A conducting circular loop is placed in a uniform magnetic field of 0.4 T with its plane perpendicular to the field. Somehow, the radius of the loop starts expanding at a constant rate of 1 mm / s. The magnitude of induced emf in the loop at an instant when the radius of the loop is 2 cm will be ___________ μ V.

Solution

Physics sir · 3 steps

The field stays fixed but the area grows, so the emf is just B times the rate at which the area grows.

1. Where flux changes. The field is perpendicular to the loop, so the flux is Φ = Bπ r². Only r changes with time, so the emf comes entirely from the growing area.

   |ε| = dΦ/dt = B · 2π r dr/dt

2. Put in numbers. Use SI units: B = 0.4 T, r = 0.02 m, dr/dt = 10⁻³ m s⁻¹. Then |ε| = 0.4 × 2π × 0.02 × 10⁻³ = 1.6π × 10⁻⁵ V.
3. So. |ε| ≈ 5.03 × 10⁻⁵ V = 50.3 μV, which rounds to 50.

The common slip: 25. Dropping the 2 when differentiating r², that is, using π r dr/dt in place of 2π r dr/dt, gives about 25, not 50.

The idea to keep. When the field is constant and the loop changes shape, emf equals B dA/dt. For a circle, dA/dt = 2π r dr/dt: the circumference times the rate at which the radius grows.

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