The steady-state temperature ( T ) along a composite rod made of two materials in series is plotted against the distance ( x ) from the hot end. The graph shows a piecewise linear drop such that at x = 0 , T = 100^ C ; at the junction x = L , T = 80^ C ; and at the cold end x = 3L , T = 20^ C . Assuming both materials have the same uniform cross-sectional area, what is the ratio of the thermal conductivity of the fir

Step-by-step solution

In steady state, the rate of heat flow (heat current) is constant throughout the composite rod. Let H₁ and H₂ be the heat currents in the first and second materials respectively. For the first material (from x = 0 to x = L ): Temperature difference T₁ = 100^ C - 80^ C = 20^ C Length L₁ = L Heat current H₁ = K₁ A (20) L For the second material (from x = L to x = 3L ): Temperature difference T₂ = 80^ C - 20^ C = 60^ C Length L₂ = 3L - L = 2L Heat current H₂ = K₂ A (60) 2L = K₂ A (30) L Since the rods are in series, H