Two identical conductors made of metals of thermal conductivities $K_{1}, K_{2}$ such that $(K_{1} < K_{2})$ are given
Statement (A): Under series connection, the effective thermal conductivity $(K_{s})$ is such that $K_{1} < K_{S} < K_{2}$
Statement (B) : Under parallel connection the effective thermal conductivity $(K_{p})$ is such that $K_{P} = \frac{K_{1} + K_{2}}{2}$

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A is false, B is true

A, B both are true

A,B both are false

A is true, B is false

answer is C.

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Detailed Solution

In series, thermal resistances add up.

$R_{s} = R_{1} + R_{2} \Rightarrow \frac{2 L}{K_{S} A} = \frac{L}{K_{1} A} + \frac{L}{K_{2} A} \Rightarrow K_{S} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}}$

Check if $K_{1} < K_{S} < K_{2}$ is true.

$\frac{2 K_{1} K_{2}}{K_{1} + K_{2}} > K_{1} \Rightarrow K_{2} > K_{1}$ , which is true

$\frac{2 K_{1} K_{2}}{K_{1} + K_{2}} < K_{2} \Rightarrow K_{1} < K_{2}$ , which is true

Hence $K_{1} < K_{S} < K_{2}$ is true

In parallel combination,

$R_{p} = \frac{R_{1} R_{2}}{R_{1} + R_{2}} \Rightarrow \frac{L}{K_{P} (2 A)} = \frac{\frac{L}{K_{1} A} \frac{L}{K_{2} A}}{\frac{L}{K_{1} A} + \frac{L}{K_{2} A}} \Rightarrow K_{P} = \frac{K_{1} + K_{2}}{2}$