Two wires `A` and `B` have the same cross section and are made of the same material, but the length of wire `A` is twice that of `B`. Then, for a given load

Options

A. the extension of `A` will be twice that of `B`
B. the extensions of `A` and `B` will be equal
C. the strain in `A` will be half that in `B`
D. the strains in `A` and `B` will be equal

Explanation

To solve the problem, we need to analyze the relationship between the two wires A and B based on their lengths, cross-sectional areas, and the material they are made of.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- Wires A and B have the same cross-sectional area (A).
- Wires A and B are made of the same material, hence they have the same Young's modulus (Y).
- The length of wire A (L_A) is twice the length of wire B (L_B), so we can write:
\[ L_A = 2L_B \]

2. **Applying the Formula for Extension**:
- The extension (ΔL) in a wire is given by the formula:
\[ \Delta L = \frac{F L}{A Y} \]
- Where:
- F = applied force (load)
- L = original length of the wire
- A = cross-sectional area
- Y = Young's modulus

3. **Calculating the Extension for Wire A**:
- For wire A:
\[ \Delta L_A = \frac{F L_A}{A Y} \]
- Substituting \(L_A = 2L_B\):
\[ \Delta L_A = \frac{F (2L_B)}{A Y} = 2 \frac{F L_B}{A Y} \]

4. **Calculating the Extension for Wire B**:
- For wire B:
\[ \Delta L_B = \frac{F L_B}{A Y} \]

5. **Finding the Ratio of Extensions**:
- Now, we can find the ratio of the extensions of wire A to wire B:
\[ \frac{\Delta L_A}{\Delta L_B} = \frac{2 \frac{F L_B}{A Y}}{\frac{F L_B}{A Y}} = 2 \]
- This shows that:
\[ \Delta L_A = 2 \Delta L_B \]
- Thus, the extension of wire A is twice that of wire B.

6. **Analyzing Strain**:
- Strain (ε) is defined as the change in length divided by the original length:
\[ \text{Strain} = \frac{\Delta L}{L} \]
- For wire A:
\[ \text{Strain}_A = \frac{\Delta L_A}{L_A} = \frac{2 \Delta L_B}{2 L_B} = \frac{\Delta L_B}{L_B} = \text{Strain}_B \]
- This indicates that the strain in both wires A and B is equal.

### Conclusion:
- The extension of wire A is twice that of wire B: \(\Delta L_A = 2 \Delta L_B\).
- The strain in wire A is equal to the strain in wire B: \(\text{Strain}_A = \text{Strain}_B\).

### Final Answers:
- **Extension**: The extension of wire A is twice that of wire B.
- **Strain**: The strain in both wires is equal.