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 a 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

Let the length of the cylinder be $l$ and let its ends be maintained at temperatures ${\theta }_1$ and ${\theta }_2.$ 

Area of the cross-section of the inner cylinder $=\pi R^2.$ 

Area of cross – section of outer cylinder $=\pi {\left(2R\right)}^2-\pi R^2=3\pi R^2$

Rate of flow of heat across inner cylinder is $Q_2=\frac{k_2\left(3\pi R^2\right)({\theta }_1-{\theta }_2)}{l} ...........\left(i\right)$

Rate of flow of heat across the outer shell is $Q_2=\frac{k_2\left(3\pi R^2\right)({\theta }_1-{\theta }_2)}{l} ..........\left(ii\right)$

Let the effective thermal conductivity of the compound cylinder be k. The rate of flow of heat across the compound cylinder is $Q=\frac{k\left(4\pi R^2\right)({\theta }_1-{\theta }_2)}{l} ..........\left(iii\right)$

Now $Q=Q_1+Q_2 .......\left(iv\right)$

Using (i), (ii) and (iii) in (iv), we get $4k=k_1+3k_2$ or $k=\frac{k_1+3k_2}{4}$

Hence the correct choice is (c)