The lengths and radii of two rods made of same materials are in the ratio of 1:2 and 2:3 respectively, if the temperature difference between the ends for the two rods be the same, then in steady state, the amount of heat flowing through them will be in the ratio

given $\frac{{\text{l}}_{\text{1}}}{{\text{l}}_{\text{2}}}\text{=}\frac{1}{2}$

K=constant= coefficient of thermal conductivity
$\frac{{\text{r}}_{\text{1}}}{{\text{r}}_{\text{2}}}\text{=}\frac{\text{2}}{\text{3}}$ ( here r is radius of the cross section)
$\text{Δθ}$=constant= temperature difference
$\frac{{\text{Q}}_{\text{1}}}{{\text{Q}}_{\text{2}}}\text{=?}$
$Power P=\frac{\text{Q}}{\text{t}}\text{=}\frac{{\text{KA(θ}}_{\text{1}}{\text{-θ}}_{\text{2}}\text{)}}{\text{l}}\text{=}\frac{KA∆\theta }{l}\text{=}\frac{K\pi r^2∆\theta }{l}$
$\frac{\text{Q}}{\text{t}}\text{α}\frac{{\text{r}}^{\text{2}}}{\text{l}}$
$\text{Pα}\frac{{\text{r}}^{\text{2}}}{\text{l}}$
$\frac{{\text{P}}_{\text{1}}}{{\text{P}}_{\text{2}}}\text{=}{\left(\frac{{\text{r}}_{\text{1}}}{{\text{r}}_{\text{2}}}\right)}^{\text{2}}\times \frac{{\text{l}}_2}{{\text{l}}_1}$
$\frac{P_1}{P_2}\text{=}{\left(\frac{2}{3}\right)}^2\times \frac{2}{1}$
$$\begin{gathered} \frac{P_1}{P_2}\text{=}\frac{4}{9}\times \frac{2}{1}=\frac{8}{9} \\ \frac{P_1}{P_2}=\frac{8}{9} is ratio of amount of heat flow per second \end{gathered}$$