• # question_answer Two wires 'A' and 'B' of the same material have their lengths in the ratio 1 : 2 and radii in the ratio 2 : 1. The two wires are connected in parallel across a battery. The ratio of the heat produced in ?A? to the heat produced in ?B? for the same time is                                                                     [MNR 1998] A) $1:2$                                     B) $2:1$C) $1:8$                                     D)  $8:1$

${{R}_{1}}=\rho \frac{{{l}_{1}}}{{{A}_{1}}}$ and ${{R}_{2}}=\rho \frac{{{l}_{2}}}{{{A}_{2}}}$Þ $\,\frac{{{R}_{1}}}{{{R}_{2}}}=\frac{{{l}_{1}}}{{{l}_{2}}}.\frac{{{A}_{2}}}{{{A}_{1}}}=\frac{{{l}_{1}}}{{{l}_{2}}}{{\left( \frac{{{r}_{2}}}{{{r}_{1}}} \right)}^{2}}$                    Given $\frac{{{l}_{1}}}{{{l}_{2}}}=\frac{1}{2}$ and $\frac{{{r}_{1}}}{{{r}_{2}}}=\frac{2}{1}$ or $\frac{{{r}_{2}}}{{{r}_{1}}}=\frac{1}{2}$Þ $\frac{{{R}_{1}}}{{{R}_{2}}}=\frac{1}{8}$            $\therefore$ Ratio of heats $\frac{{{H}_{1}}}{{{H}_{2}}}=\frac{{{V}^{2}}/{{R}_{1}}}{{{V}^{2}}/{{R}_{2}}}=\frac{{{R}_{2}}}{{{R}_{1}}}=\frac{8}{1}$