To tackle this question, we need to apply some principles of thermodynamics and black body radiation. The scenario involves a small, spherical black body being heated by sunlight focused through a lens. Let's break down the problem step by step to determine which statements are correct.
Understanding Black Body Radiation
A black body is an idealized physical object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. When it comes to temperature, the maximum temperature that a black body can reach when illuminated by sunlight can be estimated using the Stefan-Boltzmann law, which states that the power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature.
Calculating the Power Received
The power received by the small sphere from the sunlight can be calculated using the solar constant, which is approximately 1361 W/m² at the Earth's surface. The lens focuses sunlight onto the sphere, increasing the intensity of the radiation it receives. The area of the lens and the distance from the Sun will influence how much power is focused onto the sphere.
Temperature Estimation for the Sphere
Using the Stefan-Boltzmann law, we can express the power absorbed by the sphere as:
- P_absorbed = A * σ * T_s^4
Where:
- P_absorbed is the power absorbed by the sphere.
- A is the surface area of the sphere.
- σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m²K⁴).
- T_s is the temperature of the sphere in Kelvin.
Considering the Lens Effect
Since the lens has a focal length that is double its diameter, it effectively concentrates sunlight onto the sphere. This means that the power received by the sphere will be significantly higher than if it were just exposed to direct sunlight. The exact increase in temperature will depend on the radius of the sphere and the lens's diameter.
Evaluating the Statements
Now, let's evaluate the provided statements:
- (a) The maximum temperature reached by the small sphere is nearly equal to 2121 K.
- (b) The result depends on the radius of the small sphere.
- (c) If a flat thin black disk of the same radius as that of the small sphere is used, its maximum temperature would be nearly 2522 K.
- (d) If a flat thin black disk of the same radius as that of the small sphere is used, its maximum temperature would be nearly 2682 K.
Analyzing Each Statement
Statement (a) suggests a maximum temperature of 2121 K. This value seems low given the concentration of sunlight through the lens. Therefore, this statement is likely incorrect.
Statement (b) indicates that the result depends on the radius of the small sphere. This is true because the surface area and the amount of power absorbed will vary with the size of the sphere, making this statement correct.
Statements (c) and (d) both propose maximum temperatures for a flat thin black disk. The temperature of a flat disk will generally be lower than that of a sphere due to differences in geometry and heat distribution. Without precise calculations, it’s challenging to determine which of these is correct, but typically, the temperature for a flat disk would be lower than that of a sphere under the same conditions.
Final Thoughts
In conclusion, the most accurate statements based on our analysis are:
- (b) The result depends on the radius of the small sphere.
Statements (a), (c), and (d) are likely incorrect based on the principles of thermodynamics and black body radiation. To arrive at precise temperature values, further calculations involving the specific dimensions of the lens and sphere would be necessary.
