Two coils have mutual inductance 0.002 H. The current changes in the first coil according to the relation i=i₀ sin ω t, where i₀=5 A and ω=50 π rad/s. The maximum value of emf in the second coil is π/α V. The value of α is _______.

Solution

Physics sir · 3 steps

The emf in coil 2 follows di/dt of coil 1, so differentiate the sine and the peak is M times i0 times omega.

1. Mutual induction. The emf in the second coil depends on how fast the current changes in the first: e = Mdi/dt, ignoring the sign.
2. Differentiate the current. With i = i₀ sin ω t, we get di/dt = i₀ ω cos ω t. The cosine is at most 1, so the emf is largest when cos ω t = 1.

   eₘₐₓ = M i₀ ω = 0.002 × 5 × 50π = 0.5π V

3. So. eₘₐₓ = π/2 V, and comparing with π/α gives α = 2.

The common slip: 0.5. Reporting the coefficient in 0.5π gives 0.5. The question asks for α in π/α, and 0.5π = π/2, so α is 2.

The idea to keep. For a sinusoidal current i₀ sin ω t, the peak induced emf is always M i₀ ω (or L i₀ ω for self-induction). Differentiating brings out the factor ω.

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