question_answer

Three rods of identical area of cross-section and made from the same metal form the sides of an isosceles triangle \[ABC\], 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, \[\frac{{{T}_{C}}}{T}\] is equal to        [IIT 1995]

A)            \[\frac{1}{(\sqrt{2}+1)}\]

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

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

D)            \[\frac{1}{\sqrt{3}(\sqrt{2}-1)}\]

Correct Answer: B

Solution :

                   \[\because \] \[{{T}_{B}}>{{T}_{A}}\] Þ Heat will flow B to A via two paths (i) B to A (ii) and along BCA as shown.                    Rate of flow of heat in path BCA will be same i.e. \[{{\left( \frac{Q}{t} \right)}_{BC}}={{\left( \frac{Q}{t} \right)}_{CA}}\] \[\Rightarrow \frac{k(\sqrt{2}T-{{T}_{C}})A}{a}=\frac{k({{T}_{C}}-T)A}{\sqrt{2}a}\] \[\Rightarrow \frac{{{T}_{C}}}{T}=\frac{3}{1+\sqrt{2}}\]