Fig. shows a system of two concentric spheres of radii r₁ and r₂ kept at temperatures T₁ and T₂ respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to
 

We know that  $\mathrm{Q}=-\mathrm{KA}\left(\frac{\mathrm{dT}}{\mathrm{dr}}\right)$
or     $\mathrm{Q}=-\mathrm{K}4{\mathrm{\pi r}}^2\left(\frac{\mathrm{dT}}{\mathrm{dr}}\right)\left(\because \mathrm{A}=4{\mathrm{\pi r}}^2\right)$ 
       $\therefore \frac{\mathrm{dr}}{{\mathrm{r}}^2}=-\frac{4\mathrm{\pi K}}{\mathrm{Q}}\mathrm{dT}$
Integrating this expression within proper limits, we get 
          ${\int }_{{\mathrm{r}}_1}^{{\mathrm{I}}_2} \frac{\mathrm{dr}}{{\mathrm{r}}^2}=-\frac{4\mathrm{\pi K}}{\mathrm{Q}}{\int }_{{\mathrm{T}}_1}^{{\mathrm{T}}_2} \mathrm{dT}$
or   
  $\begin{array}{r}\frac{1}{{\mathrm{r}}_1}-\frac{1}{{\mathrm{r}}_2}=\frac{4\mathrm{\pi K}}{\mathrm{Q}}\left({\mathrm{T}}_1-{\mathrm{T}}_2\right)\\ \frac{r_2-r_1}{r_1{r}_2}=\frac{4\mathrm{\pi K}}{\mathrm{Q}}\left({\mathrm{T}}_1-{\mathrm{T}}_2\right)\end{array}$
or   
$\begin{array}{l}\mathrm{Q}=\frac{4\mathrm{\pi K}\left({\mathrm{T}}_1-{\mathrm{T}}_2\right)r_1{r}_2}{\left(r_2-{\mathrm{r}}_1\right)}\\ \propto \frac{r_1{r}_2}{\left({\mathrm{r}}_2-{\mathrm{r}}_1\right)}.\end{array}$