Question

Four rods of same material but different radii $r$ and length $l$ are used to connect two reservoirs of heat at different temperatures. Which one will conduct heat fastest?

• A. $r=2\text{cm},l=0.5\text{m}$
• B. $r=1\text{cm},l=0.5\text{m}$
• C. $r=2\text{cm},l=2\text{m}$
• D. $r=1\text{cm},l=1\text{m}$

Answer 1

The correct option is A $r=2\text{cm},l=0.5\text{m}$

We know that
Rate of heat transfer $q=\left(\frac{dQ}{dt}\right)=-KA\frac{dT}{dx}$
$\Rightarrow \frac{\Delta Q}{\Delta t}=K\left(\pi r^2\right)\frac{\Delta T}{l}$
[Taking magnitude into the consideration]
$\therefore \frac{\Delta Q}{\Delta t}\propto \frac{r^2}{l}$
The option for which $r$ value is highest and $l$ value is least must be the correct option. Hence option (a) is correct.