Question

A hollow sphere of inner radius $R$ and outer radius $2R$ is made of a material of thermal conductivity $K$. It is surrounded by another hollow sphere of inner radius $2R$ and outer radius $3R$ made of the same material of thermal conductivity $K$. The inside of the smaller sphere is maintained at $0^{\circ }\text{C}$ and the outside of the bigger sphere is at ${100}^{\circ }\text{C}$. The system is in steady state. Find the temperature of the interface.

• A. ${50}^{\circ }\text{C}$
• B. ${100}^{\circ }\text{C}$
• C. ${75}^{\circ }\text{C}$
• D. ${125}^{\circ }\text{C}$

Answer 1

The correct option is C ${75}^{\circ }\text{C}$

The situation is shown in the below figure. Let the temperature of the interface between the shells be $T^{\circ }\text{C}$.


Rate of heat flow in a spherical shell is given by
$H=\frac{4K\pi R_1{R}_2(T_1-T_2)}{R_2-R_1}$
Heat current flowing through both the shells are same. [Given, steady state]
$\therefore H_1=H_2$
$\Rightarrow \frac{4\pi K\left(3R\right)\left(2R\right)(100-T)}{3R-2R}=\frac{4\pi K\left(2R\right)\left(R\right)(T-0)}{2R-R}$
$\Rightarrow 6R(100-T)=2RT$
$\Rightarrow 4T=300$
$\Rightarrow T={75}^{\circ }\text{C}$
Thus, option (c) is the correct answer.