The young's modulus of a wire of length (L) and radius (r ) is Y. If the length is reduced to` L/2` and radius `r/2` , then its young's modulus will be

Options

A. (a) `Y/2`
B. (b) `Y`
C. (c) `2Y`
D. (d) `4Y`

Answer

To solve the problem, we need to understand the concept of Young's modulus and how it relates to the dimensions of a wire.

### Step-by-Step Solution:

1. **Understanding Young's Modulus**:
Young's modulus (Y) is defined as the ratio of tensile stress to tensile strain. It is a measure of the stiffness of a material. Mathematically, it is given by:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L_0} \]
where:
- \( F \) = force applied,
- \( A \) = cross-sectional area,
- \( \Delta L \) = change in length,
- \( L_0 \) = original length.

2. **Material Property**:
Young's modulus is a property of the material itself and does not depend on the dimensions of the wire (length and radius). Therefore, even if we change the dimensions of the wire, the Young's modulus remains constant as long as the material is the same.

3. **Changing Dimensions**:
In the problem, the length of the wire is reduced from \( L \) to \( \frac{L}{2} \) and the radius is reduced from \( r \) to \( \frac{r}{2} \). However, since Young's modulus is a material property, it will not change due to these alterations.

4. **Conclusion**:
Since the Young's modulus of the wire remains unchanged despite the changes in length and radius, we conclude that the new Young's modulus after changing the dimensions will still be \( Y \).

Thus, the final answer is:
\[ \text{Young's modulus after changes} = Y \]

### Final Answer:
The Young's modulus will remain \( Y \).

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