from the right-hand rule (curl the fingers in the sense of current, the thumb gives the direction of $$\vec A$$).

The torque $$\vec\tau$$ acting on a magnetic dipole in a uniform magnetic field $$\vec B$$ is given by the vector formula

$$\vec\tau = \vec\mu \times \vec B.$$

From this cross-product, the magnitude of the torque is

$$\tau = \mu\,B\,\sin\theta,$$

where $$\theta$$ is the angle between the magnetic moment $$\vec\mu$$ (or equivalently the area vector $$\vec A$$) and the magnetic field $$\vec B$$.

Now, for the circular coil described in the question:

• It has $$N$$ turns.
• Its radius is $$r$$, so its geometrical area (for one turn) is

$$A = \pi r^{2}.$$

Because the coil lies in the $$XZ$$-plane, its area vector (and hence $$\vec\mu$$) is directed along the $$\pm \hat{j}$$ (the $$Y$$) axis. The magnetic field is given as $$\vec B = B\,\hat{i},$$ i.e. along the $$X$$ axis. Therefore the angle $$\theta$$ between $$\vec\mu$$ (along $$\hat{j}$$) and $$\vec B$$ (along $$\hat{i}$$) is

$$\theta = 90^\circ,$$

for which $$\sin\theta = \sin 90^\circ = 1.$$

First, write the magnitude of the magnetic dipole moment:

$$\mu = N\,I\,A = N\,I\,(\pi r^{2}).$$

Substituting this value and $$\sin\theta = 1$$ into the torque formula, we have

$$\tau = \mu\,B\,\sin\theta = \bigl(N\,I\,\pi r^{2}\bigr)\,B\,(1).$$

Simplifying the factors, the magnitude of the torque becomes

$$\tau = B\,\pi r^{2}\,I\,N.$$

This matches exactly the expression given in Option A.

Hence, the correct answer is Option A.