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 properties and the given conditions.

### Step 1: Understand the properties of the wires
- Both wires A and B have the same cross-sectional area and are made of the same material.
- This means that the Young's modulus (Y) for both wires is the same.

### Step 2: Define the lengths of the wires
- Let the length of wire B be \( L \).
- Therefore, the length of wire A is \( 2L \) (since it is twice the length of wire B).

### Step 3: Define the loads on the wires
- Let the load (force) applied to both wires be \( F \).
- Since both wires are under the same load, the force in wire A is equal to the force in wire B, i.e., \( F_A = F_B = F \).

### Step 4: Calculate the stress in both wires
- Stress is defined as the force per unit area:
\[ \text{Stress} = \frac{F}{A} \]
- Since both wires have the same cross-sectional area \( A \) and are subjected to the same force \( F \), the stress in both wires is equal:
\[ \text{Stress}_A = \text{Stress}_B \]

### Step 5: Relate stress and strain using Young's modulus
- Young's modulus is defined as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
- Since the Young's modulus for both wires is the same and the stresses are equal, we can conclude that:
\[ \text{Strain}_A = \text{Strain}_B \]

### Step 6: Express strain in terms of change in length
- Strain is defined as the change in length divided by the original length:
\[ \text{Strain} = \frac{\Delta L}{L_0} \]
- For wire A:
\[ \text{Strain}_A = \frac{\Delta L_A}{2L} \]
- For wire B:
\[ \text{Strain}_B = \frac{\Delta L_B}{L} \]

### Step 7: Set the strains equal to each other
- Since \(\text{Strain}_A = \text{Strain}_B\):
\[ \frac{\Delta L_A}{2L} = \frac{\Delta L_B}{L} \]
- Cross-multiplying gives:
\[ \Delta L_A = 2 \Delta L_B \]
- This indicates that the change in length (extension) of wire A is twice that of wire B.

### Conclusion
Based on the analysis:
1. The extension of wire A will be twice that of wire B (True).
2. The extensions in wire A and B are equal (False).
3. The strain in wire A will be half that of wire B (False).
4. The strains in A and B are equal (True).

Thus, the correct options are 1 and 4.