question_answer

A large cylindrical rod of length L is made by joining two identical rods of copper and steel of length \[\left( \frac{L}{2} \right)\] each. The rods are completely insulated from the surroundings. If the free end of copper rod is maintained at \[100{}^\circ C\] and that of steel at \[0{}^\circ C\]then the temperature of junction is (Thermal conductivity of copper is 9 times that of steel)

A) \[90{}^\circ C~\]

B) \[50{}^\circ C\]

C) \[10{}^\circ C~\]

D) \[67{}^\circ C\]

Correct Answer: A

Solution :

Let conductivity of steel \[{{K}_{\operatorname{steel}}}=k\] then from question

Conductivity of copper\[{{K}_{\operatorname{copper}}}=9k\]

\[{{\theta }_{\operatorname{copper}}}=100{}^\circ \operatorname{C}\]

\[{{\theta }_{\operatorname{steel}}}=0{}^\circ \operatorname{C}\]

\[{{l}_{\operatorname{steel}}}={{l}_{copper}}=\frac{L}{2}\]

From formula temperature of junction;

\[\theta =\frac{{{K}_{\operatorname{copper}}}{{\theta }_{\operatorname{copper}}}{{l}_{\operatorname{steel}}}+{{K}_{\operatorname{steel}}}{{\theta }_{\operatorname{steel}}}{{l}_{\operatorname{copper}}}}{{{K}_{\operatorname{copper}}}{{l}_{\operatorname{steel}}}+{{K}_{\operatorname{steel}}}{{l}_{\operatorname{copper}}}}\]

\[=\frac{9k\times 100\times \frac{L}{2}+k\times 0\times \frac{L}{2}}{9k\times \frac{L}{2}+k\times \frac{L}{2}}\]

\[=\frac{\frac{900}{2}kL}{\frac{10kL}{2}}=90{}^\circ C\]