Question

A plastic circular loop has radius $R$, and a positive charge q is distributed uniformly around the circumference of the loop. The loop is then 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 $\overrightarrow{B}$ directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop.

Short answer

The magnitude of the magnetic torque is $\frac{q\omega R^{2}B}{2}$.

Step-by-step solution

Understand the Problem

We need to calculate the magnetic torque on a rotating charged loop within a magnetic field. The loop has a uniform charge distribution and rotates with angular speed $\omega$. The magnetic field $\overrightarrow{B}$ is parallel to the loop's plane.

Calculate the Current Equivalent

When the charged loop rotates, it behaves like a current loop. The current $I$ can be found using $I=\frac{q}{T}$, where $T=\frac{2\pi }{\omega }$ is the period of rotation. Thus, $I=\frac{q\omega }{2\pi }$.

Determine the Magnetic Dipole Moment

The magnetic dipole moment $\mu$ of a current loop is given by $\mu =NI\pi R^2$, where $N=1$ is the number of loops. Therefore, $\mu =\frac{q\omega R^2}{2}$.

Calculate the Torque

The torque $\tau$ on a magnetic dipole in a magnetic field is given by $\tau =\mu B\sin \theta$, where $\theta$ is the angle between $\mu$ and $\overrightarrow{B}$. Since the field is parallel to the plane of the loop, $\theta ={90}^{\circ }$ and $\sin \theta =1$. Thus, $\tau =\frac{q\omega R^{2}B}{2}$.

Key concepts

Circular Loop

A circular loop is a key component in this exercise. Imagine a ring or a hoop made of a flexible material, shaped into a perfect circle. For this problem, the loop is plastic, but still carries a uniform charge, denoted as $q$. The charge distribution is even along the whole circumference of the loop. This loop is being rotated around its central axis—just like a spinning hula hoop—perpendicular to the loop's own plane.
Viewing it from any point directly in front of the loop's plane, you wouldn't see much of the side-to-side movement, but rather, the loop seems to spin around like a CD. Understanding the geometry here, where the loop rotates as a whole, is crucial to visualizing the physical phenomena, like magnetic effects, that occur as a result.

Angular Speed

Angular speed, represented by $\omega$, is how fast the circular loop is spinning. If you've ever watched a carousel, imagine the speed at which it's turning. That's similar to angular speed. It's typically expressed in radians per second.
In this exercise, angular speed is constant, meaning the loop spins steadily without speeding up or slowing down. This constancy helps simplify calculations involving rotational motion. You can think of angular speed as analogous to linear speed, but while linear speed tells you how fast something is going straight, angular speed tells you about spinning motions.
Ultimately, angular speed ensures the charged loop creates an effective current, akin to a looped wire with electrical current flowing through it. This further connects to how we calculate the loop's behavior in a magnetic field.

Magnetic Field

The magnetic field is a crucial aspect in this problem, denoted by $\overrightarrow{B}$. It is the region around a magnetic material or a moving electric charge within which the force of magnetism acts.
In this exercise, the magnetic field is uniform, meaning it has the same strength and direction throughout. Additionally, it aligns parallel to the plane of the loop. This unique arrangement directly influences how the loop behaves as it rotates, affecting the forces at play.
Magnetism and moving charges have a symbiotic relationship. Thus, as the charged loop rotates, it interacts with the magnetic field, resulting in torque—a twisting effect. Understanding the direction and uniform nature of the magnetic field is critical to determining how the rotating loop behaves and the resulting magnetic effects.

Magnetic Dipole Moment

The magnetic dipole moment, signified by $\mu$, is a measure of the strength and orientation of a magnet or loop of electric current. It's a vector quantity, which means it has both a magnitude and a direction.
In this exercise, the circular loop's rotation with uniform charge distribution leads to the formation of a magnetic dipole moment. Calculate it by using the formula $\mu =\frac{q\omega R^2}{2}$. This equation helps to quantify the loop's effective "magnetism" due to the charged and rotating nature. The result is a dipole moment that has implications for other magnetic interactions—like torque.
The direction of the magnetic dipole moment typically follows the right-hand rule: if you curl your fingers in the direction of the current (or rotation in this case), your thumb will point in the direction of $\mu$. Understanding this concept is essential as it connects rotational motion to magnetic properties. It's truly a bridge between mechanical rotation and magnetic phenomena.