Two wires \(\mathrm{A}\) and \(\mathrm{B}\) having same length and material are stretched by the same force.
Their diameters are in the ratio \(1: 3\). The ratio of energy density of wire \(A\) to that of wire B when stretched, is

Strain energy per unit volume \(U=\frac{1}{2} \times\) Stress \(\times\) Strain From Hooke's law, Stress \(=\mathrm{Y} \times\) Strain
\(\Longrightarrow \mathrm{U}=\frac{1}{2 \mathrm{Y}} \times(\text { Stress })^{2}\)
where Stress \(=\frac{\mathrm{F}}{\mathrm{A}}=\frac{\mathrm{F}}{\pi \mathrm{d}^{2} / 4}\)
\(\Longrightarrow \mathrm{U} \propto \frac{1}{\mathrm{~d}^{4}}\)
Given \(: \frac{\mathrm{d}_{\mathrm{s}}}{\mathrm{d}_{\mathrm{l}}}=\frac{1}{3}\)
Thus ratio of strain energy per unit volume \(\frac{\mathrm{U}_{\mathrm{s}}}{\mathrm{U}_{1}}=\left(\frac{\mathrm{d}_{\mathrm{l}}}{\mathrm{d}_{\mathrm{s}}}\right)^{4}\) \(\Longrightarrow \frac{\mathrm{U}_{\mathrm{s}}}{\mathrm{U}_{1}}=\left(\frac{3}{1}\right)^{4}=81\)