Question

Two rods of the same material have diameters in the ratio $1:2$ and the lengths in the ratio of $2:1$. If the temperature difference between their ends is the same, then the ratio of heats conducted by them in a given time will be :

• A. 1:4
• B. 1:8
• C. 4:1
• D. 8:1

Answer 1

Answer: (B)

1:8

The heat flow is given by:
$\dot{Q}=KA\frac{dT}{dx}=KA\frac{\Delta T}{l}$
Given that ratio of diameters is $1:2$, in other words the ratio of their areas ($A\propto D^2$)
is $1:4$, and ratio of their lengths is $2:1$.
The dimensions of the rod can be assumed to be:

• Rod 1: length $2l$, area $A$
• Rod 2: length $l$, area $4A$

$\dot{Q_1}=KA\frac{\Delta T}{2l}$
$\dot{Q_2}=4KA\frac{\Delta T}{l}$
$\frac{\dot{Q_1}}{\dot{Q_2}}=\frac{1}{8}$
Given that both have the same temperature difference across each other. For same time interval (T) the ratio of heat transfer will be
$\frac{Q_1}{Q_2}=$ $\frac{\dot{Q_1}T}{\dot{Q_2}T}=\frac{1}{8}$