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?

r = 2r_o ; l = 2 l_o

r = 2r_o ; l = l_o

r = r_o ; l = l_o

r = r_o ; l = l_o

Solution

r = 2r_o ; l = l_o

Heat conduction through a rod is given by

$H = \frac{∆ Q}{∆ t} = KA (\frac{T_{1} - T_{2}}{l}) \Rightarrow H \propto \frac{r^{2}}{l}$

(a) When r = 2r_o ; l = 2l_o

$H \propto \frac{{(2 r_{o})}^{2}}{2 l_{o}} \Rightarrow H \propto \frac{2 r_{o}^{2}}{l_{o}}$

(b) When r = 2r_o;l = l_o

$H \propto \frac{{(2 r_{o})}^{2}}{l_{o}} \Rightarrow H \propto \frac{4 r_{o}^{2}}{l_{o}}$

$H \propto \frac{r_{o}^{2}}{l_{o}}$

(d) When r = r_o ; l = 2l_o

$H \propto \frac{r_{o}^{2}}{2 l_{o}}$

It is obvious that heat conduction will be more in case (b).

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