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Chapter 27: Problem 42

The plane of a \(5.0 \mathrm{cm} \times 8.0 \mathrm{cm}\) rectangular loop of wire is parallel to a \(0.19-\mathrm{T}\) magnetic field. The loop carries a current of 6.2 \(\mathrm{A}\) . (a) What torque acts on the loop? (b) What is the magnetic moment of the loop? (c) What is the maximum torque that can be obtained with the same total length of wire carrying the same current in this magnetic field?

Short Answer

Expert verified
(a) Torque is 0 Nm. (b) Magnetic moment is 0.0248 A路m虏. (c) Maximum torque is approximately 0.0028 Nm.

Step by step solution

01

Calculating the magnetic moment of the loop

The magnetic moment \( \mu \) of a loop is given by the formula \( \mu = I \times A \), where \( I \) is the current and \( A \) is the area of the loop. The area \( A \) of the rectangular loop is calculated as \( 5.0 \text{ cm} \times 8.0 \text{ cm} = 0.05 \text{ m} \times 0.08 \text{ m} = 0.004 \text{ m}^2 \). Therefore, \( \mu = 6.2 \text{ A} \times 0.004 \text{ m}^2 = 0.0248 \text{ A} \cdot \text{m}^2 \).
02

Calculating the torque on the loop

The torque \( \tau \) acting on the loop is calculated by \( \tau = \mu \times B \times \sin(\theta) \). Here, \( \mu = 0.0248 \text{ A} \cdot \text{m}^2 \), \( B = 0.19 \text{ T} \), and \( \theta = 0^\circ \) since the plane of the loop is parallel to the magnetic field, thus \( \sin(\theta) = \sin(0) = 0 \). Therefore, \( \tau = 0.0248 \cdot 0.19 \cdot 0 = 0 \).
03

Calculating maximum torque for the same wire length

The maximum torque occurs when the angle \( \theta \) is \( 90^\circ \) (\( \sin(90^\circ) = 1 \)). With the same total length of wire, the best configuration is a circular loop. The total length of the wire \( L \) is \( 2(5 + 8) \text{ cm} = 26 \text{ cm} = 0.26 \text{ m} \). The radius \( r \) for the circular loop is \( r = \frac{L}{2\pi} = \frac{0.26}{2\pi} \text{ m} \). The area \( A \) is \( \pi r^2 \). So the magnetic moment \( \mu = 6.2 \times \pi \left(\frac{0.26}{2\pi}\right)^2 = \frac{6.2\cdot0.026}{4\pi} \approx 0.0148 \text{ A} \cdot \text{m}^2 \). The maximum torque \( \tau_{max} = \mu \times B = 0.0148 \cdot 0.19 \approx 0.0028 \text{ N} \cdot \text{m} \).

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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 in understanding how magnetic fields interact with current loops. The magnetic moment (\(\mu\)) of a loop of wire is determined by the amount of current flowing through it and the area it encloses. It's calculated using the formula: \(\mu = I \times A\), where \(I\) is the current in amperes (A), and \(A\) is the area of the loop in square meters (m虏).
The larger the magnetic moment, the greater the loop's ability to exert torque in an external magnetic field. For our rectangular loop with dimensions 5.0 cm by 8.0 cm, enclosing an area of 0.004 m虏 and carrying a current of 6.2 A, the magnetic moment is 0.0248 A路m虏.
This means our loop has a magnetic moment powerful enough to interact noticeably with a magnetic field. Understanding magnetic moments helps in various fields, like designing electric motors and magnetic sensors. The loop effectively "wants" to align its magnetic moment along the direction of an external magnetic field.
Rectangular Loop
A rectangular loop of wire, as used in this exercise, is a simple yet effective shape for studying the principles of electromagnetism. By using a loop of wire, we can explore how electric currents produce magnetic fields and interact with external magnetic environments.
In our problem, the loop's dimensions are set at 5.0 cm and 8.0 cm, making it easy to calculate its area: 5.0 cm is converted to 0.05 m, and 8.0 cm to 0.08 m, resulting in an area of 0.004 m虏.
This area is crucial for determining the magnetic moment of the loop. Despite seeming simplistic, these dimensions and the loop's configuration heavily influence the magnetic interactions. For example, if the loop rotates, its orientation with respect to the magnetic field can change, affecting the torque experienced by the loop.
Such configurations are commonly found in practical applications, such as in coil designs of electric generators and transformers, where exact loop shapes affect efficiency and power production. Rectangular loops are also easy to construct and measure, making them practical for educational experiments.
Magnetic Field
A magnetic field represents the influence that a magnetic force exerts in a region around a magnet or current-carrying wire. In our exercise, the magnetic field strength is given as 0.19 T (tesla), which is a measure of its intensity.
Magnetic fields exert a force on moving charges and current-carrying conductors, giving rise to effects like magnetic torque. With our loop, the magnetic field is parallel to its plane, meaning that initially, no torque acts on it since \(\sin(0^\circ) = 0\).
However, if the loop's orientation against the field changes, the interaction becomes significant, and maximum torque can be obtained. For maximum torque, the loop must be perpendicular to the magnetic field, i.e., when \(\theta = 90^\circ\), leading to \(\sin(\theta) = 1\).
Exploring these effects of magnetic fields opens the door to understanding motors, magnetic storage devices, and MRI machines. Ultimately, magnetic fields are a fundamental aspect of electromagnetism, influencing various technological and natural phenomena.

