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: Key Formula:
The magnetic moment of a coil of \(N\) turns carrying current \(I\) with area \(A\) is \(m = NIA\).

Step 2: Apply:
Coil a: \(m_a = N_aI_a\pi r^2\). Coil b: \(m_b = N_bI_b\pi(2r)^2 = 4N_bI_b\pi r^2\).
Equal moments: \(N_aI_a = 4N_bI_b\).

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