Figure 28-45 shows a rectangular 20 -turn coil of wire, of...

Figure $28-45$ shows a rectangular 20 -turn coil of wire, of dimensions 10 $\mathrm{cm}$ by 5.0 $\mathrm{cm} .$ It carries a current of 0.10 $\mathrm{A}$ and is hinged along one long side. It is mounted in the $x y$ plane, at angle $\theta=30^{\circ}$ to the direction of a uniform magnetic field of magnitude 0.50 $\mathrm{T}$ . In unit-vector notation, what is the torque acting on the coil about the hinge line?

Figure $28-45$ shows a rectangular 20 -turn coil of wire, of dimensions 10 $\mathrm{cm}$ by 5.0 $\mathrm{cm} .$ It carries a current of 0.10 $\mathrm{A}$ and is hinged along one long side. It is mounted in the $x y$ plane, at angle $\theta=30^{\circ}$ to the direction of a uniform magnetic field
of magnitude 0.50 $\mathrm{T}$ . In unit-vector notation, what is the torque acting
on the coil about the hinge line?

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Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment represents the strength and orientation of a current-carrying loop’s magnetic field. It is calculated by multiplying the current, the number of turns, and the area of the loop, with its direction determined by the right-hand rule. This concept is critical in analyzing how loops interact with external magnetic fields.

Torque on a Magnetic Dipole

A magnetic dipole experiences a torque when placed in a uniform magnetic field, which acts to align the dipole with the field. The magnitude and direction of this torque are given by the vector cross product of the magnetic dipole moment and the magnetic field. This principle explains the rotational effect on current loops in magnetic fields.

Vector Cross Product

The vector cross product is a mathematical operation used to determine a vector that is perpendicular to two given vectors. In the context of magnetic torque, it is used to compute the direction and magnitude of the torque resulting from the interaction between the magnetic dipole moment and the magnetic field, following the right-hand rule.

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