Problem 27 - Master Resource Book in JEE Main Physics

Question

A circular current loop of magnetic moment \(M\) is in an arbitrary orientation in an external magnetic field B. The work done to rotate the loop by \(30^{\circ}\) about an axis perpendicular to its plane is \(\quad\) [NCERT Exemplar] (a) \(M B\) (b) \(\sqrt{3} \frac{M B}{2}\) (c) \(\frac{M B}{2}\) (d) zero

Step-by-Step Solution

Answer

Work done is \((1 - \frac{\sqrt{3}}{2}) M B\); none of the options explicitly match.

Step 1: Understand the Problem

The problem involves calculating the work done when a circular current loop with a magnetic moment \(M\) is rotated by \(30^{\circ}\) in an external magnetic field \(B\). We need to find the expression for the work done using the provided options.

Step 2: Identify the Formula for Work Done

The work done in rotating a magnetic moment \( \vec{M} \) in a uniform magnetic field \( \vec{B} \) by an angle \( \theta \) is given by the formula: \[ W = M B (1 - \cos \theta) \]where \(\theta\) is the angle of rotation.

Step 3: Calculate the Cosine of the Angle

Since the rotation is by \(30^{\circ}\), we calculate \(\cos 30^{\circ}\). \[ \cos 30^{\circ} = \frac{\sqrt{3}}{2} \]

Step 4: Substitute Values into the Formula

Substitute \(\cos 30^{\circ}\) into the work done formula: \[ W = M B \left(1 - \frac{\sqrt{3}}{2}\right) \]

Step 5: Simplify the Expression

Simplify the expression to find the work done:\[ W = M B \left( 1 - \frac{\sqrt{3}}{2} \right) = M B - \frac{\sqrt{3} M B}{2} \]Thus, it simplifies further to:\[ W = \left(1 - \frac{\sqrt{3}}{2}\right) M B \]

Key Concepts

Work Done in Magnetic Field

In the realm of physics, the concept of work done is crucial when it comes to understanding how energy is transferred or transformed. In a magnetic field, the work done on a magnetic moment during rotation is particularly interesting. When a magnetic moment, represented as \( \vec{M} \), is subjected to an external magnetic field \( \vec{B} \), and it is rotated through an angle \( \theta \), the work done \( W \) can be calculated using the formula:

• \( W = M B (1 - \cos \theta) \)

This equation helps us understand the relationship between the angle of rotation and the energy required to achieve such a rotation. The magnetic field acts as a force, making it necessary to apply energy – or do work – to move the magnetic moment from one position to another.
In simpler terms, the larger the angle of rotation, the more work needs to be done. Here, the cosine part of the formula \( \cos \theta \) plays a significant role in determining how much energy is required based on the orientation change. Thus, understanding this concept allows students to grasp the fundamental principles of magnetic interactions.

Circular Current Loop

A circular current loop is one of the basic and most intriguing concepts in electromagnetism. It is formed when a current flows through a wire looped into a circle, giving rise to a magnetic field. This phenomenon is particularly important because such a loop demonstrates the principles of how magnetic fields are generated and interact with other magnets and currents.
The magnetic moment \( M \) is a crucial property of the circular current loop. It represents the strength and direction of the magnetic field generated by the loop. The magnetic moment can be thought of as how "magnetized" the loop is, and it plays a significant role when the loop is placed in an external magnetic field.

• The strength of the current and the size of the loop influence the magnitude of the magnetic moment.
• The orientation of the loop in an external magnetic field affects how the magnetic forces act on the loop.

Understanding the circular current loop helps students visualize how magnets work, beyond just the bar magnets often seen in basic physics experiments.

Angle of Rotation

The angle of rotation in the context of a circular current loop in a magnetic field is crucial for calculating the work done. It is the measure of how much the loop is rotated from its initial position. The angle impacts the orientation of the magnetic moment with respect to the magnetic field.
In the given exercise, an angle of \( 30^{\circ} \) rotation was used. With such angles, trigonometric functions, such as cosine, come into play to determine how much work is needed to rotate the loop.
The formula for work done, \( W = M B (1 - \cos \theta) \), incorporates the angle of rotation; \( \cos \theta \) is vital to understand how much work is necessary. Specifically, the value \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) affects the final calculation of work.

• This concept demonstrates the intricate relationship between physics and mathematics, where angles and rotations can be expressed through functions such as cosine.
• It also underscores how even a small change in the angle of rotation can influence the energy required or work done, showcasing the delicate balance of forces in an electromagnetic field.