The shell of a space stationis a blackened sphere in which a temperature T=500k is maintained due to the operation of appliances of the station.the amount of heat given away from a unit surface area is proportional to the fourth power of thermodynamic temperature.determine the temperature T1 OF the shell if the station is enveloped by a thin spherical black screen of nearly the same radius as the radius of the shell.

To determine the temperature T1 of the shell when it is enveloped by a thin spherical black screen, we can apply the principles of thermal radiation and the Stefan-Boltzmann law. This law states that the power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature. Let's break this down step by step.

Understanding the Situation

We have a space station with a shell that maintains a temperature of T = 500 K. The shell radiates heat into space, and the amount of heat radiated per unit area is given by the Stefan-Boltzmann law:

Q = σT^4

where Q is the power radiated per unit area, σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m²K⁴), and T is the absolute temperature in Kelvin.

Introducing the Black Screen

Now, when we introduce a thin spherical black screen around the shell, we need to consider how this affects the thermal equilibrium of the system. The black screen will absorb some of the radiation emitted by the shell and will also radiate back towards the shell.

Setting Up the Equations

Let’s denote the temperature of the black screen as T1. The heat radiated by the shell can be expressed as:

Q_shell = σT^4

And the heat absorbed by the black screen from the shell is:

Q_screen = σT^4

Since the black screen is in thermal equilibrium with the shell, the power radiated by the black screen back to the shell must equal the power absorbed from the shell:

σT^4 = σT1^4

Finding the Temperature T1

Since σ is a constant and appears on both sides of the equation, we can simplify this to:

T^4 = T1^4

Now, substituting the known temperature of the shell (T = 500 K) into the equation gives us:

(500 K)^4 = T1^4

To find T1, we take the fourth root of both sides:

T1 = 500 K

Conclusion

Thus, the temperature T1 of the shell, when enveloped by the thin spherical black screen, remains at 500 K. This is because the black screen, being a perfect black body, does not change the thermal radiation balance established by the shell. In essence, the system reaches a new equilibrium where the temperatures remain equal due to the nature of black body radiation.