In a coil of resistance 8 Ω, the magnetic flux due to an external magnetic field varies with time as φ=2/3(9-t²). The value of total heat produced in the coil, till the flux becomes zero, will be _____________ J.

Solution

Physics sir · 4 steps

The emf keeps changing here, so integrate e²/R over time. Do not just multiply by time.

1. When flux is zero. φ = 2/3(9 - t²) becomes zero when t² = 9, that is at t = 3 s. So heat is produced from t = 0 to t = 3 s.
2. Induced emf. By Faraday's law, |e| = |dφ/dt| = 4/3t. The emf grows with time, so the power in the coil also grows with time.

   H = ∫₀³ e²/R dt = ∫₀³ 16t²/(9 × 8) dt = 2/9∫₀³ t² dt

3. Evaluate. ∫₀³ t² dt = 3³/3 = 9, so H = 2/9 × 9 = 2 J.
4. So. The total heat produced in the coil is 2 J.

The common slip: 1.5. Using the average emf Δφ/(Δ t) = 6/3 = 2 V as if it were constant gives 2²/8 × 3 = 1.5 J, not 2. The emf is not constant, so e²/R has to be integrated over time.

The idea to keep. When the emf depends on time, the power e²/R depends on time too. Find the time interval first, then integrate the power over it to get the heat.

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