question_answer

Three rods of same dimensions are arranged as shown in figure they have thermal conductivities\[{{\operatorname{K}}_{1}},\,\,{{K}_{2}}\,\,and\,\,{{K}_{3}}\]. The points P and Q are maintained at different temperatures for the heat to flow at the same rate along PRQ and PQ then which of the following option is correct?  

A) \[{{K}_{3}}=\frac{1}{2}\left( {{K}_{1}}+{{K}_{2}} \right)\]     

B) \[{{K}_{3}}={{K}_{1}}+{{K}_{2}}\]

C) \[{{K}_{3}}=\frac{{{K}_{1}}{{K}_{2}}}{{{K}_{1}}+{{K}_{2}}}\]                

D) \[{{K}_{3}}=2({{K}_{1}}+{{K}_{2}})\]

Correct Answer: C

Solution :

The given arrangement of rods can be redrawn as follows It is given that \[{{\operatorname{H}}_{1}}={{H}_{2}}\] \[\Rightarrow \,\,\frac{KA({{\theta }_{1}}-{{\theta }_{2}})}{2l}=\frac{{{K}_{3}}A({{\theta }_{1}}-{{\theta }_{2}})}{l}\,\,\,\Rightarrow \,\,{{K}_{3}}=\frac{K}{2}=\frac{{{K}_{1}}{{K}_{2}}}{{{K}_{1}}+{{K}_{2}}}\]