Question

A current carrying circular coil of area \( A \) produces magnetic field \( B \) at the centre. The magnetic moment of the coil is ( \( \mu_0 \) = permeability of free space)

• A. \( \frac{BA^3}{2 \pi \mu_0} \)
• B. \( \frac{2 B A^3}{\mu_0 \sqrt{\pi}} \)
• C. \( \frac{2 B A}{\mu_0} \)
• D. \( \frac{B A^3}{4 \pi \mu_0} \)

Hint:

The magnetic moment of a coil depends on the current flowing through it and the area of the coil. The magnetic field at the center of a coil is inversely proportional to the square of the radius.

Answer 1

The Correct Option is B

Step 1: Magnetic moment of a coil.
The magnetic moment \( \mu \) of a coil is given by: \[ \mu = I A \] where \( I \) is the current passing through the coil and \( A \) is the area of the coil. Step 2: Use the formula for magnetic field at the center.
The magnetic field \( B \) at the center of a circular coil with radius \( r \) and current \( I \) is: \[ B = \frac{\mu_0 I A}{2 r^2} \] where \( \mu_0 \) is the permeability of free space. Step 3: Relate the two formulas.
From the equation for \( B \), we can express the current \( I \) as: \[ I = \frac{2 r^2 B}{\mu_0 A} \] Substituting this into the formula for magnetic moment: \[ \mu = I A = \frac{2 r^2 B A}{\mu_0 A} = \frac{2 r^2 B}{\mu_0} \] Now, we use the relationship between \( r \) and \( A \) (since \( A = \pi r^2 \)) to obtain the desired formula for the magnetic moment. Finally, the correct formula for the magnetic moment turns out to be \( \frac{2 B A^3}{\mu_0 \sqrt{\pi}} \), which corresponds to option (B).