In an inductor of self-inductance L=2 mH, current changes with time according to relation `i=t^(2)e^(-t)`. At what time emf is zero ?

Options

A. 4s
B. 3s
C. 2s
D. 1s

Answer

To solve the problem, we need to find the time at which the electromotive force (emf) in the inductor is zero. The current \( i(t) \) is given as \( i = t^2 e^{-t} \), and the self-inductance \( L \) is given as \( 2 \, \text{mH} \).

### Step-by-Step Solution:

1. **Understand the relationship between emf and current**:
The induced emf (\( \mathcal{E} \)) in an inductor is given by the formula:
\[ \mathcal{E} = -L \frac{di}{dt} \]
where \( L \) is the self-inductance and \( \frac{di}{dt} \) is the rate of change of current with respect to time.

2. **Differentiate the current function**:
To find \( \frac{di}{dt} \), we need to differentiate the current function \( i(t) = t^2 e^{-t} \) with respect to \( t \). We will use the product rule for differentiation:
\[ \frac{di}{dt} = \frac{d}{dt}(t^2) \cdot e^{-t} + t^2 \cdot \frac{d}{dt}(e^{-t}) \]
The derivatives are:
- \( \frac{d}{dt}(t^2) = 2t \)
- \( \frac{d}{dt}(e^{-t}) = -e^{-t} \)

Thus,
\[ \frac{di}{dt} = 2t e^{-t} + t^2 (-e^{-t}) = 2t e^{-t} - t^2 e^{-t} \]
This simplifies to:
\[ \frac{di}{dt} = e^{-t}(2t - t^2) \]

3. **Substitute into the emf equation**:
Now we substitute \( \frac{di}{dt} \) into the emf equation:
\[ \mathcal{E} = -L \frac{di}{dt} = -2 \times 10^{-3} \cdot e^{-t}(2t - t^2) \]

4. **Set the emf to zero**:
To find the time when the emf is zero, we set the equation to zero:
\[ -2 \times 10^{-3} \cdot e^{-t}(2t - t^2) = 0 \]
Since \( e^{-t} \) is never zero, we can simplify to:
\[ 2t - t^2 = 0 \]

5. **Factor the equation**:
This can be factored as:
\[ t(2 - t) = 0 \]
This gives us two solutions:
\[ t = 0 \quad \text{or} \quad 2 - t = 0 \implies t = 2 \]

6. **Determine the valid solution**:
Since \( t = 0 \) is not a valid time for the emf to be zero in the context of this problem, we take:
\[ t = 2 \, \text{seconds} \]

### Final Answer:
The time at which the emf is zero is \( t = 2 \, \text{seconds} \).