Question

A ring consisting of two parts $ADB$ and $ACB$ of same conductivity $K$ carries an amount of heat $H$ . The $ADB$ part is now replaced with another metal keeping the temperatures $T_{1}$ and $T_{2}$ constant. The heat carried increases to $2H$ . What should be the conductivity of the new $ADB$ part? (Given $\frac{A C B}{A D B}=3$ )

• A. $\frac{7}{3}K$
• B. $2K$
• C. $\frac{5}{2}K$
• D. $3K$

Answer 1

Answer: (A)

Let $ADB=l$ and $ACB=3l$
$\frac{H_{A C B}}{H_{A D B}}=\frac{l_{A D B}}{l_{A C B}}=\frac{1}{3}$
$H_{A C B}+H_{A D B}=H$
$H_{A C B}=\frac{H}{4}$
$H_{A D B}=\frac{3 H}{4}=\frac{K A \left(T_{2} - T_{1}\right)}{l}$ ............(i)
Now the $ADB$ part is now replaced with another metal, and total heat is $2H$ .
$H_{A C B}=\frac{H}{4}$
$H_{A C B}+H′_{A D B}=2H$
In later case $H′_{A D B}=2H-H_{A C B}$
$H′_{A D B}=2H-\frac{H}{4}$
$H′_{A D B}=\frac{7 H}{4}=\frac{K_{1} A \left(T_{2} - T_{1}\right)}{l}$ ............(ii)
Comparing (i) and (ii)
$K_{1}=\frac{7 K}{3}$