Problem 70 - University Physics with Modern Physics

Question

A plastic circular loop of 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 \(\overrightarrow{\boldsymbol{B}}\) directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop.

Step-by-Step Solution

Answer

The magnetic torque magnitude is \(\frac{q R^2 \cdot \omega \cdot B}{2}\).

Step 1: Understand the Problem

We are given a circular loop with a radius \(R\) and a uniformly distributed charge \(q\). The loop rotates with an angular speed \(\omega\) around its central axis which is perpendicular to its plane. A magnetic field \(\overrightarrow{\boldsymbol{B}}\) is present parallel to the loop's plane, and we want to determine the magnetic torque on the loop.

Step 2: Magnetic Dipole Moment Calculation

The magnetic dipole moment \(\mu\) of a charged rotating loop is given by \(\mu = q \cdot A \cdot \omega / (2\pi)\), where \(A\) is the area of the loop. Since the loop is circular, \(A = \pi R^2\). Plugging in, \(\mu = \frac{q \cdot (\pi R^2) \cdot \omega}{2\pi} = \frac{q R^2 \cdot \omega}{2}\).

Step 3: Torque on a Magnetic Dipole

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 the magnetic dipole moment and the magnetic field. Since the field is parallel to the loop's plane, \(\theta = 90^\circ\), giving \(\sin(90^\circ) = 1\). Therefore, \(\tau = \mu B = \frac{q R^2 \cdot \omega}{2} B\).

Step 4: Final Expression for Torque

Substitute the expression for \(\mu\) into the torque expression: \[\tau = \frac{q R^2 \cdot \omega}{2} \cdot B = \frac{q R^2 \cdot \omega \cdot B}{2}\]. This is the magnitude of the magnetic torque on the loop.

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a fundamental concept in electromagnetism, especially in the context of loops and coils. It's essentially a measure of the strength and direction of a magnetic source that can be thought of as a magnet or a coil with a current running through it.
For a loop of radius \(R\) carrying a charge \(q\), when this loop rotates, it creates a kind of magnetic field similar to that of a small magnet. This is quantified by the magnetic dipole moment.
To calculate it, you use the formula \( \mu = \frac{q \cdot (\pi R^2) \cdot \omega}{2\pi} \). Here:

• \( \pi R^2 \) represents the area of the loop
• \(\omega\) stands for the angular speed, indicating how fast the loop is rotating
• \(q\) is the charge uniformly distributed

Simplifying this gives \( \mu = \frac{q R^2 \cdot \omega}{2} \), showing us how the rotation and the charge contribute to the magnetic moment.

Angular Speed

Angular speed, not to be confused with linear speed, is the measure of how fast something rotates or revolves relative to another point. It is denoted by \( \omega \) and usually measured in radians per second (rad/s).
In the context of rotational motion, angular speed is key to understanding phenomena such as rotational kinetic energy and torque. For our loop, it determines how frequently the loop completes a circle along its axis of rotation.
This links directly to the magnetic dipole moment because:

• Higher angular speeds (\(\omega\): many complete cycles in a short time) result in a larger magnetic dipole moment.
• As \(\omega \) increases, the strength of the loop's equivalent magnetic field also increases.

Thus, the faster the loop spins, the more significant its magnetic effects become.

Uniform Magnetic Field

A uniform magnetic field is one where the magnetic force experienced is constant in both magnitude and direction over a defined space. This consistency is important for predicting the interactions of objects with the field, like our charged, rotating loop.
In this exercise, the magnetic field \(\overrightarrow{B} \) is described as being parallel to the plane of the loop.
The uniformity here implies:

• The field lines are straight and evenly spaced.
• Any point within the designated area experiences the same magnetic force due to \(\overrightarrow{B} \).

When the magnetic dipole moment of the loop interacts with this field:

• The torque (turning effect) on the loop can be calculated as \(\tau = \mu B \,\sin(\theta)\), with \(\theta = 90^\circ\).
• This results in maximum torque since \(\sin(90^\circ) = 1\).

Hence, the uniform magnetic field and its orientation play a crucial role in determining the torque experienced by the loop.