For a coil having L=4 mH , current flow through it is `I=t^3.e^(-t)` then the time at which emf becomes zero

Options

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

Explanation

To solve the problem, we need to determine the time at which the induced electromotive force (emf) in a coil becomes zero. The given parameters are:

- Inductance of the coil, \( L = 4 \, \text{mH} = 4 \times 10^{-3} \, \text{H} \)
- Current flowing through the coil, \( I(t) = t^3 e^{-t} \)

### Step-by-Step Solution:

1. **Understand the formula for induced emf**:
The induced emf (\( \mathcal{E} \)) in a coil is given by:
\[ \mathcal{E} = -L \frac{dI}{dt} \]
where \( L \) is the inductance and \( \frac{dI}{dt} \) is the rate of change of current with respect to time.

2. **Differentiate the current function**:
We need to find \( \frac{dI}{dt} \) for the given current function \( I(t) = t^3 e^{-t} \). We will use the product rule for differentiation:
\[ \frac{dI}{dt} = \frac{d}{dt}(t^3) \cdot e^{-t} + t^3 \cdot \frac{d}{dt}(e^{-t}) \]
Calculating each derivative:
- \( \frac{d}{dt}(t^3) = 3t^2 \)
- \( \frac{d}{dt}(e^{-t}) = -e^{-t} \)

Therefore,
\[ \frac{dI}{dt} = 3t^2 e^{-t} - t^3 e^{-t} \]
Simplifying this, we get:
\[ \frac{dI}{dt} = e^{-t}(3t^2 - t^3) \]

3. **Substitute \( \frac{dI}{dt} \) into the emf equation**:
Now we substitute \( \frac{dI}{dt} \) into the emf equation:
\[ \mathcal{E} = -L \cdot e^{-t}(3t^2 - t^3) \]
Substituting \( L = 4 \times 10^{-3} \):
\[ \mathcal{E} = -4 \times 10^{-3} e^{-t}(3t^2 - t^3) \]

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

5. **Factor the equation**:
Factoring out \( t^2 \):
\[ t^2(3 - t) = 0 \]
This gives us two solutions:
\[ t^2 = 0 \quad \text{or} \quad 3 - t = 0 \]
Thus,
\[ t = 0 \quad \text{or} \quad t = 3 \]

6. **Determine the relevant time**:
Since we are interested in the time after the current starts flowing, we discard \( t = 0 \) and take:
\[ t = 3 \, \text{seconds} \]

### Final Answer:
The time at which the emf becomes zero is \( t = 3 \, \text{seconds} \).