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Chapter 27: Problem 46

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

Expert verified
The change in potential energy is -2.42 J.

Step by step solution

01

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掳 to be parallel to the field.
02

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掳. The initial potential energy \( U_i \) is given by the formula \( U = -\mu B \cos \theta \). For the antiparallel orientation, \( \cos 180掳 = -1 \), so the initial potential energy \( U_i = -\mu B (-1) = \mu B \).
03

Potential Energy for Parallel Orientation

When the magnetic moment is parallel to the field, the angle \( \theta \) is 0掳, and \( \cos 0掳 = 1 \). Therefore, the potential energy \( U_f \) for the parallel orientation is \( U_f = -\mu B (1) = -\mu B \).
04

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, \( \Delta U = U_f - U_i = -\mu B - \mu B = -2\mu B \).
05

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 \( \Delta U = -2.42 \text{ J} \).

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Key Concepts

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

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鈥檚 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鈥檚 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鈥檚 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掳, 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.

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Most popular questions from this chapter

A dc motor with its rotor and field coils connected in series has an internal resistance of 3.2 \(\Omega\). When the motor is running at full load on a 120-V line, the emf in the rotor is 105 V. (a) What is the current drawn by the motor from the line? (b) What is the power delivered to the motor? (c) What is the mechanical power developed by the motor?

A horizontal rectangular surface has dimensions 2.80 cm by 3.20 cm and is in a uniform magnetic field that is directed at an angle of 30.0\(^\circ\) above the horizontal. What must the magnitude of the magnetic field be to produce a flux of 3.10 \(\times\) 10\(^{-4}\) Wb through the surface?

A singly charged ion of \(^7\)Li (an isotope of lithium) has a mass of 1.16 \(\times\) 10\(^{-26}\) kg. It is accelerated through a potential difference of 220 V and then enters a magnetic field with magnitude 0.874 T perpendicular to the path of the ion. What is the radius of the ion's path in the magnetic field?

An electromagnet produces a magnetic field of 0.550 T in a cylindrical region of radius 2.50 cm between its poles. A straight wire carrying a current of 10.8 A passes through the center of this region and is perpendicular to both the axis of the cylindrical region and the magnetic field. What magnitude of force does this field exert on the wire?

\(\textbf{Determining Diet.}\) One method for determining the amount of corn in early Native American diets is the \(stable\) \(isotope\) \(ratio\) \(analysis\) (SIRA) technique. As corn photosynthesizes, it concentrates the isotope carbon-13, whereas most other plants concentrate carbon-12. Overreliance on corn consumption can then be correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the \(^{12}\)C and \(^{13}\)C isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed 8.50 km /s, and you want to bend them within a uniform magnetic field in a semicircle of diameter 25.0 cm for the \(^{12}\)C. The measured masses of these isotopes are 1.99 \(\times\) 10\(^{-26}\) kg (\(^{12}\)C) and 2.16 \(\times\) 10\(^{-26}\) kg (\(^{13}\)C). (a) What strength of magnetic field is required? (b) What is the diameter of the \(^{13}\)C semicircle? (c) What is the separation of the \(^{12}\)C and \(^{13}\)C ions at the detector at the end of the semicircle? Is this distance large enough to be easily observed?
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