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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 : $\frac{1}{K} = \frac{1}{{{K_1}}} + \frac{1}{{{K_2}}}$
If both Assertion and Reason are correct and the Reason is a correct explanation of the Assertion.
If both Assertion and Reason are correct but Reason is not a correct explanation of the Assertion.
If the Assertion is correct but Reason is incorrect.
If both the Assertion and Reason are incorrect.
Solution
For equivalent thermal coductivity, 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}}{{\left( {{K_2} + x} \right){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{{2K_2^2 + {K_2}x – K_2^2 – {K_2}x}}{{\left( {2{K_2} + x} \right)}}$
$ = \frac{{K_2^2}}{{2{K_2} + x}} = positive$
$So.{K_2} > K,so\,the\,value\,of\,K\,is\,smaller\,than\,$
${K_2}\,and\,{K_1}.$
