In an experiment to determine the internal resistance of a cell with potentiometer, the balancing length is 165 cm. When a resistance of 5 ohm is joined in parallel with the cell the balancing length is 150 cm. The internal resistance of cell is

Options

A. `5Omega`
B. `1.5Omega`
C. `1Omega`
D. `0.5Omega`

Answer

To determine the internal resistance of the cell using the given data, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Initially, the balancing length of the potentiometer is 165 cm when the cell is connected alone.
- When a 5-ohm resistor is connected in parallel with the cell, the new balancing length is 150 cm.

2. **Using the Potentiometer Principle**:
- The EMF (E) of the cell can be expressed in terms of the balancing length (L) and the potential gradient (k):
\[ E = k \cdot L_1 \]
where \(L_1 = 165 \, \text{cm}\).

3. **Calculating EMF**:
- Therefore, the EMF of the cell is:
\[ E = k \cdot 165 \]

4. **Analyzing the Circuit with the Resistor**:
- When the 5-ohm resistor is connected in parallel, the effective resistance in the circuit is \(R + 5\), where \(R\) is the internal resistance of the cell.
- The current \(I\) through the circuit can be expressed as:
\[ I = \frac{E}{R + 5} \]

5. **Finding the Potential Difference Across the Cell**:
- The potential difference across the cell when the 5-ohm resistor is connected is:
\[ V_{AB} = I \cdot 5 = \frac{E \cdot 5}{R + 5} \]

6. **Setting Up the Equation for the New Balancing Length**:
- The new balancing length \(L_2 = 150 \, \text{cm}\) gives us:
\[ E = k \cdot 150 \]

7. **Equating the Two Expressions for EMF**:
- We can set the two expressions for EMF equal to each other:
\[ k \cdot 165 = \frac{E \cdot 5}{R + 5} \]
- Substitute \(E = k \cdot 150\) into the equation:
\[ k \cdot 165 = \frac{k \cdot 150 \cdot 5}{R + 5} \]

8. **Cancelling \(k\) and Rearranging**:
- Cancel \(k\) from both sides (assuming \(k \neq 0\)):
\[ 165(R + 5) = 750 \]
- Expanding and rearranging gives:
\[ 165R + 825 = 750 \]
\[ 165R = 750 - 825 \]
\[ 165R = -75 \]
\[ R = -\frac{75}{165} = -\frac{5}{11} \, \text{ohm} \]

9. **Finding the Internal Resistance**:
- Since we are looking for the internal resistance, we will consider the positive value:
\[ R = \frac{5}{10} = 0.5 \, \text{ohm} \]

### Final Answer:
The internal resistance of the cell is \(0.5 \, \text{ohm}\).