The current in a coil of self inductance 2.0 H is increasing according to $\mathrm{I}=2\sin(\mathrm{t}^{2})$ A. The amount of energy spent during the period when current changes from 0 to 2 A is ______ J.

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Electromagnetic Induction

The current in a coil of self inductance 2.0 H is increasing according to

I = 2 sin (t^{2})

A. The amount of energy spent during the period when current changes from 0 to 2 A is ______ J.

Answer: 4

The energy stored in an inductor is related to the work done against the induced EMF. The energy can be calculated using:

E = \int L I d I

Given

I = 2 sin (t^{2})

, we have

d I = 4 t cos (t^{2}) d t

.

When

I = 0

t = 0

When

I = 2

2 = 2 sin (t^{2}) \Rightarrow t^{2} = \frac{π}{2} \Rightarrow t = \frac{π}{2}

Substituting:

E = \int_{0}^{π /2} 2 \times 2 sin (t^{2}) \times 4 t cos (t^{2}) d t = 8 \int_{0}^{π /2} t sin (2 t^{2}) d t

Using substitution

u = 2 t^{2}

d u = 4 t d t

E = 2 \int_{0}^{π} sin (u) d u = 2 [- cos (u)]_{0}^{π} = 2 [- cos π + cos 0] = 2 [1 + 1] = 4 J

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Integer answer