Self-inductance of a coil is 2mH, current through this coil is, `I=t^(2)e^(-t)` (t = time). After how much time will the induced emf be zero?

Options

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

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Relationship Between Current and Induced EMF
The induced electromotive force (emf) in a coil is given by the formula:
\[ \text{Induced EMF} = -L \frac{dI}{dt} \]
where \(L\) is the self-inductance of the coil and \(I\) is the current flowing through it.

### Step 2: Identify Given Values
From the problem, we have:
- Self-inductance \(L = 2 \, \text{mH} = 2 \times 10^{-3} \, \text{H}\)
- Current \(I(t) = t^2 e^{-t}\)

### Step 3: Differentiate the Current with Respect to Time
We need to find \(\frac{dI}{dt}\). To do this, we will apply the product rule of differentiation:
\[ I(t) = t^2 e^{-t} \]
Using the product rule:
\[ \frac{dI}{dt} = \frac{d}{dt}(t^2) \cdot e^{-t} + t^2 \cdot \frac{d}{dt}(e^{-t}) \]
Calculating each term:
\[ \frac{d}{dt}(t^2) = 2t \]
\[ \frac{d}{dt}(e^{-t}) = -e^{-t} \]
Thus,
\[ \frac{dI}{dt} = 2t e^{-t} + t^2 (-e^{-t}) = e^{-t}(2t - t^2) = e^{-t}t(2 - t) \]

### Step 4: Substitute \(\frac{dI}{dt}\) into the Induced EMF Equation
Now, substituting \(\frac{dI}{dt}\) into the induced EMF equation:
\[ \text{Induced EMF} = -L \frac{dI}{dt} = -2 \times 10^{-3} e^{-t} t (2 - t) \]

### Step 5: Set Induced EMF to Zero
To find when the induced EMF is zero, we set the equation to zero:
\[ -2 \times 10^{-3} e^{-t} t (2 - t) = 0 \]
Since \(e^{-t}\) is never zero, we simplify to:
\[ t(2 - t) = 0 \]
This gives us two solutions:
1. \(t = 0\)
2. \(2 - t = 0 \Rightarrow t = 2\)

### Step 6: Determine the Valid Solution
Since the problem asks for the time after which the induced EMF becomes zero, we discard \(t = 0\) (as it is the initial time). Therefore, the valid solution is:
\[ t = 2 \, \text{seconds} \]

### Final Answer
The induced EMF will be zero after \(t = 2\) seconds.

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