Question

Two rectangular blocks, having identical dimensions, can be arranged either in configuration I or in configuration II as shown in the figure. One of the blocks has thermal conductivity $k$ and the other $2 k$. The temperature difference between the ends along the $x$-axis is the same in both the configurations. It takes $9$ s to transport a certain amount of heat from the hot end to the cold end in the configuration $I$. The time to transport the same amount of heat in the configuration $II$ is :

• A. 2.0 s
• B. 3.0 s
• C. 4.5 s
• D. 6.0 s

Answer 1

Answer: (A)

$R _{1}=\frac{ L }{ \kappa A }+\frac{ L }{2 \kappa A }=\frac{3 L }{2 \kappa A }$
$\frac{1}{ R _{2}}=\frac{1}{\left(\frac{ L }{ K A }\right)}+\frac{1}{\left(\frac{ L }{2 \kappa A }\right)}=\frac{3 \kappa A }{ L }$
$R _{2}=\frac{ L }{3 \kappa A }$
$\Delta Q _{1}=\Delta Q _{2}$
$\frac{\Delta T }{ R _{1}} t _{1}=\frac{\Delta T }{ R _{2}} t _{2}$
$\Rightarrow t _{2}=\frac{ R _{2}}{ R _{1}} t _{1} $
$=2 \,\sec$