Q. A cylinder of radius $R$ made of a material of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$ made of material of thermal conductivity $K_{2}$. The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is
A) $K_{1} + K_{2}$
B) $\frac{K_{1}K_{2}}{K_{1} + K_{2}}$
C) $\frac{K_{1} + 3K_{2}}{4}$
D) $\frac{3K_{1} + K_{2}}{4}$
Explanation
(c) Both the cylinders are in parallel, for the heat flow from one end as shown
Hence $K_{eq} = \frac{K_{1}A_{1} + K_{2}A_{2}}{A_{1} + A_{2}}$
Where $A_{1} =$ Area of cross-section of inner cylinder $\propto \pi R^{2}$ and $A_{2} =$ Area of cross-section of cylindrical shell
$= \pi\left\{ (2R)^{2} - (R)^{2} \right\} = 3\pi R^{2}$
$\Rightarrow K_{eq} = \frac{K_{1}(\pi R^{2}) + K_{2}(3\pi R^{2})}{\pi R^{2} + 3\pi R^{2}} = \frac{K_{1} + 3K_{2}}{4}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.