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A cylinder of radius R of thermal conductivity K₁ is surrounded by a cylindrical shell of inner radius R and outer radius 2R made of thermal conductivity K₂. The two ends of the whole system are maintained at two different temperatures T₁ and T₂ (T₁ < T₂). The system is in steady state and there is no loss of heat across cylindrical surface. The electrical thermal conductivity of the system is

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$K = K_{1} + K_{2}$

$K = \frac{3 K_{1} + K_{2}}{4}$

$K = \frac{K_{1} + 3 K_{2}}{4}$

$K = \frac{K_{1} K_{2}}{K_{1} + K_{2}}$

answer is C.

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Detailed Solution

At steady state, $\frac{d Q}{d t} = \frac{d Q_{1}}{d t} + \frac{d Q_{2}}{d t}$

or $\frac{K (4 π R^{2}) (T_{2} - T_{1})}{L} = \frac{K_{1} (π R^{2}) (T_{2} - T_{1})}{L} + \frac{K_{2} (3 π R^{2}) (T_{2} - T_{1})}{L}$

or $4 K = K_{1} + 3 K_{2} \Rightarrow K = \frac{K_{1} + 3 K_{2}}{4}$

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