If the current through an inductor of 2 H is given by `I = t sin t A`, then the voltage across the inductor is

Options

A. `cos t + t sin t `
B. `2t cos t + 2 sin t `
C. `t cos t + sin t `
D. `2t sint + 2 cost `

Answer

To find the voltage across the inductor, we can use the formula for the voltage across an inductor, which is given by:

\[ V = L \frac{di}{dt} \]

where:
- \( V \) is the voltage across the inductor,
- \( L \) is the inductance of the inductor,
- \( \frac{di}{dt} \) is the rate of change of current with respect to time.

### Step-by-Step Solution:

1. **Identify the given values**:
- The inductance \( L = 2 \, \text{H} \).
- The current \( I(t) = t \sin(t) \, \text{A} \).

2. **Differentiate the current with respect to time**:
We need to find \( \frac{di}{dt} \). Using the product rule for differentiation:
\[ \frac{di}{dt} = \frac{d}{dt}(t \sin(t)) = \sin(t) + t \cos(t) \]
Here, we differentiate \( t \) to get \( 1 \) and \( \sin(t) \) to get \( \cos(t) \).

3. **Substitute \( \frac{di}{dt} \) into the voltage formula**:
Now, substitute \( \frac{di}{dt} \) back into the voltage equation:
\[ V = L \frac{di}{dt} = 2 \left( \sin(t) + t \cos(t) \right) \]

4. **Simplify the expression**:
\[ V = 2 \sin(t) + 2t \cos(t) \]

### Final Result:
The voltage across the inductor is:
\[ V = 2 \sin(t) + 2t \cos(t) \, \text{V} \]