The blackbody emission spectrum of object A peaks in the ul- traviolet region of the electromagnetic spectrum at a wavelength of 200 nm. That of object B peaks in the red region, at 650 nm. Which object is hotter, and, according to Wien's law, how many times hot- mer is it? According to Stefan's law, how many times more energy per unit area does the hotter body radiate per second?
The blackbody emission spectrum of object A peaks in the ul- traviolet region of the electromagnetic spectrum at a wavelength of 200 nm. That of object B peaks in the red region, at 650 nm. Which object is hotter, and, according to Wien's law, how many times hot- mer is it? According to Stefan's law, how many times more energy per unit area does the hotter body radiate per second?
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![### Blackbody Emission Spectrum and Temperature Comparison
The blackbody emission spectrum of object A peaks in the ultraviolet region of the electromagnetic spectrum at a wavelength of 200 nm. That of object B peaks in the red region, at 650 nm.
**Question:**
Which object is hotter, and according to Wien’s law, how many times hotter is it? According to Stefan’s law, how many times more energy per unit area does the hotter body radiate per second?
#### Explanation:
- **Wien's Law**: Wien’s law relates the temperature of an object to the peak wavelength of its emission spectrum. The formula is given by:
\[ \lambda_{\text{max}} T = b \]
where:
- \(\lambda_{\text{max}}\) is the peak wavelength,
- \(T\) is the absolute temperature,
- \(b\) is Wien’s displacement constant (approximately \(2.898 \times 10^{-3}\) m·K).
- **Stefan-Boltzmann Law**: This law states that the total energy radiated per unit surface area of a blackbody is directly proportional to the fourth power of the blackbody's temperature:
\[ E = \sigma T^4 \]
where:
- \(E\) is the energy radiated per unit area,
- \(\sigma\) is the Stefan-Boltzmann constant (approximately \(5.67 \times 10^{-8}\) W·m⁻²·K⁻⁴),
- \(T\) is the absolute temperature.
#### Analysis:
1. **Determine the Temperature Ratio using Wien’s Law:**
For object A:
\[ T_A = \frac{b}{\lambda_{\text{max, A}}} = \frac{2.898 \times 10^{-3}\, \text{m·K}}{200\, \text{nm}} \]
\[ T_A = \frac{2.898 \times 10^{-3}\, \text{m·K}}{200 \times 10^{-9}\, \text{m}} \]
\[ T_A = \frac{2.898 \times 10^{-3}}{200 \times 10^{-9}} \]
\[ T_A = 1.449 \times 10^4\, \text{K](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fe8677b-d2b4-4cf1-b1ed-08820154fcb5%2F3b5e6bf1-bc2d-4a29-be3e-d26620504084%2Fsgl49he_processed.png&w=3840&q=75)
Transcribed Image Text:### Blackbody Emission Spectrum and Temperature Comparison
The blackbody emission spectrum of object A peaks in the ultraviolet region of the electromagnetic spectrum at a wavelength of 200 nm. That of object B peaks in the red region, at 650 nm.
**Question:**
Which object is hotter, and according to Wien’s law, how many times hotter is it? According to Stefan’s law, how many times more energy per unit area does the hotter body radiate per second?
#### Explanation:
- **Wien's Law**: Wien’s law relates the temperature of an object to the peak wavelength of its emission spectrum. The formula is given by:
\[ \lambda_{\text{max}} T = b \]
where:
- \(\lambda_{\text{max}}\) is the peak wavelength,
- \(T\) is the absolute temperature,
- \(b\) is Wien’s displacement constant (approximately \(2.898 \times 10^{-3}\) m·K).
- **Stefan-Boltzmann Law**: This law states that the total energy radiated per unit surface area of a blackbody is directly proportional to the fourth power of the blackbody's temperature:
\[ E = \sigma T^4 \]
where:
- \(E\) is the energy radiated per unit area,
- \(\sigma\) is the Stefan-Boltzmann constant (approximately \(5.67 \times 10^{-8}\) W·m⁻²·K⁻⁴),
- \(T\) is the absolute temperature.
#### Analysis:
1. **Determine the Temperature Ratio using Wien’s Law:**
For object A:
\[ T_A = \frac{b}{\lambda_{\text{max, A}}} = \frac{2.898 \times 10^{-3}\, \text{m·K}}{200\, \text{nm}} \]
\[ T_A = \frac{2.898 \times 10^{-3}\, \text{m·K}}{200 \times 10^{-9}\, \text{m}} \]
\[ T_A = \frac{2.898 \times 10^{-3}}{200 \times 10^{-9}} \]
\[ T_A = 1.449 \times 10^4\, \text{K
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