The current in a coil of self inductance 2.0 H is increasing according to I = 2 sin(t²) A. The amount of energy spent during the period when current changes from 0 to 2 A is ____________ J.

Solution

Physics sir · 4 steps

Ignore the sin(t²): the energy in an inductor depends only on the final current, not on how it got there.

1. Where energy goes. As the current rises, the source does work against the induced emf LdI/dt. That work is stored in the coil's magnetic field.

   W = ∫₀^I L I dI = 1/2LI²

2. Why time drops out. The integral runs over current, not time. So the rule I = 2 sin(t²) does not matter. Only the start value 0 and the end value 2 A count.
3. Put in numbers. W = 1/2 × 2.0 × 2² = 4 J.
4. So. The energy spent in building the current from 0 to 2 A is 4 J.

The common slip: 8. Dropping the 1/2 and using LI² gives 8 J instead of 4 J.

The idea to keep. Energy stored in an inductor is 1/2LI² and depends only on the current. A time dependence in the question is usually a distraction unless it asks for power or emf.

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