Question

A rectangular slab consists of two cubes of copper and brass of equal sides having thermal conductivities in the ratio \(4 : 1\). If the free face of brass is at \(0^\circ \text{C}\) and that of copper is at \(100^\circ \text{C}\), then the temperature of their interface is

• A. \(80^\circ \text{C}\)
• B. \(20^\circ \text{C}\)
• C. \(60^\circ \text{C}\)
• D. \(40^\circ \text{C}\)

Hint:

In steady-state heat conduction problems with different materials, use the equality of heat flow and ratio of thermal conductivities to find interface temperature.

Answer 1

The Correct Option is A

Step 1: Let the thermal conductivity of copper be \(K_C = 4K\), and that of brass be \(K_B = K\). Let \(T\) be the temperature at the interface. Since both cubes have equal cross-sectional area and equal length, and the system is in steady-state, the rate of heat flow through both materials is the same. \[ \frac{K_C (100 - T)}{L} = \frac{K_B (T - 0)}{L} \Rightarrow 4(100 - T) = T \Rightarrow 400 - 4T = T \Rightarrow 5T = 400 \Rightarrow T = 80^\circ \text{C} \] % Final Answer \[ \boxed{80^\circ \text{C}} \]