Three heat-conducting rods of identical length and uniform cross-sectional area are joined at a common point to form a Y-shaped junction. The thermal conductivities of the three rods are K , 2K , and 3K . The free ends of the rods with conductivities K , 2K , and 3K are maintained at steady temperatures of 100^ C , 40^ C , and 0^ C respectively. Assuming no heat loss from the lateral surfaces, the steady-state temper

Step-by-step solution

Let the steady-state temperature of the central junction be T_J . According to Kirchhoff's junction rule for heat (conservation of energy), the net rate of heat flow into the junction must be zero in steady state. H = 0 H₁ + H₂ + H₃ = 0 Using the formula for heat current H = KA(T_ end - T_J) L , we can write: K A (100 - T_J) L + 2K A (40 - T_J) L + 3K A (0 - T_J) L = 0 Since A and L are identical for all rods, we can divide the entire equation by KA L : (100 - T_J) + 2(40 - T_J) + 3(0 - T_J) = 0 Expanding the terms