If the ratio of lengths, radii and Young's moduli of steel and brass wires in the figure are \(a, b\) and \(c\) respectively, then the corresponding ratio of increase in their lengths is

\(\frac{{{\ell _s}}}{{{\ell _b}}} = a,\frac{{{r_s}}}{{{r_b}}} = b,\frac{{{y_s}}}{{{y_b}}} = c,\frac{{\Delta \ell s}}{{\Delta {\ell _b}}} = ?\)

\(As,y = \frac{{F\ell }}{{A\Delta \ell }} \Rightarrow \Delta \ell  = \frac{{F\ell }}{{Ay}}\)

\(\Delta {\ell _s} = \frac{{3mg{\ell _s}}}{{\pi _s^2.{y_s}}}\left[ {{F_s} = \left( {M + 2M} \right)g} \right]\)

\(\Delta {\ell _b} = \frac{{2Mg{\ell _b}}}{{\pi r_b^2{y_b}}}\left[ {{F_b} = 2Mg} \right]\)

\(\therefore \frac{{\Delta {\ell _s}}}{{\Delta {\ell _b}}} = \frac{{\frac{{3Mg{\ell _s}}}{{\pi r_s^2.{y_s}}}}}{{\frac{{2Mg.{\ell _b}}}{{\pi {b^2}.{y_b}}}}} = \frac{{3a}}{{2{b^2}C}}\)