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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 3R made of material of thermal conductivity $K_{2}$ .The ends of the combined system are maintained at two different temperature. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity is

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a

$K_{1} + K_{2}$

b

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

c

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

d

$\frac{K_{1} + 8 K_{2}}{9}$

answer is D.

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Detailed Solution

$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$

but $R = \frac{l}{K A}$

$∴ \frac{K A}{L} = \frac{K_{1} A_{1}}{L_{1}} + \frac{K_{2} A_{2}}{L_{2}}$

Here $l_{1} = l_{2} = l$ and

$A_{1} = π R^{2}, A_{2} = π (3 R)^{2} - π R^{2}$

$A = π (3 R)^{2}$

$∴ K = \frac{K_{1} + 8 K_{2}}{9}$

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A cylinder of radius R made of a material of thermal conductivity K1, is surrounded by a cylindrical shell of inner radius R and outer radius 3R made of material of thermal conductivity K2.The ends of the combined system are maintained at two different temperature. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity is