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)

Heat conduction through $a$ rod is rate of change of heat $\left(\frac{\Delta Q}{\Delta t}\right)$.
$\therefore H=\frac{\Delta Q}{\Delta t}=KA\left(\frac{T_1-T_2}{l}\right)$
$\Rightarrow H\propto \frac{r^2}{l}...$ (i)
(a) When $r=2r_0,l=2l_0$
$H\propto \frac{{\left(2r_0\right)}^2}{2l_0}$
$\Rightarrow H\propto \frac{2r_0^2}{l_0}$
(b) When $r=2r_0,l=l_0$
$H\propto \frac{{\left(2r_0\right)}^2}{l_0}$
$\Rightarrow H\propto \frac{4r_0^2}{l_0}$
(c) When $r=r_0,l=l_0$
$H\propto \frac{r_0^2}{l_0}$
(d) When $r=r_0,l=2l_0$
$H\propto \frac{r_0^2}{2l_0}$
It is obvious that heat conduction will be more in, case (b).