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Chapter 29: Problem 25

A current loop with magnetic dipole moment \(\mu\) is placed in a uniform magnetic field \(\mathbf{B},\) with its moment making angle \(\theta\) with the field. With the arbitrary choice of \(U=0\) for \(\theta=90^{\circ},\) prove that the potential energy of the dipole-field system is \(U=-\mu \cdot\) B. You may imitate the discussion in Chapter 26 of the potential energy of an electric dipole in an electric field.

Short Answer

Expert verified
The potential energy \(U\) of a magnetic dipole in a magnetic field is given by \(U = -\mu . \mathbf{B}\)

Step by step solution

01

Understanding and Applying Torque

Firstly, it's important to recall that the torque acting on a magnetic dipole of moment \( \mu \) in an external magnetic field \( \mathbf{B} \) is expressed as \( \tau = \mu \times \mathbf{B} \). In terms of magnitude, this becomes \( \tau = \mu B \sin \theta \), where \( \theta \) is the angle between \( \mu \) and \( \mathbf{B} \).
02

Determining Work Done

The work done (dW) to rotate the dipole by an infinitesimally small angle \( d\theta \) is equal to the torque multiplied by the angle. Therefore, \( dW = \tau d\theta = \mu B \sin \theta d\theta \). However, to bring the dipole from \( \theta \) to \( \theta = 90^{\circ} \), we need to do negative work, hence \( dW = -\mu B \sin \theta d\theta \).
03

Calculating Potential Energy

The potential energy (U) is then given by the work done to rotate the dipole from \( \theta = 0^{\circ} \) to \( \theta \). Therefore, integrate \( dW \) from 0 to \( \theta \), obtaining: \( U = -\int_0^{\theta} \mu B \sin \theta' d\theta' = -\mu B \cos \theta \). Here, the potential energy at \( \theta =90^{\circ} \) was chosen as zero (a reference point), so the constant of integration disappears and we get the required relation \( U = -\mu B \cos\theta = -\mu . \mathbf{B} \). Here, \( -\mu . \mathbf{B} \) signifies the dot product of the magnetic moment and the magnetic field.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Potential Energy
Potential energy is essentially the energy stored within a system due to force fields. For a magnetic dipole in a magnetic field, potential energy depends on orientation. The key formula is:
  • \(U = - \mu \cdot \mathbf{B}\)
This represents the interaction energy between the magnetic moment (\(\mu\)) and the magnetic field (\(\mathbf{B}\)). When their directions align, the potential energy is minimized. Conversely, potential energy is maximized when they are perpendicular.

In the context of this problem, choosing an angle of \(90^{\circ}\) as a reference point (\(U = 0\)) simplifies calculations. Hence, for any other angle \(\theta\), the potential energy can be found using integration and trigonometric identities.
Uniform Magnetic Field
A uniform magnetic field has the same strength and direction at every point. Imagine field lines that are perfectly parallel and evenly spaced.

When a magnetic dipole is placed in such a field, it interacts uniformly. This uniformity ensures that the equations of motion and energy are predictable and consistent across the dipole's exposure to the field.
  • Strength is constant: \(\mathbf{B}\) is uniform everywhere.
  • Direction is constant: Lines do not diverge or converge.
Understanding these properties is essential for correctly calculating potential energy and torque on the dipole.
Torque
Torque is a rotational force that acts on an object. For a magnetic dipole in a magnetic field, torque helps determine the rotational tendency of the dipole. The formula for torque \(\tau\) on a magnetic dipole is:
  • \( \tau = \mu \times \mathbf{B} \)
  • In terms of magnitude, this is \( \tau = \mu B \sin \theta \).
\(\theta\) is the angle between the magnetic moment (\(\mu\)) and the magnetic field (\(\mathbf{B}\)).

Torque is maximal when \(\mu\) is perpendicular to \(\mathbf{B}\). Conversely, it is zero when \(\mu\) is aligned with \(\mathbf{B}\), meaning there's no rotational force acting to change its orientation.
Magnetic Moment
A magnetic moment is a vector quantity that represents the strength and orientation of a magnetic source, like a loop of current. It plays a central role in how systems interact with magnetic fields.

For a current loop:
  • Magnetic moment \(\mu\) is related to current and the area of the loop.
  • Direction follows the right-hand rule, perpendicular to the plane of the loop.
The interaction with external fields depends on the orientation of \(\mu\) within the field \(\mathbf{B}\). This determines potential energy and torque, revealing the dynamics of the dipole鈥檚 motion and energy in a uniform magnetic field.

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Most popular questions from this chapter

Consider an electron orbiting a proton and maintained in a fixed circular path of radius \(R=5.29 \times 10^{-11} \mathrm{m}\) by the Coulomb force. Treating the orbiting charge as a current loop, calculate the resulting torque when the system is in a magnetic field of \(0.400 \mathrm{T}\) directed perpendicular to the magnetic moment of the electron.

At the Fermilab accelerator in Batavia, Illinois, protons having momentum \(4.80 \times 10^{-16} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) are held in a circular orbit of radius \(1.00 \mathrm{km}\) by an upward magnetic field. What is the magnitude of this field?

A current of \(17.0 \mathrm{mA}\) is maintained in a single circular loop of \(2.00 \mathrm{m}\) circumference. A magnetic field of \(0.800 \mathrm{T}\) is directed parallel to the plane of the loop. (a) Calculate the magnetic moment of the loop. (b) What is the magnitude of the torque exerted by the magnetic field on the loop?

A proton moving at \(4.00 \times 10^{6} \mathrm{m} / \mathrm{s}\) through a magnetic field of 1.70 T experiences a magnetic force of magnitude \(8.20 \times 10^{-13} \mathrm{N} .\) What is the angle between the proton's velocity and the field?

The needle of a magnetic compass has magnetic moment \(9.70 \mathrm{mA} \cdot \mathrm{m}^{2} .\) At its location, the Earth's magnetic field is \(55.0 \mu \mathrm{T}\) north at \(48.0^{\circ}\) below the horizontal. (a) Identify the orientations of the compass needle that represent minimum potential energy and maximum potential energy of the needle-field system. (b) How much work must be done on the needle to move it from the former to the latter orientation?
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