∙A plastic circular loop has radius R, and a positive charge...

\bullet A plastic circular loop has radius $R,$ and a positive charge $q$ is distributed uniformly around the circumference of the loop. The loop is now rotated around its central axis, perpendicular to the plane of the loop, with angular speed $\omega .$ If the loop is in a region where there is a uniform magnetic field $\frac{1}{B}$ directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop.

Key Concepts

Magnetic Torque on a Dipole

A magnetic dipole in an external magnetic field experiences a torque that tends to align the dipole with the field. This torque is mathematically given as the cross product of the magnetic moment and the magnetic field vectors. Its magnitude depends on the strength of the magnetic moment, the magnitude of the field, and the sine of the angle between them. This principle is fundamental in understanding how magnetic dipoles behave in various magnetic field configurations.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that represents the strength and orientation of a current loop's magnetic effect. It is defined as the product of the current flowing in the loop and the area enclosed by the loop. In problems involving a rotating charged loop, the moving charge is equivalent to a current, thereby creating a dipole moment that interacts with external magnetic fields.

Current from Rotating Charges

When a charge is uniformly distributed along a loop and the loop rotates, the revolving charge mimics a current. This effective current can be determined by dividing the total charge by the period of rotation. This concept links the rotational motion of charges to the generation of a magnetic field, thereby bridging electricity and magnetism.

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