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

Answer: (B)

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
$\therefore \, \, \, \, \, \, 3\theta = 180^{\circ} \, or \, \, \theta=60^{\circ}$
NOTE
Rate of heat flow
$ \, \, \, \, \, (H) =\frac{Temperature \, difference \, (TD)}{ Thermal \, resistance \, (R}$
where, $ \, \, \, R=\frac{l}{KA}$
K = Thermal conductivity of the rod.
This is similar to the current flow through a resistance (R) where current (i) = Rate of flow of charge
$\frac{Potential \, difference \, (PD)}{Electrical \, resistance \, (R)}$
Here, R $=\frac{1}{\sigma A}$ where $\sigma$ = Electrical conductivity