If an increase in length of copper wire is 0.5% due to stretching, the percentage increase in its resistance will be

Options

A. ` 0.1%`
B. `0.2% `
C. ` 1%`
D. `2%`

Explanation

To find the percentage increase in resistance when the length of a copper wire is increased by 0.5%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between length and resistance**:
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity of the material, \( L \) is the length of the wire, and \( A \) is the cross-sectional area.

2. **Initial conditions**:
Let the initial length of the wire be \( L \) and the initial cross-sectional area be \( A \). The initial resistance \( R \) can be expressed as:
\[ R = \frac{\rho L}{A} \]

3. **Calculate the new length after the increase**:
The length of the wire is increased by 0.5%. Therefore, the new length \( L' \) can be calculated as:
\[ L' = L + 0.5\% \text{ of } L = L + 0.005L = 1.005L \]

4. **Volume conservation**:
The volume of the wire remains constant, which means:
\[ V = A \cdot L = A' \cdot L' \]
where \( A' \) is the new cross-sectional area. Rearranging gives:
\[ A' = \frac{A \cdot L}{L'} = \frac{A \cdot L}{1.005L} = \frac{A}{1.005} \]

5. **Calculate the new resistance**:
The new resistance \( R' \) can be expressed as:
\[ R' = \frac{\rho L'}{A'} = \frac{\rho (1.005L)}{\frac{A}{1.005}} = \frac{\rho (1.005L) \cdot 1.005}{A} = \frac{\rho (1.005)^2 L}{A} \]
This simplifies to:
\[ R' = R \cdot (1.005)^2 \]

6. **Calculate the percentage increase in resistance**:
To find the percentage increase in resistance, we first find the ratio of the new resistance to the old resistance:
\[ \frac{R'}{R} = (1.005)^2 \]
Calculating \( (1.005)^2 \):
\[ (1.005)^2 = 1.010025 \]
The percentage increase in resistance is:
\[ \text{Percentage Increase} = \left( \frac{R' - R}{R} \right) \times 100 = \left( (1.010025 - 1) \times 100 \right) = 1.0025\% \]
Rounding this gives approximately:
\[ \text{Percentage Increase} \approx 1\% \]

### Final Answer:
The percentage increase in resistance when the length of the copper wire is increased by 0.5% is approximately **1%**.