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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
$r = r_0$; $l = 2l_0$
D
$r = r_0$; $l = l_0$
Detailed Solution
Rate of heat flow through a rod: $\dfrac{Q}{t} = \dfrac{KA\,\Delta T}{l} = \dfrac{\Delta T}{R_{heat}}$, where the thermal resistance is $R_{heat} = \dfrac{l}{KA} = \dfrac{l}{K\pi r^2}$
The material and the temperature difference are the same, so $\dfrac{Q}{t} \propto \dfrac{r^2}{l}$.
$r = 2r_0$, $l = 2l_0$: $\dfrac{4r_0^2}{2l_0} = 2\dfrac{r_0^2}{l_0}$
$r = 2r_0$, $l = l_0$: $\dfrac{4r_0^2}{l_0} = 4\dfrac{r_0^2}{l_0}$
$r = r_0$, $l = 2l_0$: $\dfrac{1}{2}\dfrac{r_0^2}{l_0}$
$r = r_0$, $l = l_0$: $\dfrac{r_0^2}{l_0}$
So the rod with $r = 2r_0$ and $l = l_0$ conducts the most heat.
The material and the temperature difference are the same, so $\dfrac{Q}{t} \propto \dfrac{r^2}{l}$.
$r = 2r_0$, $l = 2l_0$: $\dfrac{4r_0^2}{2l_0} = 2\dfrac{r_0^2}{l_0}$
$r = 2r_0$, $l = l_0$: $\dfrac{4r_0^2}{l_0} = 4\dfrac{r_0^2}{l_0}$
$r = r_0$, $l = 2l_0$: $\dfrac{1}{2}\dfrac{r_0^2}{l_0}$
$r = r_0$, $l = l_0$: $\dfrac{r_0^2}{l_0}$
So the rod with $r = 2r_0$ and $l = l_0$ conducts the most heat.