A wire of length l is bent to form a circular coil of some...

A wire of length $l$ is bent to form a circular coil of some turns. A current $I$ is then established in the coil and it is placed in a uniform magnetic field $B$. The maximum torque that acts on the coil is :
(a) $I B l^{2}$
(b) $4 \pi I B l^{2}$
(c) $\frac{n^{2} B}{4 \pi}$
(d) zero

Key Concepts

Design Optimization under Geometrical Constraints

In problems where a fixed amount of material (wire length) is used to form a coil, there is a trade-off between the number of turns and the area per turn. Optimizing the parameters such as the number of turns and the loop area is crucial for maximizing the coil’s magnetic dipole moment and, hence, the torque it experiences in a magnetic field.

Coil Geometry from Fixed Wire Length

When forming a coil from a wire of fixed length, the geometry of the coil (such as the loop radius and, consequently, its area) is constrained by the wire length. For a single circular loop, the area is determined by the circumference, linking the physical dimensions of the loop directly to the magnetic dipole moment.

Magnetic Torque on a Current Loop

The torque experienced by a current-carrying loop in a magnetic field is given by the cross product ? = ? × B, where ? is the magnetic dipole moment and B is the external magnetic field. When the dipole moment, ? = I·A (with I being the current and A the area of the loop), is perpendicular to the field, the torque is maximized, leading to ?_max = I·A·B.

Magnetic Dipole Moment

The magnetic dipole moment is a measure of the strength of a current loop's magnetic effect and is defined as ? = I·A. This quantity determines how strongly the loop interacts with external magnetic fields and is essential when calculating forces and torques acting on it.

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