Three rods of identical cross-section and lengths are made of three different materials of thermal conductivity K₁ , K₂ and K₃ , respecrtively. They are joined together at their ends to make a long rod (see figure). One end of the long rod is maintained at 100ᵒC and the other at 0ᵒC (see figure). If the joints of the rod are at 70ᵒC and 20ᵒC in steady state and there is no loss of energy from the surface of the rod, the correct relationship between K₁ , K₂ and K₃ is :

Solution

Physics sir · 3 steps

In series the same heat flows through every rod, so a bigger temperature drop means a smaller K.

1. Same heat current. In steady state no heat is stored at a joint and none leaks out, so the heat current H = (KA Δ T)/L is the same in all three rods. Here A and L are also the same, so K Δ T must be the same for each rod.

   K₁(100-70) = K₂(70-20) = K₃(20-0) ⇒ 30K₁ = 50K₂ = 20K₃

2. Pick a common value. Set each product equal to 300. Then K₁ = 10, K₂ = 6 and K₃ = 15, so K₁ : K₂ : K₃ = 10 : 6 : 15.
3. So. K₁ : K₃ = 10 : 15 = 2 : 3 and K₂ : K₃ = 6 : 15 = 2 : 5. That is option (a).

The idea to keep. For rods in series with the same area and length, K Δ T is constant. Each rod's K is inversely proportional to its temperature drop: the rod with the biggest drop conducts worst.

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