Question

The magnetic moments associated with two closely wound circular coils a and b of radius \(r\) and \(2r\) respectively are equal if (where \(N_a\), \(I_a\) and \(N_b\), \(I_b\) are number of turns and current in A and B respectively)

• A. \(2N_aI_a = N_bI_b\)
• B. \(N_a = 2N_b\)
• C. \(N_aI_a = 4N_bI_b\)
• D. \(4N_aI_a = N_bI_b\)

Hint:

Magnetic moment is \(NIA\) with \(A=\pi r^2\).

Answer 1

The Correct Option is C

Step 1: Plan:
Compare the areas first.

Step 2: Steps:
The radius of b is twice that of a, so its area is $4$ times larger. For equal moments, $NI$ of coil a must make up for the smaller area, i.e. $N_aI_a$ must be $4$ times $N_bI_b$.

Final Answer:
The condition is $N_aI_a = 4N_bI_b$, option (C). \[ \boxed{N_aI_a = 4N_bI_b} \]