A mete bridge is used to determine the resistance of an unknown wire by measuring the balance point length l. If the wire is replaced by another wire of same material but with double the length and half the thickness the balancing point is expected to be

Options

A. `1/(8l)`
B. `1/(4l)`
C. `8l`
D. 16l

Answer

To solve the problem, we need to determine the new balancing point length \( L' \) when the wire is replaced by another wire of the same material, but with double the length and half the thickness.

### Step-by-Step Solution:

1. **Understanding the Initial Setup**:
- Let the original wire have a length \( L \) and a thickness \( r \) (radius).
- The resistance \( R \) of the wire can be expressed as:
\[ R = \frac{\rho L}{A} \]
where \( A \) is the cross-sectional area of the wire.

2. **Calculating the Cross-Sectional Area**:
- The cross-sectional area \( A \) of a wire with radius \( r \) is given by:
\[ A = \pi r^2 \]
- Therefore, the resistance can be rewritten as:
\[ R = \frac{\rho L}{\pi r^2} \]

3. **Replacing the Wire**:
- The new wire has double the length, so \( L' = 2L \).
- The thickness (radius) is half, so the new radius \( r' = \frac{r}{2} \).

4. **Calculating the New Cross-Sectional Area**:
- The new cross-sectional area \( A' \) is:
\[ A' = \pi (r')^2 = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} = \frac{\pi r^2}{4} \]

5. **Calculating the New Resistance**:
- The resistance \( R' \) of the new wire can now be calculated:
\[ R' = \frac{\rho L'}{A'} = \frac{\rho (2L)}{\frac{\pi r^2}{4}} = \frac{8\rho L}{\pi r^2} \]
- Since the original resistance \( R \) is \( \frac{\rho L}{\pi r^2} \), we can express \( R' \) in terms of \( R \):
\[ R' = 8R \]

6. **Using the Meter Bridge Principle**:
- According to the meter bridge principle, the ratio of the resistances is equal to the ratio of the lengths:
\[ \frac{R}{R'} = \frac{L}{L'} \]
- Substituting \( R' = 8R \):
\[ \frac{R}{8R} = \frac{L}{L'} \implies \frac{1}{8} = \frac{L}{L'} \]

7. **Finding the New Balancing Point**:
- Rearranging gives:
\[ L' = 8L \]

### Conclusion:
The new balancing point \( L' \) when the wire is replaced is expected to be \( 8L \).