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Chapter 27: Problem 40

The plane of a 5.0 cm \(\times\) 8.0 cm rectangular loop of wire is parallel to a 0.19-T magnetic field. The loop carries a current of 6.2 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
The torque is 0 N路m, the magnetic moment is 0.0248 A路m虏, and the maximum torque would use a square loop.

Step by step solution

01

Understand the problem requirements

We have a rectangular loop of wire with dimensions 5.0 cm by 8.0 cm placed in a uniform magnetic field of 0.19 T, carrying a current of 6.2 A. We need to calculate the torque acting on the loop, the magnetic moment of the loop, and the maximum torque possible with the same wire length.
02

Calculate the magnetic moment (part b)

The magnetic moment, denoted by \( \mu \), is given by the formula \( \mu = I \times A \), where \( I \) is the current and \( A \) is the area of the loop. First, calculate the area \( A = 5.0 \text{ cm} \times 8.0 \text{ cm} = 40 \text{ cm}^2 = 0.004 \text{ m}^2 \). Then, the magnetic moment \( \mu = 6.2 \text{ A} \times 0.004 \text{ m}^2 = 0.0248 \text{ A路m}^2 \).
03

Calculate the torque on the loop (part a)

The torque \( \tau \) on the loop is calculated using the formula \( \tau = \mu \times B \sin \theta \). Since the plane of the loop is parallel to the magnetic field, \( \theta = 0^{\circ} \), and thus, \( \sin 0^{\circ} = 0 \). Therefore, \( \tau = 0.0248 \text{ A路m}^2 \times 0.19 \text{ T} \times 0 = 0 \text{ N路m} \).
04

Calculate the maximum torque with the same wire length (part c)

The total length of the wire is \( 2 \times (0.05 \text{ m} + 0.08 \text{ m}) = 0.26 \text{ m} \). To maximize the area \( A' \) with this length, we form a square. For a square, side \( s = \frac{0.26}{4} \), thus \( A' = s^2 = \left(\frac{0.26}{4}\right)^2 \text{ m}^2 \). The maximum torque is \( \tau' = \mu' \cdot B \cdot \sin 90^{\circ} \), where \( \mu' = I \cdot A' \). Substitute in the values to find \( \tau' \).

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Key Concepts

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

Torque in Magnetic Fields
When a current-carrying loop is placed in a magnetic field, it experiences a torque. This torque is the result of the magnetic forces acting on the current in the loop. It's calculated using the formula:
  • \( \tau = \mu \times B \times \sin \theta \)
Here, \( \mu \) is the magnetic moment, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the magnetic moment vector and the magnetic field.

In the provided exercise, the loop's plane is parallel to the magnetic field, meaning \( \theta = 0^{\circ} \). This results in \( \sin 0^{\circ} = 0 \), so the torque is zero.

In a scenario where \( \theta = 90^{\circ} \), the torque is maximized because \( \sin 90^{\circ} = 1 \). This is vital for understanding how to position loops in applications like electric motors for maximum efficiency.
Rectangular Loop in Magnetic Field
A rectangular loop in a magnetic field offers a clear visual for examining the interactions between current and magnetic force. The shape of the loop influences how effectively it interacts with the magnetic field.

A loop with sides measuring 5.0 cm by 8.0 cm, as in the given problem, forms a rectangle that can maximize the magnetic force when the orientation is optimal. The area of the loop, important for calculating magnetic moment, is given by multiplying its sides: \( A = 5.0 \text{ cm} \times 8.0 \text{ cm} = 0.004 \text{ m}^2 \).

This concept illustrates the importance of the area and shape in determining the loop's interaction with the magnetic field. By reshaping this loop with the same perimeter into a different configuration, such as a square, one can achieve different magnetic properties and achieve maximum torque as needed.
Magnetic Force Calculations
Magnetic force calculations are essential in determining how loops behave in a magnetic field. While the torque depends on factors like area and angle, the magnetic moment \( \mu \) is critical.
  • \( \mu = I \times A \)
Current \( I \) in amperes and area \( A \) contribute directly to \( \mu \). In our exercise, the loop's current is 6.2 A with an area of 0.004 m虏, producing a magnetic moment of 0.0248 A路m虏.

By understanding these calculations, one can manipulate variables such as current, area, and shape to optimize systems. For instance, adjusting a loop into a square shape with the same wire length maximizes both the area and subsequently, the obtainable torque. This insight aids in various applications, from magnetic sensors to electric motor design.

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Most popular questions from this chapter

\(\textbf{Determining Diet.}\) 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?

A particle with charge -5.60 nC is moving in a uniform magnetic field \(\overrightarrow{B} =\) -(1.25 T)\(\hat{k}\). The magnetic force on the particle is measured to be \(\overrightarrow{F} =\) -(3.40 \(\times\) 10\(^{-7}\)N)\(\hat{\imath}\) + (7.40 \(\times\) 10\(^{-7}\)N)\(\hat{\jmath}\). (a) Calculate all the components of the velocity of the particle that you can from this information. (b) Are there components of the velocity that are not determined by the measurement of the force? Explain. (c) Calculate the scalar product \(\vec{v}\) \(\cdot\) \(\overrightarrow{F}\). What is the angle between \(\vec{v}\) and \(\overrightarrow{F}\)?

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?

A plastic circular loop has radius \(R\), and a positive charge q is distributed uniformly around the circumference of the loop. The loop is then rotated around its central axis, perpendicular to the plane of the loop, with angular speed \(\omega\). If the loop is in a region where there is a uniform magnetic field \(\overrightarrow{B}\) directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop.

Singly ionized (one electron removed) atoms are accelerated and then passed through a velocity selector consisting of perpendicular electric and magnetic fields. The electric field is 155 V/m and the magnetic field is 0.0315 T. The ions next enter a uniform magnetic field of magnitude 0.0175 T that is oriented perpendicular to their velocity. (a) How fast are the ions moving when they emerge from the velocity selector? (b) If the radius of the path of the ions in the second magnetic field is 17.5 cm, what is their mass?
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