Problem 40 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

The torque is 0 N·m, the magnetic moment is 0.0248 A·m², and the maximum torque would use a square loop.

Step 1: 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.

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

Step 3: 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, \( \tau = 0.0248 \text{ A·m}^2 \times 0.19 \text{ T} \times 0 = 0 \text{ N·m} \).

Step 4: 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 \( \tau' \).

Key Concepts

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 m², producing a magnetic moment of 0.0248 A·m².

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.