A length L of wire carries a current i. Show that if the...

A length $L$ of wire carries a current $i$. Show that if the wire is formed into a circular coil, then the maximum torque in a given magnetic field is developed when the coil has one turn only, and that maximum torque has the magnitude $\tau=L^{2} i B / 4 \pi$.

Key Concepts

Optimization of Coil Design for Maximum Torque

Maximizing the torque on a current-carrying coil in a magnetic field involves optimizing the configuration to maximize the magnetic moment. Given a fixed length of wire, forming a single-turn coil maximizes the area enclosed by the loop, and consequently, the magnetic moment. This principle underlies why a one-turn configuration yields the highest torque, as it leverages the entire length of the wire to create the largest possible loop area under the specified constraints.

Geometric Constraints of a Coil

This concept involves understanding the relationship between the physical dimensions of a coil and its electrical properties. When a fixed length of wire is used to form a coil, the size of the loop (its radius and area) and the number of turns are interdependent. Larger areas generally result in higher magnetic moments, but the dimensions are constrained by the total wire length, making the relationship pivotal for designing coils with optimal magnetic characteristics.

Magnetic Moment

The magnetic moment is a measure of the strength and orientation of a magnet's influence and, for a current-carrying loop, it is defined as the product of the current and the area enclosed by the loop. It encapsulates how a given current distribution generates its own magnetic field and interacts with external magnetic fields, thereby determining the magnitude of the exerted torque.

Torque on a Current-Carrying Loop

This concept refers to the rotational force experienced by a current-carrying loop placed in a magnetic field. The torque arises because the magnetic forces acting on different segments of the loop produce a net rotational effect. The magnitude of this torque depends on the magnetic moment of the loop as well as the strength and orientation of the magnetic field, and it reaches a maximum when the field is perpendicular to the plane of the loop.

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