The current in a coil of self inductance 2.0 H is increasing according to I = 2sin(t2)A. The amount of energy spent during the period when current changes from 0 to 2 A is _______ J.Correct answer is '4'. Can you explain this answer?

Question

Answer

I = 2sin(t²) ⇒ dI = 4tsin(t²) dt
So, if I = 0 ⇒ t = 0
and I = 2 ⇒ 2 = 2 sin t²

= 2[-cosπ + cos 0] = 4

Answer

Calculation of Energy Spent in the Coil:

Energy spent in a coil during a change in current can be calculated using the formula:

$$ W = \frac{1}{2} L \int_{i_1}^{i_2} i \cdot di $$

Where,
W = Energy spent
L = Self-inductance of the coil
i1 = Initial current
i2 = Final current

Given data:
Self-inductance (L) = 2.0 H
Initial current (i1) = 0 A
Final current (i2) = 2 A
Current function: $ I = 2 \sin(t^2) $ A

Calculation:
1. Calculate the time when the current changes from 0 to 2 A:
$$ 2 \sin(t^2) = 2 $$
$$ \sin(t^2) = 1 $$
$$ t^2 = \frac{\pi}{2} $$
$$ t = \sqrt{\frac{\pi}{2}} $$

2. Substitute the time value in the current function to find the current at t = $ \sqrt{\frac{\pi}{2}} $:
$$ I = 2 \sin(\frac{\pi}{2}) = 2 \times 1 = 2 $$ A

3. Substitute the values in the energy formula:
$$ W = \frac{1}{2} \times 2 \times \int_{0}^{2} i \cdot di $$
$$ W = \int_{0}^{2} i \cdot di $$
$$ W = \frac{1}{2} \times \int_{0}^{2} 2 \cdot di $$
$$ W = \frac{1}{2} \times [2i]_{0}^{2} $$
$$ W = \frac{1}{2} \times (2 \times 2 - 0) $$
$$ W = \frac{1}{2} \times 4 $$
$$ W = 2 $$ J

Therefore, the energy spent during the period when the current changes from 0 to 2 A in the coil is 4 J.

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