A wire of length 25.0 cm carrying a current of 4.51 mA is to be formed into a circular coil an

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A wire of length 25.0 cm carrying a current of 4.51 mA is...

A wire of length $25.0 \mathrm{~cm}$ carrying a current of $4.51 \mathrm{~mA}$ is to be formed into a circular coil and placed in a uniform magnetic field $\vec{B}$ of magnitude $5.71 \mathrm{mT}$. If the torque on the coil from the field is maximized, what are (a) the angle between $\vec{B}$ and the coil's magnetic dipole moment and (b) the number of turns in the coil? (c) What is the magnitude of that maximum torque?

Key Concepts

Wire Length Constraint and Coil Geometry

When forming a coil from a fixed wire length, there is a trade?off between the number of turns and the area of each turn. The circumference of each loop determines the radius (via 2?r for a circle) and consequently the area of that loop. In a coil with multiple turns, the total magnetic dipole moment is the product of the current, the number of turns, and the area of a single turn. However, because the wire length is fixed, increasing the number of turns reduces the radius—and thus the area—of each loop. This relationship is crucial in optimizing the coil design to maximize the magnetic dipole moment and hence the torque.

Magnetic Dipole Moment

This concept describes how a current?carrying loop creates a dipole moment that characterizes its magnetic strength and orientation. The magnetic dipole moment is given by the product of the current and the area of the loop. When the loop is formed into a circle, the area is ?r², and if there are multiple turns, the net moment is the sum over all turns.

Torque on a Current Loop

A current loop placed in a uniform magnetic field experiences a torque because of the interaction between its magnetic dipole moment and the external field. The torque is given by the cross product ? = m × B, and its magnitude is ? = mB sin?, where ? is the angle between the magnetic field vector and the dipole moment vector.

Maximizing Torque Condition

For the maximum torque to be achieved, the sine term in the torque expression must be maximized. Since sin? reaches its maximum value of 1 at 90°, the optimum condition is when the magnetic field is perpendicular to the coil's dipole moment.

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