Three rods of same dimensions are arranged as shown in figure they have thermal conductivities \[{{K}_{1}},{{K}_{2}}\]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:
A) \[{{K}_{3}}=\frac{1}{2}({{K}_{1}}+{{K}_{2}})\]
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 relation for the heat flow is given by \[\frac{Q}{t}=\frac{KA({{Q}_{2}}-{{Q}_{1}})}{l}\] Now equivalent conductivity of PRQ is given by \[=\frac{l+l}{\frac{l}{{{K}_{1}}}+\frac{l}{{{K}_{2}}}}=\frac{2{{K}_{1}}{{K}_{2}}}{{{K}_{1}}+{{K}_{2}}}\] \[{{K}_{3}}=\frac{{{K}_{1}}{{K}_{2}}}{{{K}_{1}}+{{K}_{2}}}\]
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