According to the Stefan-Boltzmann law, how much energy is radiated into space per unit time by each square meter of the Sun's surface? If the Sun's radius is 696,000 km, what is the total power output of the Sun?
According to the Stefan-Boltzmann law, how much energy is radiated into space per unit time by each square meter of the Sun's surface? If the Sun's radius is 696,000 km, what is the total power output of the Sun?
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![### Understanding the Stefan-Boltzmann Law and the Sun's Power Output
#### Problem Statement:
**According to the Stefan-Boltzmann law, how much energy is radiated into space per unit time by each square meter of the Sun’s surface? If the Sun’s radius is 696,000 km, what is the total power output of the Sun?**
#### Explanation of Concepts:
**Stefan-Boltzmann Law:**
The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body in unit time (\( E \)) is directly proportional to the fourth power of the black body's absolute temperature (\( T \)). Mathematically, it can be expressed as:
\[ E = \sigma T^4 \]
where \( \sigma \) (sigma) is the Stefan-Boltzmann constant, approximately equal to \( 5.67 \times 10^{-8} \, \text{Wm}^{-2}\text{K}^{-4} \).
**Application to the Sun:**
To find the total power output of the Sun, we need to consider the entire surface area of the Sun and apply the Stefan-Boltzmann law across that area.
1. **Energy Radiated by Each Square Meter:**
Let \( T \) be the surface temperature of the Sun in Kelvin. By using the Stefan-Boltzmann law, the energy radiated per unit area per unit time by the Sun is given by \( E = \sigma T^4 \).
2. **Total Power Output of the Sun:**
The total power output (P) of the Sun is the energy radiated per unit area per unit time multiplied by the total surface area of the Sun. The surface area \( A \) of the Sun can be calculated using the formula for the surface area of a sphere (\( 4 \pi R^2 \)):
\[ A = 4 \pi R^2 \]
where \( R \) is the radius of the Sun.
Given:
- Radius of the Sun (\( R \)) = 696,000 km = \( 696,000 \times 10^3 \) meters = \( 6.96 \times 10^8 \) meters.
Combining these formulas, the total power output of the Sun (P) is:
\[ P = E \times A = \sigma T^4 \times 4 \pi](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fe8677b-d2b4-4cf1-b1ed-08820154fcb5%2F6287faf8-74fd-43dc-9fed-65f00ed63a28%2Flcflkjj_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding the Stefan-Boltzmann Law and the Sun's Power Output
#### Problem Statement:
**According to the Stefan-Boltzmann law, how much energy is radiated into space per unit time by each square meter of the Sun’s surface? If the Sun’s radius is 696,000 km, what is the total power output of the Sun?**
#### Explanation of Concepts:
**Stefan-Boltzmann Law:**
The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body in unit time (\( E \)) is directly proportional to the fourth power of the black body's absolute temperature (\( T \)). Mathematically, it can be expressed as:
\[ E = \sigma T^4 \]
where \( \sigma \) (sigma) is the Stefan-Boltzmann constant, approximately equal to \( 5.67 \times 10^{-8} \, \text{Wm}^{-2}\text{K}^{-4} \).
**Application to the Sun:**
To find the total power output of the Sun, we need to consider the entire surface area of the Sun and apply the Stefan-Boltzmann law across that area.
1. **Energy Radiated by Each Square Meter:**
Let \( T \) be the surface temperature of the Sun in Kelvin. By using the Stefan-Boltzmann law, the energy radiated per unit area per unit time by the Sun is given by \( E = \sigma T^4 \).
2. **Total Power Output of the Sun:**
The total power output (P) of the Sun is the energy radiated per unit area per unit time multiplied by the total surface area of the Sun. The surface area \( A \) of the Sun can be calculated using the formula for the surface area of a sphere (\( 4 \pi R^2 \)):
\[ A = 4 \pi R^2 \]
where \( R \) is the radius of the Sun.
Given:
- Radius of the Sun (\( R \)) = 696,000 km = \( 696,000 \times 10^3 \) meters = \( 6.96 \times 10^8 \) meters.
Combining these formulas, the total power output of the Sun (P) is:
\[ P = E \times A = \sigma T^4 \times 4 \pi
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