When a conducting wire is connected in the right gap and known resistance in the left gap, the balancing length is 60cm. The balancing length becomes 42.4 cm when the wire is stretched so that its length increases by

Options

A. `10%`
B. `20%`
C. `25%`
D. `42.7 %`

Answer

To solve the problem step by step, we need to analyze the situation involving a meter bridge and the resistance of a wire.

### Step-by-Step Solution:

1. **Understanding the Initial Setup**:
- We have a meter bridge with a known resistance \( R \) connected in one gap and a conducting wire in the other gap.
- The initial balancing length is given as \( L_1 = 60 \, \text{cm} \).

2. **Using the Balance Condition**:
- According to the meter bridge principle, the ratio of the resistances is equal to the ratio of the lengths:
\[ \frac{R}{r} = \frac{L_1}{100 - L_1} \]
- Here, \( r \) is the resistance of the conducting wire.

3. **Calculating the Initial Resistance Ratio**:
- Substituting \( L_1 \):
\[ \frac{R}{r} = \frac{60}{100 - 60} = \frac{60}{40} = \frac{3}{2} \]
- This gives us the first equation:
\[ \frac{R}{r} = 1.5 \quad \text{(Equation 1)} \]

4. **Analyzing the Stretched Wire**:
- When the wire is stretched, the new balancing length is \( L_2 = 42.4 \, \text{cm} \).
- The new resistance \( r' \) of the stretched wire can be analyzed similarly:
\[ \frac{R}{r'} = \frac{L_2}{100 - L_2} \]

5. **Calculating the New Resistance Ratio**:
- Substituting \( L_2 \):
\[ \frac{R}{r'} = \frac{42.4}{100 - 42.4} = \frac{42.4}{57.6} \]
- Calculating this gives:
\[ \frac{R}{r'} = 0.736 \quad \text{(Equation 2)} \]

6. **Finding the Ratio of Resistances**:
- Now, we can find the ratio of the new resistance to the old resistance:
\[ \frac{r'}{r} = \frac{0.736}{1.5} = 0.49067 \]
- This means:
\[ r' = 2.073 \cdot r \]

7. **Relating Lengths and Resistances**:
- The resistance of a wire is given by:
\[ r = \frac{\rho l}{A} \]
- When the wire is stretched, its length increases to \( l' \) and its area decreases. The volume remains constant:
\[ A \cdot l = A' \cdot l' \]
- Thus, we can express \( A' \) in terms of \( A \) and the lengths:
\[ A' = \frac{A \cdot l}{l'} \]

8. **Substituting into Resistance Equation**:
- The new resistance can be expressed as:
\[ r' = \frac{\rho l'}{A'} = \frac{\rho l' \cdot l}{A \cdot l'} = \frac{\rho l'^2}{A \cdot l} \]

9. **Setting Up the Equation**:
- From the ratio of resistances, we have:
\[ 2.073 \cdot r = \frac{\rho l'^2}{\rho l} \implies 2.073 = \frac{l'^2}{l} \]

10. **Finding the New Length**:
- Taking the square root:
\[ \frac{l'}{l} = \sqrt{2.073} \approx 1.44 \]
- Thus:
\[ l' \approx 1.427 \cdot l \]

11. **Calculating Percentage Increase**:
- The percentage increase in length is given by:
\[ \text{Percentage Increase} = \frac{l' - l}{l} \times 100 \]
- Substituting the values:
\[ = \frac{1.427l - l}{l} \times 100 = 0.427 \times 100 \approx 42.7\% \]

### Final Answer:
The percentage increase in the length of the wire is approximately **42.7%**.