The magnetic moments associated with two closely wound circular coils A and B of radius r_A=10 cm and r_B=20 cm respectively are equal if : (Where N_A, I_A and N_B, I_B are number of turn and current of A and B respectively)

Solution

Physics sir · 3 steps

Moment goes as area, so the coil with half the radius needs four times the NI to keep up.

1. Moment of a coil. A closely wound coil of N turns carrying current I has magnetic moment m = NIA. For a circle the area is A = π r², so the moment grows with the square of the radius.

   N_A I_A π r_A² = N_B I_B π r_B²

2. Put in radii. Cancel π and use r_A = 10 cm, r_B = 20 cm: N_A I_A × 100 = N_B I_B × 400. Coil B has four times the area of coil A.
3. So. Dividing by 100 gives N_A I_A = 4 N_B I_B, which is option (d).

The idea to keep. For a coil, m = NIπ r². Doubling the radius quadruples the moment, so to match a bigger coil, the smaller one must carry four times the product NI.

Answer from the NTA final answer key. Solution by PrepWiser.