Question

Six identical rods are arranged as shown in the figure and the system is at steady state, the temperature of the junction B will be

• A. 200°C
• B. 140°C
• C. 80°C
• D. 20°C

Answer 1

Answer: (B)

140°C

Answer 2

The system consists of six identical rods, each with thermal resistance $R$. Rod AB connects junction A (at $200^\circ C$) to junction B. Rod CD connects junction C to junction D (at $20^\circ C$). A diamond-shaped structure connects B and C, formed by four identical rods. Due to symmetry, the diamond structure between B and C acts as a single rod of resistance $R$.

Thus, the system can be viewed as three identical resistances ($R_{AB}$, $R_{diamond, BC}$, and $R_{CD}$) connected in series. The total thermal resistance from A to D is $R_{total} = R + R + R = 3R$.

The heat flow rate $H$ from A to D is given by: $H = \frac{T_A - T_D}{R_{total}} = \frac{200^\circ C - 20^\circ C}{3R} = \frac{180^\circ C}{3R} = \frac{60^\circ C}{R}$

Now, we can find the temperature at junction B by considering the heat flow through rod AB: $H = \frac{T_A - T_B}{R_{AB}}$ $\frac{60^\circ C}{R} = \frac{200^\circ C - T_B}{R}$ $60^\circ C = 200^\circ C - T_B$ $T_B = 200^\circ C - 60^\circ C = 140^\circ C$