The wire is stretched to increase the length by 1%. Find the percentage change in the resistance.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem of finding the percentage change in resistance when a wire is stretched to increase its length by 1%, we can follow these steps:

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

### Step 2: Relate volume to length and area
Since the volume of the wire remains constant when it is stretched, we can express this relationship as:
\[ V = A \times L \]
If the length \( L \) increases, the area \( A \) must decrease to keep the volume constant.

### Step 3: Express the new length and area
If the length increases by 1%, we can express the new length \( L' \) as:
\[ L' = L + 0.01L = 1.01L \]
Let the new area be \( A' \). Since the volume is constant:
\[ A \times L = A' \times L' \]
Substituting \( L' \):
\[ A \times L = A' \times (1.01L) \]
This simplifies to:
\[ A' = \frac{A}{1.01} \]

### Step 4: Substitute the new values into the resistance formula
Now we can find the new resistance \( R' \):
\[ R' = \frac{\rho L'}{A'} = \frac{\rho (1.01L)}{\frac{A}{1.01}} = \frac{\rho (1.01L) \times 1.01}{A} = \frac{\rho (1.01^2 L)}{A} \]
Thus, we can express the new resistance in terms of the original resistance \( R \):
\[ R' = R \times (1.01^2) \]

### Step 5: Calculate the percentage change in resistance
The percentage change in resistance \( \Delta R \) can be calculated as:
\[ \Delta R = \frac{R' - R}{R} \times 100\% \]
Substituting \( R' \):
\[ \Delta R = \frac{R(1.01^2) - R}{R} \times 100\% = (1.01^2 - 1) \times 100\% \]
Calculating \( 1.01^2 \):
\[ 1.01^2 = 1.0201 \]
Thus:
\[ \Delta R = (1.0201 - 1) \times 100\% = 0.0201 \times 100\% = 2.01\% \]

### Conclusion
The percentage change in resistance when the wire is stretched to increase its length by 1% is approximately **2.01%**.

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