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 step by step, we will use the principles of a potentiometer and the relationships between voltage, resistance, and length.

### Step-by-Step Solution:

1. **Understanding the Potentiometer Principle**:
In a potentiometer, the potential difference (voltage) across a length of wire is directly proportional to the length of the wire. This can be expressed as:
\[ V = k \cdot L \]
where \( V \) is the voltage, \( k \) is a constant (proportionality constant), and \( L \) is the length of the wire.

2. **Setting Up the Equations**:
For the first balance at length \( l_1 \) with resistance \( R \):
\[ V = k \cdot l_1 \quad \text{(1)} \]
For the second balance at length \( l_2 \) with resistance \( R + r \):
\[ V' = k \cdot l_2 \quad \text{(2)} \]

3. **Relating Voltage to Resistance**:
According to Ohm's law, the voltage across a resistor can also be expressed as:
\[ V = I \cdot R \quad \text{and} \quad V' = I' \cdot (R + r) \]
where \( I \) and \( I' \) are the currents through the resistors \( R \) and \( R + r \), respectively.

4. **Equating the Two Voltages**:
Since both equations (1) and (2) represent the same potential difference, we can equate them:
\[ k \cdot l_1 = I \cdot R \quad \text{and} \quad k \cdot l_2 = I' \cdot (R + r) \]

5. **Finding the Wavelength**:
From the first equation, we can express \( k \):
\[ k = \frac{V}{l_1} = \frac{I \cdot R}{l_1} \]
Substituting \( k \) in the second equation:
\[ \frac{I \cdot R}{l_1} \cdot l_2 = I' \cdot (R + r) \]

6. **Subtracting the Equations**:
Rearranging gives:
\[ I \cdot R \cdot \frac{l_2}{l_1} = I' \cdot (R + r) \]
By subtracting the first equation from the second, we can isolate \( r \):
\[ (R + r) - R = \frac{I \cdot R}{l_1} \cdot (l_2 - l_1) \]
Thus,
\[ r = \frac{I \cdot R}{l_1} \cdot (l_2 - l_1) \]

7. **Final Expression for Resistance \( r \)**:
We can express \( r \) in terms of \( R \), \( l_1 \), and \( l_2 \):
\[ r = \frac{R}{l_1} \cdot (l_2 - l_1) \]

### Conclusion:
The value of resistance \( r \) can be expressed as:
\[ r = R \cdot \frac{(l_2 - l_1)}{l_1} \]