Current in a coil of self-inductance `2.0 H` is increasing as `I = 2 sin t^(2)`. The amount of energy spent during the period when the current changes from `0` to `2 A` is

Options

A. `1 J`
B. `2 J`
C. `3 J`
D. `4 J`

Explanation

To solve the problem, we need to calculate the energy spent when the current in a coil changes from 0 to 2 A, given that the self-inductance \( L \) of the coil is \( 2.0 \, H \).

### 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 self-inductance and \( I \) is the current.

2. **Substitute the Given Values**:
From the problem, we know:
- Self-inductance \( L = 2.0 \, H \)
- The maximum current \( I = 2 \, A \)

Now, substituting these values into the formula:
\[ U = \frac{1}{2} \times 2 \times (2)^2 \]

3. **Calculate \( I^2 \)**:
Calculate \( (2)^2 \):
\[ (2)^2 = 4 \]

4. **Substitute \( I^2 \) Back into the Energy Formula**:
Now substitute \( I^2 \) back into the energy formula:
\[ U = \frac{1}{2} \times 2 \times 4 \]

5. **Simplify the Expression**:
Now simplify the expression:
\[ U = \frac{1}{2} \times 2 \times 4 = 1 \times 4 = 4 \, J \]

6. **Conclusion**:
The amount of energy spent when the current changes from 0 to 2 A is:
\[ U = 4 \, J \]

### Final Answer:
The energy spent during the period when the current changes from 0 to 2 A is **4 joules**.

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