In Fig. 28-32, a rectangular loop carrying current lies in...

In Fig. 28-32, a rectangular loop carrying current lies in the plane of a uniform magnetic field of magnitude $0.050 \mathrm{~T}$. The loop consists of a single turn of flexible conducting wire that is wrapped around a flexible mount such that the dimensions of the rectangle can be changed. (The total length of the wire is not changed.) As edge length $x$ is varied from approximately zero to its maximum value of approximately $4.0 \mathrm{~cm}$, the magnitude $\tau$ of the torque on the loop changes. The maximum value of $\tau$ is $4.80 \times 10^{-8} \mathrm{~N} \cdot \mathrm{m}$. What is the current in the loop?

Key Concepts

Optimization of Enclosed Area with a Fixed Perimeter

For a loop made of a fixed length of wire, the maximum area that can be enclosed is achieved when the loop is shaped as a square. This principle arises from a common optimization problem in calculus, ensuring that for a given perimeter the area is maximized. In applications involving current loops, maximizing the area enhances the magnetic dipole moment, which in turn affects the torque experienced in a magnetic field.

Magnetic Dipole Moment

The magnetic dipole moment quantifies the strength and the orientation of a current loop's magnetic field. It is defined as the product of the current flowing through the loop and the area enclosed by that loop. This concept is fundamental because it connects the geometrical configuration of the loop to its interaction with external magnetic fields.

Torque on a Current Loop in a Magnetic Field

When a current-carrying loop is placed in an external magnetic field, it experiences a torque that tends to align the loop's magnetic dipole moment with the field. This torque is given by the cross product of the magnetic moment and the magnetic field, and its maximum value occurs when the moment and the field are perpendicular.

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