A 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:

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the relationship between the potential difference \( V \) across the external resistance \( R \) and the resistance \( R \) itself, given a cell with an electromotive force (emf) \( \epsilon \) and internal resistance \( r \).

### Step-by-Step Solution:

1. **Understanding the Circuit**:
- We have a cell with emf \( \epsilon \) and internal resistance \( r \).
- This cell is connected to a variable external resistance \( R \).
- We need to find the potential difference \( V \) across the external resistance \( R \) as \( R \) is varied.

2. **Finding the Current in the Circuit**:
- The total resistance in the circuit is the sum of the internal resistance \( r \) and the external resistance \( R \):
\[ R_{\text{total}} = R + r \]
- Using Ohm's Law, the current \( I \) flowing through the circuit can be expressed as:
\[ I = \frac{\epsilon}{R + r} \]

3. **Calculating 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 \( I \):
\[ V = \left(\frac{\epsilon}{R + r}\right) \cdot R \]
- Simplifying this, we get:
\[ V = \frac{\epsilon R}{R + r} \]

4. **Analyzing the Expression for \( V \)**:
- The expression \( V = \frac{\epsilon R}{R + r} \) shows how \( V \) changes as \( R \) varies.
- As \( R \) increases, we can analyze the behavior of \( V \):
- If \( R \) approaches infinity, the term \( R + r \) will also approach infinity, leading to:
\[ V \to \epsilon \quad \text{(as \( R \to \infty \))} \]
- If \( R \) is very small (approaching 0), then:
\[ V \to 0 \quad \text{(as \( R \to 0 \))} \]

5. **Graphical Representation**:
- The relationship between \( V \) and \( R \) is hyperbolic because \( V \) is proportional to \( R \) but divided by a term that increases with \( R \).
- As \( R \) increases, \( V \) increases initially and approaches \( \epsilon \) but never exceeds it.
- Therefore, the plot of \( V \) versus \( R \) will be a hyperbola that approaches the line \( V = \epsilon \) as \( R \) increases.

6. **Choosing the Correct Option**:
- Based on the analysis, we can eliminate options that suggest a linear relationship.
- The correct option is the one that shows a hyperbolic curve approaching \( \epsilon \) as \( R \) increases.

### Final Answer:
The correct plot of potential difference \( V \) across \( R \) as \( R \) is increased is represented by option 3.