A uniform wire (Young's modulus \(2 \times 10^{11}\, Nm^{-2}\) ) is subjected to longitudinal tensile stress of \(5 \times 10^7\,Nm^{-2}\) . If the over all volume change in the wire is \(0.02\%,\) the fractional decrease in the radius of the wire is close to

\(Stress\left( {\frac{F}{A}} \right) = 5 \times {10^7}N{m^{ - 2}}\)

\(\Delta V = 0.02\%  = 2 \times {10^{ - 4}}{m^3}\)

\(\frac{{\Delta r}}{r} = ?\)

\(\gamma  = \frac{{stress}}{{strain}} \Rightarrow strain\left( {\frac{{\Delta \ell }}{{{\ell _0}}}} \right) = \frac{\gamma }{{stress}}\,\,...\left( i \right)\)

\(\Delta V = 2\pi {\ell _0}\Delta r - \pi {r^2}\Delta \ell \)                       \(...\left( {ii} \right)\)

From eqns \((i)\) and \((ii)\) putting the value of 

\(\Delta \ell ,{\ell _0}\,and\,\Delta V\,and\,solving\,we\,get\)

\(\frac{{\Delta r}}{r} = 0.25 \times {10^{ - 4}}\)