question_answer

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?                             [AIPMT (S) 2005]

A)   \[r=2{{r}_{0}};\,l=2{{l}_{0}}\]

B)       \[r=2{{r}_{0}};\,l={{l}_{0}}\]

C)  \[r={{r}_{0}};\,l={{l}_{0}}\]

D)                   \[r={{r}_{0}};\,l=2{{l}_{0}}\]

Correct Answer: B

Solution :

Key Idea: 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 \[r=2{{r}_{0}};\,l=2{{l}_{0}}\]

\[H\propto \frac{{{(2{{r}_{0}})}^{2}}}{2{{l}_{0}}}\]

\[\Rightarrow \]   \[H\propto \frac{2r_{0}^{2}}{{{l}_{0}}}\]

[b] When \[r=2{{r}_{0}};\,l={{l}_{0}}\]

\[H\propto \frac{2{{({{r}_{0}})}^{2}}}{{{l}_{0}}}\]

\[\Rightarrow \]   \[H\propto \frac{4r_{0}^{2}}{{{l}_{0}}}\]

[c] When \[r={{r}_{0}};\,l={{l}_{0}}\]

\[H\propto \frac{r_{0}^{2}}{{{l}_{0}}}\]

[d] When \[r={{r}_{0}};\,l=2{{l}_{0}}\]

\[H\propto \frac{r_{0}^{2}}{2{{l}_{0}}}\]

It is obvious that heat conduction will be more is case [b].

Note:    It is fact that the temperature of whole rod does not come equal when heat is being continuously supplied due to the reason that temperature difference in the rod for the heat flow is same as we require a potential difference across a resistance for the current flow through it.