Question

Which of the following circular rods, (given radius $r$ and length $l$ each made of the same material and whose ends are maintained at the same temperature will conduct most heat?

• A. $r=2r_{0};\,\,l=2l_{0}$
• B. $r=2r_{0};l=l_{0}$
• C. $r=r_{0};l=l_{0}$
• D. $r=r_{0};l=2l_{0}$

Answer 1

Answer: (B)

$r=2r_{0};l=l_{0}$

Answer 2

Heat conduction through a rod is given by
$H=\frac{\Delta Q}{\Delta t}=K A\left(\frac{T_{1}-T_{2}}{l}\right)$
$\Rightarrow H \propto \frac{r^{2}}{l} \ldots$(i)
(a) When $r=2 r_{0} ; l=2 l_{0}$
$H \propto \frac{\left(2 r_{0}\right)^{2}}{2 l_{0}}$
$\Rightarrow H \propto \frac{2 r_{0}^{2}}{l_{0}}$
(b) When $r=2 r_{0} ; l=l_{0}$
$H \propto \frac{\left(2 r_{0}\right)^{2}}{l_{0}}$
$\Rightarrow H \propto \frac{4 r_{0}^{2}}{l_{0}}$
(c) When $r=r_{0} ; l=l_{0}$
$H \propto \frac{r_{0}^{2}}{l_{0}}$
(d) When $r=r_{0} ; l=2 l_{0}$
$H \propto \frac{r_{0}^{2}}{2 l_{0}}$
It is obvious that heat conduction will be more in case (b).