Assertion :The equivalent thermal conductivity of two plates of same thickness in contact is less than the smaller value of thermal conductivity Reason: For two plates of equal thickness in contact the equivalent thermal conductivity is given by:
$\frac{1}{K} = {\frac{1}{K}}_{1} + {\frac{1}{K}}_{2}$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Both Assertion and Reason are incorrect

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
For equivalent thermal conductivity, the relation is
$\frac{1}{K_{R}} = \frac{1}{K_{1}} + \frac{1}{K_{2}}$ ; If $K_{1} = K_{2} = K$
$\frac{1}{K_{R}} = \frac{1}{K} + \frac{1}{K} = \frac{2}{K} \Rightarrow K_{R} = \frac{K}{2}$
Which is less than $K$
If $K_{1} > K_{2}$ suppose $K_{1} = K_{2} + x$
$\frac{1}{K} = \frac{1}{K_{1}} + \frac{1}{K_{2}} = \frac{K_{2} + K_{1}}{K_{1} K_{2}}$
$\Rightarrow$ $\frac{1}{K} = \frac{K_{2} + K_{2} + x}{(K_{2} + x) K_{2}}$ $\Rightarrow$ $K = \frac{{K_{2}}^{2} + K_{2} x}{2 K_{2} + x}$
Now $K_{2} - K = K_{2} - \frac{{K_{2}}^{2} + K_{2} x}{2 K_{2} + x}$
$= \frac{{K_{2}}^{2} + K_{2} x - {K_{2}}^{2} - K_{2} x}{2 K_{2} + x}$
$= \frac{{K_{2}}^{2}}{2 K_{2} + x} =$ Positive
So, $K_{2} > K$ , so the value of $K$ is smaller than $K_{2}$ and $K_{1}$

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Q. Assertion :The equivalent thermal conductivity of two plates of same thickness in series is less than the smaller value of thermal conductivity. Reason: For two plates of equal thickness in series the equivalent thermal conductivity is given by

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Assertion :The equivalent thermal conductivity of two plates of same thickness in contact is less than the smaller value of thermal conductivity Reason: For two plates of equal thickness in contact the equivalent thermal conductivity is given by:1K=1K1+1K2