Question

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 $\frac{1}{K}=\frac{1}{K_1}+\frac{1}{K_2}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

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
$\frac{1}{K_R}=\frac{1}{K_1}+K_2;If{K}_1=K_2=K$
$\frac{1}{K_R}=\frac{1}{K}+\frac{1}{K}=\frac{2}{K}\Rightarrow K_R=\frac{K}{2}$
Which is less than K,
If $K_1>K_2$ suppose $K_1=K_2+x$
$\frac{1}{K}=\frac{1}{K_1}+\frac{1}{K_2}=\frac{K_2+K_1}{K_1{K}_2}$
$\Rightarrow \frac{1}{K}=\frac{K_2+K_2+x}{(K_2+X)K_2}\Rightarrow K=\frac{K_2^2+K_{2}x}{2K_2+x}$
Now, $K_2-K=K_2-\frac{K_2^2+K_{2}x}{2K_2+x}$
$=\frac{2K_2^2+K_{2}x-K_2^2-K_{2}x}{(2K_2+x)}=\frac{K_2^2}{2K_2+x}$ = possitive
So, K_2 > K, so the value of K is smaller than $K_2$ and $K_1$

OR

If the term $\Delta T$ is compared with potential and power radiated with current then the resistance of each will be $\frac{l}{kA}$

When in series and $l,A$ are equal then

$\frac{1}{k_{eq}}=\frac{1}{k_2}+\frac{1}{k_2}$

Here $\frac{1}{k_{eq}}$ is greater than both $\frac{1}{k_1}$ and $\frac{1}{k_2}$

So $k_{eq}$ is less than $k_1\&k_2$