# Three rods made of the same material and having: Heat Conduction

Problem: 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°C and 90°C respectively. The temperature of junction of the three rods will be?  (IIT JEE 2001)

1. 45°C
2. 60°C
3. 30°C
4. 20°C

Answer: The answer is (B) i.e., temperture of junction of the three rods is 60°C.

Solution: Let T be the temperature of the junction. The rate of heat flow to the junction is \begin{align} \frac{\mathrm{d}Q_\text{in}}{\mathrm{d}t}&=\kappa A \frac{(90-T)}{l}+\kappa A \frac{(90-T)}{l}\nonumber\\ &=2\kappa A \frac{(90-T)}{l}, \end{align} and the rate of heat flow out of the junction is \begin{align} \frac{\mathrm{d}Q_\text{out}}{\mathrm{d}t}=\kappa A \frac{(T-0)}{l}=\frac{\kappa AT}{l}. \end{align} In steady state, the rate of inflow of heat at the junction is equal to the rate of outflow of heat i.e., \begin{align} \frac{\mathrm{d}Q_\text{in}}{\mathrm{d}t}=\frac{\mathrm{d}Q_\text{out}}{\mathrm{d}t}. \end{align} Solve above equations to get temperature of junction of the three rods T = 60°C.