Heat and Thermodynamics 3 Question 22
22. Three rods of identical cross-sectional area and made from the same metal form the sides of an isosceles triangle $A B C$, right angled at $B$. The points $A$ and $B$ are maintained at temperatures $T$ and $(\sqrt{2}) T$ respectively. In the steady state, the temperature of the point $C$ is $T _c$. Assuming that only heat conduction takes place, $T _c / T$ is
$(1995,2$ M)
(a) $\frac{1}{2(\sqrt{2}-1)}$
(b) $\frac{3}{\sqrt{2}+1}$
(c) $\frac{1}{\sqrt{3}(\sqrt{2}-1)}$
(d) $\frac{1}{(\sqrt{2}+1)}$
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Solution:
- The diagramatic representation of the given problem is shown in figure. Since, $T _B>T _A$ the heat will flow from $B$ to $A$. Similarly, heat will also flow from $B$ to $C$ and $C$ to $A$.
Applying the conduction formula
$$ \begin{gathered} \frac{\Delta Q}{\Delta t}=\frac{K A}{l}(\Delta T) \\ \left(\frac{\Delta T}{\sqrt{2} a}\right) _{C A}=\left(\frac{\Delta T}{a}\right) _{B C} \Rightarrow \frac{T _C-T}{\sqrt{2} a}=\frac{\sqrt{2} T-T _C}{a} \\ 3 T=T _C(\sqrt{2}+1) \Rightarrow \frac{T _C}{T}=\frac{3}{(\sqrt{2}+1)} \end{gathered} $$
