Thermal Properties of Matter - Result Question 31
32. 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?
[2005]
(a) $r=r_0 ; l=l_0$
(c) $r=r_0 ; l=2 l_0$
(b) $r=2 r_0 ; l=l_0$
(d) $r=2 r_0 ; l=2 l_0$
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Answer:
Correct Answer: 32. (b)
Solution:
- (b) From given option
(i) $r=2 r_0, l=2 l_0$
$ \therefore R=\frac{2 \ell_0}{K \pi(2 r_0)^{2}}=\frac{\ell_0}{2 K \pi r_0^{2}} $
(ii) $r=2 r_0, l=l_0$
$ \therefore R=\frac{\ell_0}{K \pi(2 r_0)^{2}}=\frac{\ell_0}{4 K \pi r_0^{2}} $
(iii) $r=r_0, l=2 l_0$
$ \therefore R=\frac{2 \ell_0}{K \pi r_0^{2}}=\frac{2 \ell_0}{K \pi r_0^{2}} $
(iv) $r=r_0, l=l_0$
$ \therefore R=\frac{\ell_0}{K \pi r_0^{2}}=\frac{\ell_0}{K \pi r_0^{2}} $
It is clear that for option (b) resistance is minimum, hence, heat flow will be maximum.
(i) Rate of heat flow is directly proportional to area
(ii) inversely proportional to length.
$\therefore$ Heat flow will be maximum when $r$ is maximum and $\ell$ is minimum. We know that $Q=\frac{T_H-T_L}{R}$
Also, Thermal resistance $R=\frac{\ell}{K A}=\frac{\ell}{K \pi r^{2}}$
Heat flow will be maximum when thermal resistance is minimum.
