How much work is required to rotate the current loop (Fig. 27–23) in a uniform magnetic field $\overrightarrow{B}$ from (a) θ = 0° ($\overrightarrow{\mu }$ ∣∣ $\overrightarrow{B}$) to θ = 180°, (b) θ = 90° to θ = -90°.

Verified step by step guidance

1. Understand the problem: The work required to rotate a current loop in a uniform magnetic field is related to the change in potential energy of the magnetic dipole moment (μ) in the field. The potential energy is given by U = -μ·B·cos(θ), where θ is the angle between the magnetic moment (μ) and the magnetic field (B).
2. Set up the work equation: The work done to rotate the loop is equal to the change in potential energy, W = ΔU = U_final - U_initial. For each part of the problem, calculate the initial and final potential energies using U = -μ·B·cos(θ).
3. For part (a): When θ changes from 0° to 180°, calculate U_initial = -μ·B·cos(0°) and U_final = -μ·B·cos(180°). Substitute these values into W = U_final - U_initial to find the work required.
4. For part (b): When θ changes from 90° to -90°, calculate U_initial = -μ·B·cos(90°) and U_final = -μ·B·cos(-90°). Again, substitute these values into W = U_final - U_initial to determine the work required.
5. Simplify the expressions: Use the trigonometric values for cos(0°), cos(180°), cos(90°), and cos(-90°) to simplify the calculations. Note that cos(0°) = 1, cos(180°) = -1, cos(90°) = 0, and cos(-90°) = 0. This will help you finalize the expressions for the work in both parts (a) and (b).

Key Concepts

Work in a Magnetic Field

In physics, work is defined as the energy transferred when a force is applied over a distance. When dealing with magnetic fields, the work done on a magnetic dipole (like a current loop) in a magnetic field is related to the angle between the dipole moment and the magnetic field. The work can be calculated using the formula W = -μ·B·cos(θ), where μ is the magnetic moment, B is the magnetic field strength, and θ is the angle between them.

Magnetic Moment

The magnetic moment is a vector quantity that represents the strength and direction of a magnetic source. For a current loop, the magnetic moment (μ) is given by the product of the current (I) flowing through the loop and the area (A) of the loop, expressed as μ = I·A. The orientation of the magnetic moment relative to an external magnetic field influences the torque and work done when the loop is rotated.

Torque in a Magnetic Field

Torque (τ) is the rotational equivalent of force and is crucial when analyzing the motion of a current loop in a magnetic field. The torque experienced by a magnetic dipole in a magnetic field is given by τ = μ × B, where the cross product indicates that the torque depends on the angle between the magnetic moment and the magnetic field. This torque causes the loop to rotate, and understanding it is essential for calculating the work done during the rotation.