If the length of a wire is doubled, then its Young's modulus

Options

A. become double
B. become half
C. increases but becomes not exactly double
D. remains unchanged

Answer

To solve the question regarding the effect of doubling the length of a wire on its Young's modulus, we can follow these steps:

### Step-by-step Solution:

1. **Understanding Young's Modulus**:
Young's modulus (Y) is defined as the ratio of stress to strain in a material. Mathematically, it is expressed as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
where stress is defined as the force applied per unit area, and strain is the relative change in length.

2. **Defining Stress and Strain**:
- **Stress (σ)**:
\[ \sigma = \frac{F}{A} \]
where \( F \) is the force applied and \( A \) is the cross-sectional area of the wire.
- **Strain (ε)**:
\[ \epsilon = \frac{\Delta L}{L_0} \]
where \( \Delta L \) is the change in length and \( L_0 \) is the original length of the wire.

3. **Doubling the Length of the Wire**:
If the original length of the wire is \( L_0 \) and we double it, the new length \( L \) becomes:
\[ L = 2L_0 \]

4. **Effect on Stress and Strain**:
- When the length is doubled, if we apply the same force \( F \), the stress remains the same because it depends on the force and the cross-sectional area, which has not changed.
- The strain will change because the new length is \( 2L_0 \). If the wire stretches by \( \Delta L \), the new strain becomes:
\[ \epsilon' = \frac{\Delta L}{2L_0} \]
This means that for the same amount of stretch, the strain is halved.

5. **Calculating New Young's Modulus**:
Since stress remains the same and strain is halved, the Young's modulus can be recalculated as:
\[ Y' = \frac{\sigma}{\epsilon'} = \frac{F/A}{\Delta L/(2L_0)} = 2 \cdot \frac{F/A}{\Delta L/L_0} = 2Y \]
However, this calculation assumes a constant force and does not reflect the intrinsic property of the material.

6. **Conclusion**:
Young's modulus is an intrinsic property of the material and does not change with the dimensions of the wire. Therefore, even if the length of the wire is doubled, the Young's modulus remains unchanged.

### Final Answer:
The Young's modulus remains unchanged when the length of the wire is doubled.