To determine the temperature of the shell of the space station when it is surrounded by a thin spherical black screen, we can apply the principles of thermal radiation and equilibrium. The key concept here is that the shell and the black screen will reach a thermal equilibrium where they exchange energy through radiation.
Understanding Thermal Radiation
All objects emit thermal radiation based on their temperature. The Stefan-Boltzmann Law describes this relationship, stating that the power radiated by a black body per unit area is proportional to the fourth power of its absolute temperature. The formula is given by:
P = σT^4
where P is the power radiated, σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m²K^4), and T is the temperature in Kelvin.
Initial Conditions
In our scenario, the space station's shell is initially at a temperature of T = 500 K. When the black screen is introduced, it will absorb some of the radiation emitted by the shell. Since the black screen is nearly the same radius, we can assume it will also radiate energy back toward the shell.
Setting Up the Equilibrium
At thermal equilibrium, the power radiated by the shell must equal the power absorbed from the black screen. Let’s denote the temperature of the black screen as T_screen. The power radiated by the shell can be expressed as:
P_shell = σ(T_shell)^4 = σ(500 K)^4
For the black screen, which is a perfect black body, the power absorbed from the shell will be equal to the power it radiates back to the shell:
P_screen = σ(T_screen)^4
Equating Powers
At equilibrium, we have:
σ(500 K)^4 = σ(T_screen)^4
Since σ is a constant and appears on both sides, we can simplify this to:
(500 K)^4 = (T_screen)^4
Solving for the Temperature
Taking the fourth root of both sides gives us:
T_screen = 500 K
This indicates that the temperature of the black screen will also be 500 K. Therefore, the shell of the space station will maintain its temperature of 500 K even when enveloped by the black screen, as both objects reach thermal equilibrium at the same temperature.
Conclusion
In summary, when the space station's shell is surrounded by a thin black screen, both will stabilize at the same temperature of 500 K due to the principles of thermal radiation and equilibrium. This scenario illustrates how energy exchange works in a closed system, emphasizing the balance between emitted and absorbed radiation.
