Which of the four resistances \(P, Q, R\) and \(S\) generate the greatest amount of heat when a current flows from \(A\) to \(B\) ?

\(\mathrm{R}_{2}=\mathrm{R}+\mathrm{S}=1 \,\Omega+2 \,\Omega=3 \,\Omega\)

\(\mathrm{I}_{1} \mathrm{R}_{1}=\mathrm{I}_{2} \mathrm{R}_{2}\)

\(\mathrm{I}_{1}=\frac{\mathrm{R}_{2}}{\mathrm{R}_{1}} \mathrm{I}_{2}=\frac{3}{6} \mathrm{I}_{2}=\frac{\mathrm{I}_{2}}{2}\)

or \(\mathrm{I}_{2}=2 \mathrm{I}_{1}\)

Heat flow \(\mathrm{H}=\mathrm{I}^{2} \mathrm{Rt}\)

For \(Q, H_{Q}=I_{1}^{2} Q t=\frac{I_{2}^{2}}{4} \times 4 t=I_{2}^{2} t\)

For \(\mathrm{S}, \mathrm{H}_{\mathrm{S}}=\mathrm{I}_{2}^{2} \mathrm{St}=\mathrm{I}_{2}^{2} \cdot 2 \mathrm{t}=2 \mathrm{I}_{2}^{2} \mathrm{t}\)

\(\therefore \) Greatest amount of heat generated by \(S\).