Consider a cylindrical shell of length $l$, internal and external radii $r_1$ and $r_2$, respectively. Let its inner surface be maintained at a steady temperature $ \theta_1$ and the outer surface at a steady temperature $ \theta_2 \: (\theta_2 < \theta_1).$

Consider an elementary cylindrical shell of thickness $dr$, at temperature difference $d \theta$

Considering temperature of the layer $ \theta$ at a distance $r$ from the axis,

$ \int_{r_1}^r \large\frac{dr}{r}$$ = - K \large\frac{2 \pi l}{t1}$$ \int_{\theta_1}^{\theta} \: d\theta$

$ \Rightarrow ln \bigg( \large\frac{r}{r_1} \bigg)$$ = K2 \pi l(\theta_1-\theta) / H$………….(i)

and $ \int_{r_1}^r \large\frac{dr}{r}$$ = - K \large\frac{2 \pi l}{H}$$ \int_{\theta_1}^{\theta_2} \: d\theta$

$ \Rightarrow ln \bigg(\large\frac{r_2}{r_1} \bigg)$$ = K2 \pi l(\theta_1-\theta_2)/H$………(ii)

Dividing (i) by (ii), $\large\frac{ln \bigg( \large\frac{r}{r_1} \bigg)}{ ln \bigg( \large\frac{r_2}{r_1} \bigg) }$$ = \large\frac{(\theta_1-\theta) }{ (\theta_1 – \theta_2)}$

$ \Rightarrow \theta = \theta_1 – (\theta_1 – \theta_2) \large\frac{ln \bigg( \large\frac{r}{r_1} \bigg)}{ ln\bigg( \large\frac{r_2}{r_1} \bigg)}$

Ans : (D)