Three identical rods $A, B$ and $C$ are placed end to end. A temperature difference is maintained between the free ends of $A$ and $C$. The thermal conductivity of $B$ is thrice that of $C$ and half of that of $A$. The effective thermal conductivity of the system will be $\left(K_{A}\right.$ is the thermal conductivity of $\left.\operatorname{rod} A\right)$

• Correct answer
  A

  $\frac{1}{3} K_{A}$
• B

  $3 K_{A}$
• C

  $2 K_{A}$
• D

  $\frac{2}{3} K_{A}$

Solution

$\begin{array}{ll}\text { Given, } & K_{B}=K_{A} / 2, \\ \text { and } & K_{B}=3 K_{C} \\ \therefore & K_{C}=K_{A} / 6\end{array}$ Rods are in series form so $$ \begin{gathered} \frac{L}{K}=\frac{l_{1}}{K_{A}}+\frac{l_{2}}{K_{B}}+\frac{l_{3}}{K_{C}} \\ \frac{3 l}{K}=\frac{l}{K_{A}}+\frac{l}{K_{A} / 2}+\frac{l}{K_{A} / 6} \\ \frac{3 l}{K}=\frac{9 l}{K_{A}} \\ K=\frac{K_{A}}{3} \end{gathered} $$

Hence, the correct answer is option: 'A'.