Question

A coil with magnetic moment 1.45 A \(\cdot\) m\(^2\) is oriented initially with its magnetic moment antiparallel to a uniform 0.835-T magnetic field. What is the change in potential energy of the coil when it is rotated 180\(^\circ\) so that its magnetic moment is parallel to the field?

Short answer

The change in potential energy is -2.42 J.

Step-by-step solution

Understanding the Problem

We have a magnetic moment (\( \mu = 1.45 \text{ A} \cdot \text{m}^2 \)) and a uniform magnetic field (\( B = 0.835 \text{ T} \)). Initially, the magnetic moment is antiparallel to the field, and we need to find the change in potential energy when the moment is rotated 180° to be parallel to the field.

Potential Energy for Antiparallel Orientation

The initial orientation of the magnetic moment is antiparallel to the magnetic field, which means the angle \( \theta \) is 180°. The initial potential energy \( U_i \) is given by the formula \( U = -\mu B \cos \theta \). For the antiparallel orientation, \( \cos 180° = -1 \), so the initial potential energy \( U_i = -\mu B (-1) = \mu B \).

Potential Energy for Parallel Orientation

When the magnetic moment is parallel to the field, the angle \( \theta \) is 0°, and \( \cos 0° = 1 \). Therefore, the potential energy \( U_f \) for the parallel orientation is \( U_f = -\mu B (1) = -\mu B \).

Calculate Change in Potential Energy

The change in potential energy \( \Delta U \) is the difference between the final potential energy \( U_f \) and the initial potential energy \( U_i \). Thus, \( \Delta U = U_f - U_i = -\mu B - \mu B = -2\mu B \).

Substitute the Given Values

Substitute \( \mu = 1.45 \text{ A} \cdot \text{m}^2 \) and \( B = 0.835 \text{ T} \) into the equation: \( \Delta U = -2(1.45)(0.835) \). This evaluates to \( \Delta U = -2.42 \text{ J} \).

Key concepts

Magnetic Moment

The magnetic moment is an essential concept when dealing with magnetic fields and materials. It is a vector quantity and denotes the strength and direction of a magnetic source. Think of it as a measure of how much torque a system will experience in a magnetic field such as a coil or magnet.
Let’s look at some practical characteristics of the magnetic moment:

• The unit for magnetic moment is ampere meter squared (\( ext{A} \cdot ext{m}^2 \)).
• It is usually aligned with the magnetic field lines when it is in equilibrium.
• A magnetic moment causes a system to align with the external magnetic field similar to how a compass needle aligns with the Earth’s magnetic field.

In the context of the given problem, the magnetic moment (\( ext{1.45 A} \cdot ext{m}^2 \)) refers to the coil's ability to interact with the magnetic field. It’s this interaction that affects the system's potential energy, which changes as the coil rotates.

Uniform Magnetic Field

A uniform magnetic field is one in which the magnetic field lines are parallel and evenly spaced. This specific arrangement means that the field has the same strength and direction at every point within it.
Understanding the characteristics of a uniform magnetic field can help clarify how it interacts with other magnetic entities like coils:

• The magnetic field strength is expressed in teslas (\( ext{T} \)).
• Uniform fields are commonly used in basic magnetic field experiments and applications to ensure predictable interactions.
• The problem involves a uniform magnetic field of \( 0.835 \ T \), meaning every point in that field exerts the same magnetic influence on the coil.

In the problem, the uniform magnetic field interacts consistently with the coil's magnetic moment, enabling straightforward calculation of changes in potential energy as the coil rotates.

Rotation of Coil

The rotation of a coil within a magnetic field is a fundamental experiment that illustrates how magnetic fields influence objects with a magnetic moment.
Two primary principles are essential here:

• When the magnetic moment of the coil is aligned opposite to the magnetic field, the potential energy is maximized.
• As the coil rotates to align the magnetic moment with the magnetic field, the system's potential energy decreases.

This rotation ultimately results in a change in potential energy, critical for understanding energy dynamics in magnetic systems.
In this exercise, the coil initially starts antiparallel, making it have high potential energy as calculated by \( U_i = \mu B \). Upon rotating 180°, aligning with the magnetic field, the final potential energy becomes \( U_f = -\mu B \), leading to a net energy change calculated as \( \Delta U = -2\mu B \), demonstrating the concept of energy transformation through mechanical movement.