If a copper wire is stretched to make it 0.1% longer, then the percentage change in resistance is approximately -

Options

A. `0.1%`
B. `0.2%`
C. `0.4%`
D. `0.8%`

Explanation

To solve the problem of finding the percentage change in resistance when a copper wire is stretched to make it 0.1% longer, 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. **Determine the Change in Length**:
If the wire is stretched to make it 0.1% longer, we can express this as:
\[ \Delta L = 0.001L \]
Therefore, the new length \( L' \) becomes:
\[ L' = L + \Delta L = L + 0.001L = 1.001L \]

3. **Determine the Change in Area**:
When the wire is stretched, its volume remains constant. The volume \( V \) of the wire can be expressed as:
\[ V = L \cdot A \]
When the length changes to \( L' \), the new area \( A' \) can be found using:
\[ V = L' \cdot A' = 1.001L \cdot A' \]
Setting the volumes equal gives:
\[ L \cdot A = 1.001L \cdot A' \]
Simplifying this, we find:
\[ A' = \frac{A}{1.001} \]

4. **Calculate the New Resistance**:
The new resistance \( R' \) can be expressed as:
\[ R' = \frac{\rho L'}{A'} = \frac{\rho (1.001L)}{(A/1.001)} = \frac{\rho (1.001^2 L)}{A} = 1.001^2 \cdot R \]

5. **Calculate the Percentage Change in Resistance**:
The percentage change in resistance can be calculated using:
\[ \text{Percentage Change} = \left( \frac{R' - R}{R} \right) \times 100 \]
Substituting \( R' \):
\[ \text{Percentage Change} = \left( \frac{1.001^2 R - R}{R} \right) \times 100 = (1.001^2 - 1) \times 100 \]
Calculating \( 1.001^2 \):
\[ 1.001^2 \approx 1.002001 \]
Therefore:
\[ \text{Percentage Change} \approx (1.002001 - 1) \times 100 \approx 0.2001\% \]

6. **Conclusion**:
The percentage change in resistance is approximately \( 0.2\% \).

### Final Answer:
The percentage change in resistance is approximately **0.2%**.