The current in a coil of self inductance 5 henry in increasing according to `I = 2 sin^(2)t`. The amount of energy spen during the period when current changes from 0 to 2 amperes is

Options

A. `10 J`
B. `5 J`
C. `100 J`
D. `2 J`

Explanation

To solve the problem, we need to find the amount of energy stored in the coil when the current changes from 0 to 2 amperes. The self-inductance \( L \) of the coil is given as 5 henries, and the current \( I \) is given by the equation \( I = 2 \sin^2(t) \).

### Step-by-Step Solution:

1. **Identify the Formula for Energy Stored in an Inductor:**
The energy \( U \) stored in an inductor is given by the formula:
\[ U = \frac{1}{2} L I^2 \]
where \( L \) is the inductance and \( I \) is the current.

2. **Determine the Maximum Current:**
The maximum current \( I \) is given as 2 A. This is the final current we are interested in.

3. **Calculate the Energy at Maximum Current:**
Substitute \( L = 5 \) H and \( I = 2 \) A into the energy formula:
\[ U = \frac{1}{2} \times 5 \times (2)^2 \]
Simplifying this:
\[ U = \frac{1}{2} \times 5 \times 4 = \frac{20}{2} = 10 \text{ joules} \]

4. **Determine the Initial Energy:**
When the current is 0 A, the energy stored in the inductor is:
\[ U_i = \frac{1}{2} L (0)^2 = 0 \text{ joules} \]

5. **Calculate the Energy Spent:**
The energy spent when the current changes from 0 A to 2 A is the difference between the final energy and the initial energy:
\[ U_{\text{spent}} = U_f - U_i = 10 - 0 = 10 \text{ joules} \]

### Final Answer:
The amount of energy spent during the period when the current changes from 0 to 2 amperes is **10 joules**.

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