If K₁ and K₂ are the thermal conductivities, L₁ and L₂ are the lengths and A₁ and A₂ are the cross sectional areas of steel and copper rods respectively such that K₂/K₁=9, A₁/A₂=2, L₁/L₂=2. Then, for the arrangement as shown in the figure, the value of temperature T of the steel - copper junction in the steady state will be:

Solution

Physics sir · 4 steps

In steady state the same heat current flows through steel and copper, so equate the two and solve for T.

1. Same heat current. The rods are insulated and in series, so no heat leaks out sideways. In steady state the heat per second leaving the steel equals the heat per second entering the copper. Steel runs from 450°C down to T, and copper runs from T down to 0°C.

   (K₁ A₁ (450 - T))/L₁ = (K₂ A₂ (T - 0))/L₂

2. Use the ratios. Take K₁ = K, A₂ = A, L₂ = L. Then K₂ = 9K, A₁ = 2A, L₁ = 2L. For steel, (K₁ A₁)/L₁ = (K × 2A)/2L = KA/L. For copper, (K₂ A₂)/L₂ = 9KA/L.
3. Solve for T. Cancel KA/L from both sides: 450 - T = 9T. So 10T = 450, which gives T = 45°C.
4. So. The junction sits at 45°C, option (c). It is close to the cold end because copper conducts 9 times better, so it needs only a small temperature drop to carry the same heat.

The idea to keep. Rods in series carry the same heat current. Each temperature drop is proportional to that rod's thermal resistance L/KA, so the better conductor takes the smaller drop.

Answer from the NTA final answer key. Solution by PrepWiser.