Two plates A and B have thermal conductivities $$84$$ W m$$^{-1}$$ K$$^{-1}$$ and $$126$$ W m$$^{-1}$$ K$$^{-1}$$ respectively. They have same surface area and same thickness. They are placed in contact along their surfaces. If the temperatures of the outer surfaces of A and B are kept at $$100°$$C and $$0°$$C respectively, then the temperature of the surface of contact in steady state is _____ °C.

Two plates A and B are in series (thermal contact). In steady state, the heat flow rate through both is equal.

Let $$T$$ be the temperature at the contact surface. Both plates have the same area $$A$$ and thickness $$L$$.

Heat flow: $$\frac{K_A A (100 - T)}{L} = \frac{K_B A (T - 0)}{L}$$

$$K_A(100 - T) = K_B \times T$$

$$84(100 - T) = 126T$$

$$8400 - 84T = 126T$$

$$8400 = 210T$$

$$T = \frac{8400}{210} = 40°\text{C}$$

The temperature at the contact surface is $$\mathbf{40}$$°C.