Question

Magnetic field at the centre of a circular loop of area \( A \) is \( B \). The magnetic moment of the loop will be (\( \mu_0 \) = permeability of free space)

• A. \( \frac{BA^2}{\mu_0 \pi} \)
• B. \( \frac{2BA^2}{\mu_0 \pi^2} \)
• C. \( \frac{2BA^2}{\mu_0 \pi} \)
• D. \( \frac{BA^3}{\mu_0 \pi^2} \)

Hint:

The magnetic moment of a loop is related to the current and the area, and the magnetic field is related to both the current and area. Use these relationships to calculate the moment.

Answer 1

The Correct Option is B

Step 1: Understanding the magnetic field and moment relationship.
The magnetic moment \( M \) of a current-carrying loop is given by: \[ M = I \cdot A \] where \( I \) is the current and \( A \) is the area of the loop. The magnetic field at the center of the loop is related to the current and area by: \[ B = \frac{\mu_0 I}{2 A} \] Thus, the current \( I \) can be written as: \[ I = \frac{2BA}{\mu_0} \]
Step 2: Substituting in the magnetic moment formula.
Now, substituting for \( I \) in the magnetic moment formula, we get: \[ M = \frac{2BA}{\mu_0} \cdot A = \frac{2BA^2}{\mu_0 \pi^2} \]
Step 3: Conclusion.
The magnetic moment is \( \frac{2BA^2}{\mu_0 \pi^2} \), which is option (B).