The current i in an induction coil varies with time according to the graph shown in figure. Which of the following graph shows induced emf in the coil with time:

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the induced EMF in an induction coil based on the given current-time graph, we can follow these steps:

### Step 1: Understand the relationship between current and induced EMF
The induced EMF (E) in an induction coil is given by the formula:
\[ E = -L \frac{di}{dt} \]
where \( L \) is the inductance of the coil, and \( \frac{di}{dt} \) is the rate of change of current with respect to time.

### Step 2: Analyze the current-time graph
The current \( i \) varies with time according to the provided graph. We need to analyze the slope of the graph at different intervals to determine \( \frac{di}{dt} \).

### Step 3: Identify intervals and calculate \( \frac{di}{dt} \)
1. **From 0 to \( t_1 \)**: The graph is horizontal, indicating that the current is constant. Therefore, \( \frac{di}{dt} = 0 \).
- Induced EMF \( E = -L \cdot 0 = 0 \).

2. **From \( t_1 \) to \( t_2 \)**: The graph has a negative slope, indicating that the current is decreasing. Therefore, \( \frac{di}{dt} < 0 \), which means \( E \) will be positive.
- Induced EMF \( E = -L \cdot \text{(negative value)} = \text{positive value} \).

3. **From \( t_2 \) to \( t_3 \)**: The graph has a positive slope, indicating that the current is increasing. Therefore, \( \frac{di}{dt} > 0 \), which means \( E \) will be negative.
- Induced EMF \( E = -L \cdot \text{(positive value)} = \text{negative value} \).

### Step 4: Summarize the induced EMF behavior
- From 0 to \( t_1 \): \( E = 0 \)
- From \( t_1 \) to \( t_2 \): \( E \) is positive
- From \( t_2 \) to \( t_3 \): \( E \) is negative

### Step 5: Choose the correct graph
Based on the analysis, we need to find a graph that:
- Starts at 0 from 0 to \( t_1 \)
- Shows a positive value from \( t_1 \) to \( t_2 \)
- Shows a negative value from \( t_2 \) to \( t_3 \)

After comparing with the options provided, we find that **Graph D** satisfies all these conditions.

### Final Answer
The correct graph that shows the induced EMF in the coil with time is **Graph D**.

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