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Q.
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?
Heat conduction through $a$ rod is rate of change of heat $\left(\frac{\Delta Q}{\Delta t}\right)$.
$\therefore H=\frac{\Delta Q}{\Delta t}=K A\left(\frac{T_{1}-T_{2}}{l}\right) $
$\Rightarrow H \propto \frac{r^{2}}{l}\,\,\,\,...$ (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).