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Chapter 29: Problem 20

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?

Short Answer

Expert verified
Magnetic moment (m) is 0.0170 Am^2 and the torque (\(\tau\)) is 0 Nm.

Step by step solution

01

Calculate the area of the loop

The magnetic moment (\textbf{m}) of a loop is determined by the product of the current (I) and the area (A) it encloses. With a given circumference (C = 2.00 \(\mathrm{m}\)), we can calculate the area by first finding the radius (r) using the circumference formula \(C = 2\pi r\). Solving for r gives \(r = \frac{C}{2\pi}\). Then the area of a circle is given by \(A = \pi r^2\).
02

Compute the magnetic moment

Using the formula \(\textbf{m} = I \times A\), the magnetic moment would be the product of the current and the area of the loop. Remember to convert the current from milliamperes to amperes before calculating.
03

Calculate the torque exerted by the magnetic field

Torque (\(\tau\)) is given by the formula \(\tau = \textbf{m}B\sin(\theta)\), where B is the magnetic field strength and \(\theta\) is the angle between the magnetic moment and the magnetic field direction. Since the magnetic field is parallel to the plane of the loop, \(\theta = 0^\circ\) or \(\pi\) radians, and \(\sin(\theta)\) is 0. Accordingly, no torque would be exerted because the loop is in alignment with the field.

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Key Concepts

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

Understanding a Magnetic Field
Imagine a magnetic field as a region where a magnetic force can be felt. It's invisible to the naked eye, but its effects are very real. The magnetic field is represented by the symbol 'B' and is measured in teslas (T). These fields emerge around materials like magnets or flow of electric charges, such as current through a wire.

In our exercise, a magnetic field strength of 0.800 T interacts with a current-carrying loop. This interaction is central to many technologies, such as electric motors and generators. Simply, the magnetic field provides a playground where electromagnetic forces come to life and exert their influence.
Electromagnetic Torque on a Current Loop
Electromagnetic torque, often symbolized as \(\tau\), is the twist that a magnetic field can apply to a current-carrying object. This torque is crucial in rotating machines and meters. It's given by the formula \(\tau = \textbf{m}B\sin(\theta)\) where 'm' is the magnetic moment, 'B' is the magnetic field strength, and \(|\sin(\theta)|\) represents the sine of the angle between the magnetic moment direction and the magnetic field lines.

In a situation where the magnetic field is parallel to the plane of the loop, like in our textbook example, the angle is either 0 degrees or 180 degrees. When you take the sine of these angles, you get zero, implying no torque is generated. This can be helpful in certain applications to maintain stability. Understanding this helps in designing electromechanical systems that require precise control of movement.
Circular Current Loop and Magnetic Moment
A circular current loop generates its own magnetic field and possesses a quantity known as the magnetic moment. The magnetic moment \(\textbf{m}\) is a vector quantity that represents the strength and orientation of a magnetic source, determined by multiplying the current \(I\) by the area \(A\) it encircles, formulated as \(\textbf{m} = I \times A\).

To solve our exercise, we first find the loop's radius from the given circumference and then the area. Knowing these, we compute the loop's magnetic moment, which gives us a measure of how much torque we'd expect if the loop were subjected to a magnetic field at a non-parallel angle. Magnetic moments play a significant role in magnetic storage media, atomic physics, and even in medical imaging techniques such as MRI.

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Most popular questions from this chapter

An electron is accelerated through 2400 \(\mathrm{V}\) from rest and then enters a uniform 1.70 - \(\mathrm{T}\) magnetic field. What are (a) the maximum and \((\mathrm{b})\) the minimum values of the magnetic force this charge can experience?

What is the required radius of a cyclotron designed to accelerate protons to energies of 34.0 \(\mathrm{MeV}\) using a magnetic field of 5.20 \(\mathrm{T} ?\)

The magnetic field of the Earth at a certain location is directed vertically downward and has a magnitude of 50.0\(\mu \mathrm{T}\) . A proton is moving horizontally toward the west in this field with a speed of \(6.20 \times 10^{6} \mathrm{m} / \mathrm{s}\) . (a) What are the direction and magnitude of the magnetic force the field exerts on this charge? (b) What is the radius of the circular arc followed by this proton?

A cyclotron designed to accelerate protons has an outer radius of 0.350 \(\mathrm{m}\) . The protons are emitted nearly at rest from a source at the center and are accelerated through 600 \(\mathrm{V}\) each time they cross the gap between the dees. The dees are between the poles of an electromagnet where the field is 0.800 \(\mathrm{T}\) . (a) Find the cyclotron frequency. (b) Find the speed at which protons exit the cyclotron and (c) their maximum kinetic energy. (d) How many revolutions does a proton make in the cyclotron? (e) For what time interval does one proton accelerate?

A proton moving at \(4.00 \times 10^{6} \mathrm{m} / \mathrm{s}\) through a magnetic field of 1.70 \(\mathrm{T}\) 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?
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