Question

A metal rod of length 2 m has cross-sectional areas 2A and A as shown in figure. The two ends are maintained at temperatures ${100}^{\circ }C$ and ${70}^{\circ }C$. The temperature of middle point C is

• A. ${80}^{\circ }C$
• B. ${85}^{\circ }C$
• C. ${90}^{\circ }C$
• D. ${95}^{\circ }C$

Answer 1

The correct option is C ${90}^{\circ }C$

Here, $K_1=K_2,l_1=l_2=1m,A_1=2A_2=2A$
where $A_2=A$

Also $T_1={100}^{\circ }C$ and $T_2={70}^{\circ }C$

Using $\frac{\Delta Q}{\Delta t}=\frac{KA\left(\Delta T\right)}{l}$
Let temperature at C be $T$, then

$(\frac{\Delta Q}{\Delta t}{)}_{BC}=$ $(\frac{\Delta Q}{\Delta t}{)}_{CA}$

$\therefore$ $\frac{K2A(100-T)}{1}=\frac{KA(T-70)}{1}$

$200-2T=T-70$
$\Rightarrow T={90}^{\circ }C$