A conducting circular loop is placed in X-Y plane in presence of magnetic field B⃗=(3 t³ ĵ+3 t² k̂) in SI unit. If the radius of the loop is 1 m, the induced emf in the loop, at time, t=2 s is n π V. The value of n is ___________.

Solution

Physics sir · 3 steps

The loop lies flat in the XY plane, so only the k̂ part of the field threads through it.

1. Which component counts. A loop in the XY plane has its area vector along k̂. The flux is B⃗ · A⃗, so the ĵ component lies in the plane of the loop and adds nothing.
2. Write the flux. The area is π r² = π × 1² = π m². So φ = 3t² × π = 3π t² in webers.

   |ε| = dφ/dt = d/dt(3π t²) = 6π t

3. So. At t = 2 s, |ε| = 6π × 2 = 12π V. So n = 12.

The common slip: 36. Using the ĵ component instead gives d/dt(3t³)×π = 9t²π = 36π, so 36, but that component lies in the plane of the loop and gives zero flux.

The idea to keep. Flux needs the field component along the area vector. For a loop in a plane, keep only the component perpendicular to that plane, then differentiate with time.

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