A steel wire is stretched by `1 kg wt`. If the radius of the wire is doubled, its Young's modulus will

Options

A. remain unchanged
B. become half
C. become double
D. become four times

Explanation

To solve the problem regarding the effect of doubling the radius of a steel wire on its Young's modulus when stretched by a weight, we can follow these steps:

### Step-by-Step Solution:

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

2. **Define Stress and Strain**:
- **Stress (σ)** is given by:
\[ \sigma = \frac{F}{A} \]
where \( F \) is the force applied (in this case, the weight of 1 kg) and \( A \) is the cross-sectional area of the wire.
- **Strain (ε)** is defined as:
\[ \epsilon = \frac{\Delta L}{L_0} \]
where \( \Delta L \) is the change in length and \( L_0 \) is the original length of the wire.

3. **Effect of Doubling the Radius**:
- If the radius of the wire is doubled, the new radius \( r' = 2r \).
- The cross-sectional area \( A \) of the wire is given by:
\[ A = \pi r^2 \]
- If the radius is doubled, the new area \( A' \) becomes:
\[ A' = \pi (2r)^2 = \pi \cdot 4r^2 = 4A \]
- Thus, when the radius is doubled, the cross-sectional area increases by a factor of 4.

4. **Calculate New Stress**:
- The stress with the new area \( A' \) becomes:
\[ \sigma' = \frac{F}{A'} = \frac{F}{4A} = \frac{\sigma}{4} \]
- This shows that the stress is reduced to one-fourth of its original value.

5. **Young's Modulus Calculation**:
- Since Young's modulus is a property of the material and does not depend on the dimensions of the wire, it remains constant regardless of the changes in radius or area.
- Therefore, even though the stress changes, the Young's modulus \( Y \) remains unchanged.

### Conclusion:
The Young's modulus of the steel wire will remain unchanged when the radius is doubled.

### Final Answer:
**The Young's modulus will remain unchanged.**