Problem 116 - University Physics with Modern Physics

Question

The rate at which radiant energy from the sun reaches the earth's upper atmosphere is about 1.50 \(\mathrm{kW} / \mathrm{m}^{2}\) . The distance from the earth to the sun is \(1.50 \times 10^{11} \mathrm{m},\) and the radius of the sun is \(6.96 \times 10^{8} \mathrm{m}\) (a) What is the rate of radiation of energy per unit area from the sun's surface? (b) If the sun radiates as an ideal black-body, what is the temperature of its surface?

Step-by-Step Solution

Answer

(a) Calculate power per unit area on Sun's surface using sphere surface equation. (b) Use Stefan-Boltzmann law to derive Sun's surface temperature.

Step 1: Convert Given Values

Identify and convert the given values into usable forms. We have the solar constant, i.e., the intensity of radiation reaching the Earth's atmosphere as \( 1.50 \; \mathrm{kW/m^2} \), or \( 1500 \; \mathrm{W/m^2} \). The distance from the Earth to the Sun is \( r = 1.50 \times 10^{11} \; \mathrm{m} \). The radius of the Sun is \( R_s = 6.96 \times 10^8 \; \mathrm{m} \).

Step 2: Calculate Total Power Emitted by the Sun

Determine the total power \( P \) emitted by the Sun. The power emitted is spread over a sphere with radius equal to the distance between Earth and the Sun. The surface area of this sphere is \( 4\pi r^2 \). Thus, \( P = 1.50 \times 10^{3} \times 4 \pi (1.50 \times 10^{11})^2 \).

Step 3: Calculate Power per Unit Area on Sun's Surface

Find the power per unit area on the surface of the Sun. Use the formula for power per unit area: \( rac{P}{4\pi R_s^2} \). Substitute \( R_s = 6.96 \times 10^8 \; \mathrm{m} \) and the previously calculated \( P \).

Step 4: Apply Stefan-Boltzmann Law

Use the Stefan-Boltzmann Law to find the temperature of the Sun's surface. The law states \( P/A = \sigma T^4 \), where \( P/A \) is the power per unit area, \( \sigma = 5.67 \times 10^{-8} \; \mathrm{W/m^2K^4} \). Solve for \( T \) by rearranging the formula to \( T = \left( \frac{P/A}{\sigma} \right)^{1/4} \), and substituting \( P/A \) from the previous step.

Key Concepts

Solar Constant

The solar constant is a measure of the intensity of solar radiation per unit area reaching the Earth's upper atmosphere. This value is essential for understanding the earth's climate system and the balance of energy that drives it. The average solar constant is about 1,500 watts per square meter (W/m²).

Factors influencing the solar constant include:

• The Earth’s orbital distance, which is approximately 150 million kilometers from the Sun.
• The output variations of the solar energy from the Sun itself.
• Atmospheric conditions that may slightly alter the values received.

The solar constant also serves as a baseline for calculating the amount of solar power accessible for use by solar panels and other solar energy applications.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law defines the power radiated by a black body in terms of its temperature. This law is a cornerstone of studying radiation and energy transfer processes. In formula terms, it is expressed as:\[P/A = \sigma T^4\]where \(P/A\) is the power per unit area, \(\sigma\) is the Stefan-Boltzmann constant (approximately \(5.67 \times 10^{-8} \, \mathrm{W/m^2K^4}\)), and \(T\) is the absolute temperature in Kelvin.

This law illustrates that the energy emitted by a body increases with the fourth power of the temperature. It’s crucial in astrophysics and climate science, helping to estimate the effective temperature of celestial bodies like stars and planets.

• The law implies that as a body gets hotter, it radiates energy faster.
• It is primarily applicable to "black bodies", perfect emitters, and absorbers of radiation.

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 analyzing radiation, black bodies serve as a perfect model for studying the emission of energy. Black body radiation refers to the phenomenon where such objects emit thermal radiation when heated.

A prominent aspect of black body radiation includes:

• Every physical body emits more energy as the temperature rises, becoming a significant source of radiation itself.
• The color of the emitted light shifts from red to blue as the temperature increases, described as the black body spectrum.

Understanding this radiation pattern allows scientists to discern the temperature and composition of stars and other celestial bodies by observing the emitted radiation. This principle is essential in producing accurate thermal cameras and other radiometric devices.

Energy Emission

Energy emission describes the release of energy in the form of electromagnetic radiation from a surface or a body. This concept is fundamental to the operational processes in fields such as thermodynamics and astrophysics. Solar energy, primarily, is the result of nuclear fusion processes at the core of the Sun, leading to continuous energy output to the surrounding space.

Key points about energy emission include:

• The quantity and intensity of energy emitted (or radiated) are dependent on the energy source’s nature, temperature, and surface area.
• In celestial terms, stars, including our Sun, emit energy across the electromagnetic spectrum, which includes visible, infrared, and ultraviolet light.
• Emission rates and patterns inform scientists about star life cycles and physical properties.

Energy emission is a concept that bridges various scientific disciplines, illuminating both natural phenomena and technological applications, such as solar power harvesting.