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Most popular questions from this chapter

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 . Overneliance 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 \(^{10} \mathrm{C}\) and \(^{13} \mathrm{C}\) isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed 8.50 \(\mathrm{km} / \mathrm{s}\) , and you want to bend them within a uniform magnetic field in a semicircle of diameter 25.0 \(\mathrm{cm}\) for the 12 \(\mathrm{C}\) . The measured masses of these isotopes are \(1.99 \times 10^{-25} \mathrm{kg}\left(^{12} \mathrm{C}\right)\) and \(2.16 \times 10^{-26} \mathrm{kg}\left(^{13} \mathrm{C}\right) .\) (a) What strength of magnetic field is required? (b) What is the diameter of the \(^{13} \mathrm{C}\) semicircle? (c) What is the separation of the \(^{12}\mathrm{C}\) and \(^{13}\mathrm{C}\) ions at the detector at the end of the semicircle? Is this distance large enough to be easily observed?

A particle with charge \(6.40 \times 10^{-19} \mathrm{C}\) travels in a circular orbit with radius 4.68 \(\mathrm{mm}\) due to the force exerted on it by a magnetic field with magnitude 1.65 \(\mathrm{T}\) and perpendicular to the orbit. (a) What is the magnitude of the linear momentum \(\vec{p}\) of the partcle? (b) What is the magnitude of the angular momentum \(\overrightarrow{\boldsymbol{L}}\) of the particle?

A wire 25.0 \(\mathrm{cm}\) long lies along the \(z\) -axis and carries a current of 9.00 \(\mathrm{A}\) in the \(+z\) -direction. The magnetic field is uniform and has components \(B_{x}=-0.242 \mathrm{T}, B_{y}=-0.985 \mathrm{T}\) , and \(B_{z}=\) \(-0.336 \mathrm{T} .\) (a) Find the components of the magnetic force on the wire. (b) What is the magnitude of the net magnetic force on the wire?

A circular loop of wire with area \(A\) lies in the \(x y\) -plane. As viewed along the \(z\) -axis looking in the \(-z\) -direction toward the origin, a current \(I\) is circulating clockwise around the loop. The torque produced by an extemal magnetic field \(\vec{B}\) is given by \(\vec{\tau}=D(4 \hat{z}-3 \hat{y}),\) where \(D\) is a positive constant, and for this orientation of the loop the magnetic potential energy \(U=-\vec{\mu} \cdot \vec{B}\) is negative. The magnitude of the magnetic field is \(B_{0}=13 D / L A\) (a) Determine the vector magnetic moment of the current loop. (b) Determine the components \(B_{x}, B_{y}\) , and \(B_{z}\) of \(\vec{B}\) .

An electromagnet produces a magnetic field of 0.550 T in a cylindrical region of radius 2.50 \(\mathrm{cm}\) between its poles. A straight wire carrying a current of 10.8 \(\mathrm{A}\) 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 is exerted on the wire?
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