In a potentiometer experiment if the voltage across resistance R and R + r are balanced at the length `l_1 and l_2` respectively the value of resistance r will be

Options

A. `((l_1-l_2)/(l_2))R`
B. `((l_1-l_2)/(l_1))R`
C. `((l_2-l_1)/(l_1))R`
D. `((l_2-l_1)/(l_2))R`

Answer

To solve the problem, we need to analyze the potentiometer experiment and derive the expression for the resistance \( r \) based on the given lengths \( l_1 \) and \( l_2 \).

### Step-by-Step Solution:

1. **Understanding the Potentiometer Setup**:
- In a potentiometer, the voltage drop across a wire of length \( l \) is proportional to the length of the wire. The resistance per unit length is denoted as \( \lambda \) (in ohms/meter).
- When a jockey is placed at length \( l_1 \), it balances the resistance \( R \).
- When the jockey is placed at length \( l_2 \), it balances the resistance \( R + r \).

2. **Setting Up the Equations**:
- For the first case (length \( l_1 \)):
\[ V = I \cdot R \quad \text{(Ohm's Law)} \]
The voltage drop across the potentiometer wire of length \( l_1 \) is:
\[ V = \lambda \cdot l_1 \]
Therefore, we can write:
\[ R = \lambda \cdot l_1 \quad \text{(1)} \]

- For the second case (length \( l_2 \)):
The voltage drop across the potentiometer wire of length \( l_2 \) is:
\[ V = \lambda \cdot l_2 \]
Thus, we have:
\[ R + r = \lambda \cdot l_2 \quad \text{(2)} \]

3. **Subtracting the Equations**:
- Now, we will subtract equation (1) from equation (2):
\[ (R + r) - R = \lambda \cdot l_2 - \lambda \cdot l_1 \]
Simplifying this gives:
\[ r = \lambda \cdot (l_2 - l_1) \quad \text{(3)} \]

4. **Expressing \( \lambda \)**:
- From equation (1), we can express \( \lambda \):
\[ \lambda = \frac{R}{l_1} \]
- Substituting this value of \( \lambda \) into equation (3):
\[ r = \frac{R}{l_1} \cdot (l_2 - l_1) \]

5. **Final Expression**:
- Therefore, the value of the resistance \( r \) is:
\[ r = \frac{R \cdot (l_2 - l_1)}{l_1} \]

### Conclusion:
The value of resistance \( r \) is given by:
\[ r = \frac{R \cdot (l_2 - l_1)}{l_1} \]