The current in a coil varies with time as shown in the figuro. The variation of induced emf with time would be

Options

A.
B.
C.
D.

Answer

To solve the problem of finding the variation of induced EMF with time based on the given current variation in a coil, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: The induced EMF (ε) in a coil is given by Faraday's law of electromagnetic induction, which states:
\[ \varepsilon = -L \frac{di}{dt} \]
where \(L\) is the inductance of the coil and \(\frac{di}{dt}\) is the rate of change of current with respect to time.

2. **Analyze the Current-Time Graph**: Examine the graph of current (I) versus time (t) provided in the question. Identify the different segments of the graph to determine how the current changes over time.

3. **Segment 1: From 0 to \(T/4\)**:
- The current increases linearly.
- Here, \(\frac{di}{dt}\) is constant and positive.
- Therefore, the induced EMF (ε) will be negative (since the formula has a negative sign) and constant.

4. **Segment 2: From \(T/4\) to \(T/2\)**:
- The current remains constant (horizontal line).
- Thus, \(\frac{di}{dt} = 0\).
- Consequently, the induced EMF (ε) will be zero.

5. **Segment 3: From \(T/2\) to \(3T/4\)**:
- The current decreases linearly.
- Here, \(\frac{di}{dt}\) is constant and negative.
- Therefore, the induced EMF (ε) will be positive and constant.

6. **Segment 4: From \(3T/4\) to \(T\)**:
- The current remains constant again.
- Thus, \(\frac{di}{dt} = 0\).
- Therefore, the induced EMF (ε) will again be zero.

7. **Plotting the Induced EMF**: Based on the analysis:
- From 0 to \(T/4\), ε is negative and constant.
- From \(T/4\) to \(T/2\), ε is zero.
- From \(T/2\) to \(3T/4\), ε is positive and constant.
- From \(3T/4\) to \(T\), ε is zero.

8. **Select the Correct Option**: Compare the derived graph of induced EMF with the provided options in the question. The correct option should match the behavior we have plotted.

### Final Answer:
The variation of induced EMF with time will show a negative constant value from \(0\) to \(T/4\), zero from \(T/4\) to \(T/2\), a positive constant value from \(T/2\) to \(3T/4\), and zero again from \(3T/4\) to \(T\).