Question

In the figure shown below, two rods are maintained at the same temperature difference between the ends. As a result, heat is flowing through two rods made of the same material. If the diameters of the rods are in the ratio $1:4$ and their lengths are in the ratio $3:2$, then find the ratio of the rates of flow of heat through them.

• A. $1:24$
• B. $1:8$
• C. $1:4$
• D. $1:6$

Answer 1

The correct option is A $1:24$

Given that,
$(\Delta T{)}_1=(\Delta T{)}_2$ and $k_1=k_2$
Ratio of diameters of rods $1$ and $2$, $\left(\frac{d_1}{d_2}\right)=\frac{1}{4}$
Ratio of lengths of rods $1$ and $2$, $\left(\frac{l_1}{l_2}\right)=\frac{3}{2}$
We know that,
Rate of flow of heat in rod $1$ and $2$,
$q_1=\frac{k_1{A}_1(T_1-T_2)}{l_1}$;
$q_2=\frac{k_2{A}_2(T_1-T_2)}{l_2}$
$\therefore \frac{q_1}{q_2}=\left(\frac{A_1}{A_2}\right)\times \left(\frac{l_2}{l_1}\right)$
$\Rightarrow \frac{q_1}{q_2}={\left(\frac{d_1}{d_2}\right)}^2\times \left(\frac{l_2}{l_1}\right)=\frac{1}{16}\times \frac{2}{3}=\frac{1}{24}$