Four wires of the same diameter are connected in turn between two points, maintained at a constant potential difference. Their resistivities are; $\rho $ and $L$ (wire $1$ )., $1.2\,\rho $ and $1.2\,L$ (wire $2$ ), $0.9\,\rho $ and $0.9\,L$ (wire $3$ ) and $\rho $ and $1.5\,L$ (wire $4$ ). Rank the wires according to the rates at which energy is dissipated as heat, greatest first
AIIMS 2010, Medium
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$\mathrm{R}_{1}=\frac{\rho \mathrm{L}}{\mathrm{A}}$
$\mathrm{R}_{2}=\frac{(1.2 \rho)(1.2 \mathrm{L})}{\mathrm{A}}=\frac{1.44 \rho \mathrm{L}}{\mathrm{A}}$
$\mathrm{R}_{3}=\frac{(0.9 \mathrm{\rho })(0.9 \mathrm{L})}{\mathrm{A}}=\frac{0.81 \rho \mathrm{L}}{\mathrm{A}}$
$\mathrm{R}_{4}=\frac{\rho(1.5 \mathrm{L})}{\mathrm{A}}=\frac{1.5 \mathrm{\rho L}}{\mathrm{A}}$
$\therefore $ $\mathrm{R}_{3}<\mathrm{R}_{1}<\mathrm{R}_{2}<\mathrm{R}_{4}$
For constant potential, $\mathrm{P} \propto \frac{1}{\mathrm{R}}$
$\therefore P_{3} < P_{1} < P_{2} < P_{4}$
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