`2A` cell having an emf `epsilon` and internal resistance `r` is connected across a variable external resistance `R`. As the resistance `R` is increased, the plot of potential difference `V` across `R` is given by

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Explanation

To solve the problem, we need to analyze the circuit involving a cell with an electromotive force (emf) \( \epsilon \) and internal resistance \( r \), connected to a variable external resistance \( R \). We want to find the relationship between the potential difference \( V \) across the external resistance \( R \) as \( R \) is varied.

### Step-by-Step Solution:

1. **Understand the Circuit Configuration**:
- We have a cell with emf \( \epsilon \) and internal resistance \( r \).
- The external resistance \( R \) is variable.
- The total resistance in the circuit is \( R + r \).

2. **Apply Ohm's Law**:
- According to Ohm's law, the current \( I \) flowing through the circuit can be expressed as:
\[ I = \frac{\epsilon}{R + r} \]

3. **Calculate the Potential Difference \( V \)**:
- The potential difference \( V \) across the external resistance \( R \) can be calculated using Ohm's law:
\[ V = I \cdot R \]
- Substituting the expression for \( I \):
\[ V = \left(\frac{\epsilon}{R + r}\right) R \]
- Therefore, we have:
\[ V = \frac{\epsilon R}{R + r} \]

4. **Analyze the Relationship**:
- The equation \( V = \frac{\epsilon R}{R + r} \) shows that \( V \) depends on \( R \).
- As \( R \) increases, the term \( R + r \) also increases, which means that \( V \) will initially increase but will eventually decrease as \( R \) becomes much larger than \( r \).

5. **Determine the Graphical Representation**:
- To understand how \( V \) changes with \( R \), we can analyze the behavior of the function:
- For small values of \( R \), \( V \) will increase.
- As \( R \) approaches infinity, \( V \) approaches \( \epsilon \) but never exceeds it.
- The graph will show an initial increase and then a leveling off as \( R \) increases.

6. **Conclusion**:
- The plot of potential difference \( V \) across the external resistance \( R \) will start from zero, increase to a maximum value (which is less than \( \epsilon \)), and then approach \( \epsilon \) asymptotically as \( R \) increases.