Problem 42 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

(a) Torque is 0 Nm. (b) Magnetic moment is 0.0248 A·m². (c) Maximum torque is approximately 0.0028 Nm.

Step 1: 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, \( \mu = 6.2 \text{ A} \times 0.004 \text{ m}^2 = 0.0248 \text{ A} \cdot \text{m}^2 \).

Step 2: 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, \( \tau = 0.0248 \cdot 0.19 \cdot 0 = 0 \).

Step 3: 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 \( \tau_{max} = \mu \times B = 0.0148 \cdot 0.19 \approx 0.0028 \text{ N} \cdot \text{m} \).

Key Concepts

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 (m²).
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 m² and carrying a current of 6.2 A, the magnetic moment is 0.0248 A·m².
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 m².
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.