a
(a) Consider a concentric spherical shell of radius \(r\) and thickness \(dr\) as shown in fig.
The radial rate of flow of heat through this shell in steady state will be \(H = \frac{{dQ}}{{dt}} = - KA\frac{{dT}}{{dr}} = - K\,(4\pi {r^2})\frac{{dT}}{{dr}}\)

==> \(\int_{\,{r_1}}^{\,{r_2}} {\frac{{dr}}{{{r^2}}} = - \frac{{4\pi K}}{H}\int_{\,{T_1}}^{{T_1}} {dT} } \)

Which on integration and simplification gives

\(H = \frac{{dQ}}{{dt}} = \frac{{4\pi K{r_1}{r_2}({T_1} - {T_2})}}{{{r_2} - {r_1}}}\)

==> \(\frac{{dQ}}{{dt}} \propto \frac{{{r_1}{r_2}}}{{({r_2} - {r_1})}}\)