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 an inductor when the current through it is given, we can use the formula for the voltage across an inductor:

### Step-by-Step Solution:

1. **Understand the given information**:
- The inductance \( L \) is given as \( 2 \, \text{H} \).
- The current \( I(t) \) through the inductor is given by \( I(t) = t \sin t \, \text{A} \).

2. **Use the formula for voltage across an inductor**:
\[ V = L \frac{dI}{dt} \]
where \( V \) is the voltage across the inductor, \( L \) is the inductance, and \( \frac{dI}{dt} \) is the rate of change of current with respect to time.

3. **Differentiate the current \( I(t) \)**:
- We need to find \( \frac{dI}{dt} \) for \( I(t) = t \sin t \).
- Using the product rule for differentiation:
\[ \frac{dI}{dt} = \frac{d}{dt}(t) \cdot \sin t + t \cdot \frac{d}{dt}(\sin t) \]
- This simplifies to:
\[ \frac{dI}{dt} = 1 \cdot \sin t + t \cdot \cos t = \sin t + t \cos t \]

4. **Substitute \( \frac{dI}{dt} \) into the voltage formula**:
\[ V = L \frac{dI}{dt} = 2 \left( \sin t + t \cos t \right) \]

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

6. **Final result**:
The voltage across the inductor is:
\[ V = 2t \cos t + 2 \sin t \]