Question

Three rods of same dimensions have thermal conductivities $3\text{K}$, $\text{2K}$ and $\text{K}$. They are arranged as shown in the figure, with their ends at ${100}^{\circ }\text{C},{50}^{\circ }\text{C}$ and $0^{\circ }\text{C}.$ The temperature of their junction point is:

• A. ${75}^{\circ }\text{C}$
• B. ${\frac{200}{3}}^{\circ }\text{C}$
• C. ${40}^{\circ }\text{C}$
• D. ${\frac{100}{3}}^{\circ }\text{C}$

Answer 1

The correct option is B ${\frac{200}{3}}^{\circ }\text{C}$

Let temperature of the junction be $\theta$. As there are no sources and sinks at the junction, we can say that:
Net heat entering the junction = Net heat leaving the junction
Assume the direction of heat flow as shown below:


Then,
$H_1=H_2+H_3$
$3K(100-\theta )=2K(\theta -50)+K(\theta -0)$
$\Rightarrow -300K+3K\theta -100K+2K\theta +K\theta =0$
$\Rightarrow 6\theta =400$
$\Rightarrow \theta =\frac{400}{6}={\frac{200}{3}}^{\circ }\text{C}$
Hence, our initial assumption is correct and option (b) is the correct answer.