Question

Three rods made of the same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at $0^{\circ }C$ and ${90}^{\circ }C$ respectively. The temperature of junction of the three rods will be

• A. ${45}^{\circ }C$
• B. ${60}^{\circ }C$
• C. ${30}^{\circ }C$
• D. ${20}^{\circ }C$

Answer 1

The correct option is B ${60}^{\circ }C$

Let $theta$ be the temperature of the junction (say B). Thermal resistance of all the three rods is equal. Rate of heat flow through AB + Rate of heat flow through CB = Rate of heat flow through BD
$\therefore \frac{{90}^{\circ }-\theta }{R}+\frac{{90}^{\circ }-\theta }{R}=\frac{\theta -0}{R}$

Here, R = thermal resistance
$\therefore 3\theta ={180}^{\circ }or\theta ={69}^{\circ }C$

NOTE
Rate of heat flow
$\left(H\right)=\frac{Temperaturedifference\left(TD\right)}{Thermalresistance\left(R\right)}$
Were, $R=\frac{1}{KA}$

K = thermal conductivity of the rod.
This is similar to the current flow through a resistance (R) where Current (i) = Rate of flow charge
$=\frac{Potentialdifference\left(PD\right)}{Electricalresistance\left(R\right)}$
Here, $R=\frac{1}{\sigma A}$ where $\sigma$ = Electrical conductivity.