The Poisson’s ratio of a material is 0.5. If a force is applied to a wire of this material, there is a decrease in cross-sectional area by 4%. The percentage increase in length is-A. 1%B. 2%C. 2.5%D. 4%

To solve this question, we will use the relationship between Poisson's ratio, the change in cross-sectional area, and the change in length when a material is subjected to stress.
Step 1: Understanding Poisson’s Ratio
Poisson’s ratio ν\nu is the ratio of the lateral strain (change in width or cross-sectional area) to the longitudinal strain (change in length) for a material under uniform stress. Mathematically:
ν=−Lateral StrainLongitudinal Strain.\nu = -\frac{\text{Lateral Strain}}{\text{Longitudinal Strain}}.
For a wire, lateral strain refers to the change in the cross-sectional area or radius, and longitudinal strain refers to the change in length.
Step 2: Given Information
• Poisson’s ratio ν=0.5\nu = 0.5.
• The decrease in cross-sectional area is 4%, which means the lateral strain is -4% (a decrease).
We need to find the percentage increase in length, which corresponds to the longitudinal strain.
Step 3: Relating Lateral Strain and Longitudinal Strain
The change in cross-sectional area (ΔA\Delta A) is related to the lateral strain. For small changes, the lateral strain ϵlateral\epsilon_{\text{lateral}} can be approximated as the change in area divided by the original area. If the original area is A0A_0, the percentage change in area ΔA\Delta A is:
ϵlateral=ΔAA0=−4%.\epsilon_{\text{lateral}} = \frac{\Delta A}{A_0} = -4\%.
Poisson's ratio connects the lateral strain with the longitudinal strain, and we can write:
ν=−ϵlateralϵlongitudinal.\nu = -\frac{\epsilon_{\text{lateral}}}{\epsilon_{\text{longitudinal}}}.
Substituting the given Poisson’s ratio ν=0.5\nu = 0.5 and ϵlateral=−0.04\epsilon_{\text{lateral}} = -0.04 (since -4% is -0.04):
0.5=0.04ϵlongitudinal.0.5 = \frac{0.04}{\epsilon_{\text{longitudinal}}}.
Step 4: Solving for Longitudinal Strain
Now, solve for the longitudinal strain ϵlongitudinal\epsilon_{\text{longitudinal}}:
ϵlongitudinal=0.040.5=0.08.\epsilon_{\text{longitudinal}} = \frac{0.04}{0.5} = 0.08.
This corresponds to an 8% increase in length (since longitudinal strain is the percentage change in length).
Step 5: Conclusion
The percentage increase in length is 8%. However, based on the options given, none of the options match this exactly. If we consider that there might be some rounding in the choices or slight error in the problem's formulation, it's important to check if the options reflect the actual calculation or approximations. Based on the setup, the result is about 8%, which does not match the options provided.