question_answer

Three rods of identical cross-sectional area and made from the same metal from the sides of an isosceles traingle ABC, 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  

A) \[\frac{1}{2\left( \sqrt{2}-1 \right)}\]     

B) \[\frac{3}{\sqrt{2}+1}\]

C) \[\frac{1}{\sqrt{3}\left( \sqrt{2}-1 \right)}\]

D)        \[\frac{1}{\sqrt{2}+1}\]

Correct Answer: B

Solution :

[b] Heat flow from B to A,A to C and C to B (for steady state t  condition, \[\Delta Q/\Delta t\]is same)      Where \[\frac{\Delta Q}{\Delta t}=\frac{k\,A\Delta T}{\ell }\] For sides AC and CB \[{{\left( \frac{\Delta T}{\sqrt{2}a} \right)}_{AC}}={{\left( \frac{\Delta T}{a} \right)}_{CB}}\] \[\Rightarrow \,\,\,\frac{T-{{T}_{C}}}{\sqrt{2}A}=\frac{{{T}_{C}}-\sqrt{2}T}{a}\]  \[\Rightarrow T-{{T}_{C}}=\sqrt{2}{{T}_{C}}-2T\] \[\Rightarrow 3T={{T}_{C}}(\sqrt{2}+1)\Rightarrow \frac{{{T}_{C}}}{T}=\frac{3}{\sqrt{2}+1}\]