The graph shows the variation of magnetic field B that exists through a conducing loop, perpendicular to the plane of the loop. The emf's induced in the loop during equal intervals a, b, c, d and e are related as

Options

A. `E_(b) lt E_(d) lt E_(e)`
B. `E_(a) gt E_(c)`
C. `E_(c) gt E_(a)`
D. `E_(b) gt E_(d) = E_(e)`

Answer

To solve the problem regarding the induced EMFs in a conducting loop due to the variation of the magnetic field over time, we will follow these steps:

### Step 1: Understand the relationship between magnetic field and induced EMF
According to Faraday's law of electromagnetic induction, the induced EMF (ε) in a loop is directly related to the rate of change of magnetic flux (Φ) through the loop. The formula is given by:
\[ \text{EMF} = -\frac{d\Phi}{dt} \]
where \(\Phi = B \cdot A \cdot \cos(\theta)\). Since the magnetic field is perpendicular to the plane of the loop, \(\cos(\theta) = 1\).

### Step 2: Express EMF in terms of magnetic field change
Since the area (A) of the loop is constant, we can simplify the expression for EMF:
\[ \text{EMF} = A \cdot \frac{dB}{dt} \]
This means that the induced EMF is proportional to the rate of change of the magnetic field (dB/dt).

### Step 3: Analyze the graph
From the graph provided in the question, we need to identify the intervals (A, B, C, D, E) and observe the changes in the magnetic field (B) over these intervals:
- **Interval A-B**: There is a significant increase in the magnetic field, indicating a high rate of change (dB/dt is large).
- **Interval B-C**: The magnetic field remains constant, so there is no change (dB/dt = 0).
- **Interval C-D**: There is a decrease in the magnetic field, indicating a negative rate of change (dB/dt is negative).
- **Interval D-E**: The magnetic field decreases further, indicating a continued negative rate of change.

### Step 4: Compare the induced EMFs
Based on the analysis of the graph:
- The EMF during interval A-B (E_A) will be the highest because of the maximum change in magnetic field.
- The EMF during interval B-C (E_B) will be zero since there is no change in the magnetic field.
- The EMFs during intervals C-D (E_C) and D-E (E_D) will be negative but will have different magnitudes.

Thus, we can summarize the relationship as:
\[ E_A > E_D > E_E > E_C > E_B \]

### Step 5: Identify the correct option
From the analysis, we can conclude that the induced EMFs are related as follows:
\[ E_B < E_C < E_D < E_A \]
This leads us to the conclusion that option D (E_B is greater than E_A) is incorrect. Instead, we find that E_A is the greatest and E_B is the least.

### Final Answer
The induced EMFs in the loop during intervals A, B, C, D, and E are related as:
\[ E_A > E_D > E_E > E_C > E_B \]