Question

The Sun radiates like a perfect black body with an emissivity of exactly 1. (a) Calculate the surface temperature of the Sun, given that it is a sphere with a 7.00×10⁸-m radius that radiates 3.80×10²⁶ W into 3-K space. (b) How much power does the Sun radiate per square meter of its surface? (c) How much power in watts per square meter is that value at the distance of Earth, 1.50×10¹¹ m away? (This number is called the solar constant.)

a) 5800 K, 6.00×10⁷ W/m², 1.37×10³ W/m²
b) 6000 K, 5.80×10⁷ W/m², 1.34×10³ W/m²
c) 6200 K, 5.60×10⁷ W/m², 1.30×10³ W/m²
d) 6400 K, 5.40×10⁷ W/m², 1.26×10³ W/m²

Answer 1

Using the Stefan-Boltzmann law and the given values for the Sun's power output and radius, we calculate the Sun's surface temperature, the power it radiates per square meter, and the solar constant at the distance of Earth.

Step-by-step explanation:

Calculating the Solar Constants

The question involves calculating certain solar constants using the properties of black body radiation and geometric calculations. To solve for the surface temperature of the Sun, we use the Stefan-Boltzmann law which relates the total energy radiated per unit surface area of a black body to the fourth power of the black body's temperature. Given the power radiated by the Sun and its radius, we can find the temperature and the power per square meter on its surface. Then, by considering the radius of Earth's orbit, we can determine the solar constant, the power per square meter received at Earth's distance.

Part (a): The surface temperature of the Sun can be calculated using the Stefan-Boltzmann law with the given power and area. Part (b): The power per square meter radiated by the Sun is found by dividing the total power by the Sun's surface area. Part (c): The solar constant is the power received per square meter at the distance of the Earth, determined by spreading the Sun's total power over a sphere with a radius equal to the Earth-Sun distance.

The correct answers to the multiple-choice question are: (a) 5800 K, (b) 6.00×10⁷ W/m², (c) 1.37×10³ W/m².