Given that the solar spectrum corresponds to a temperature of 5800 K and peaks at a wavelength of 500 nm, use Wien's law to determine the wavelength corresponding to the peak of the black-body curve in the core of the Sun, where the temperature is 10' K
Given that the solar spectrum corresponds to a temperature of 5800 K and peaks at a wavelength of 500 nm, use Wien's law to determine the wavelength corresponding to the peak of the black-body curve in the core of the Sun, where the temperature is 10' K
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![### Determining Peak Wavelength Using Wien's Law
Given that the solar spectrum corresponds to a temperature of 5800 K and peaks at a wavelength of 500 nm, use Wien’s law to determine the wavelength corresponding to the peak of the black-body curve in the core of the Sun, where the temperature is 10^7 K.
### Explanation and Calculation
**Wien’s Law:**
\[
\lambda_{\text{max}} = \frac{b}{T}
\]
where:
- \(\lambda_{\text{max}}\) is the peak wavelength,
- \(b\) is Wien’s constant (\(2.897 \times 10^{-3}\) m·K),
- \(T\) is the temperature in Kelvin (K).
**Given Data:**
- Temperature of solar spectrum, \(T_1 = 5800\) K,
- Peak wavelength of solar spectrum, \(\lambda_1 = 500\) nm,
- Temperature of Sun’s core, \(T_2 = 10^7\) K.
**Determine peak wavelength for Sun’s core:**
1. Convert 500 nm to meters:
\[
500 \, \text{nm} = 500 \times 10^{-9} \, \text{m}
\]
2. Calculate \(b\) using the given temperature and wavelength:
\[
b = \lambda_1 \times T_1 = 500 \times 10^{-9} \times 5800 = 2.9 \times 10^{-3} \, \text{m·K}
\]
3. Use Wien’s constant \(b = 2.9 \times 10^{-3} \, \text{m·K}\) for the core temperature:
\[
\lambda_{\text{max}} = \frac{b}{T_2} = \frac{2.9 \times 10^{-3}}{10^7} = 2.9 \times 10^{-10} \, \text{m}
\]
Therefore, the peak wavelength corresponding to the core of the Sun at \(10^7\) K is approximately \(0.29 \, \text{nm}\).
---
This information and calculation is useful for understanding the thermal radiation behavior of different temperatures and can be applied to various astrophysical and scientific contexts involving black-body radiation](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbdf4ad1c-8d6e-4c6e-ab0d-20fae6a9f061%2F93f57c59-80fe-4aa5-91db-e5fcfd4e7a98%2Fg221cok_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining Peak Wavelength Using Wien's Law
Given that the solar spectrum corresponds to a temperature of 5800 K and peaks at a wavelength of 500 nm, use Wien’s law to determine the wavelength corresponding to the peak of the black-body curve in the core of the Sun, where the temperature is 10^7 K.
### Explanation and Calculation
**Wien’s Law:**
\[
\lambda_{\text{max}} = \frac{b}{T}
\]
where:
- \(\lambda_{\text{max}}\) is the peak wavelength,
- \(b\) is Wien’s constant (\(2.897 \times 10^{-3}\) m·K),
- \(T\) is the temperature in Kelvin (K).
**Given Data:**
- Temperature of solar spectrum, \(T_1 = 5800\) K,
- Peak wavelength of solar spectrum, \(\lambda_1 = 500\) nm,
- Temperature of Sun’s core, \(T_2 = 10^7\) K.
**Determine peak wavelength for Sun’s core:**
1. Convert 500 nm to meters:
\[
500 \, \text{nm} = 500 \times 10^{-9} \, \text{m}
\]
2. Calculate \(b\) using the given temperature and wavelength:
\[
b = \lambda_1 \times T_1 = 500 \times 10^{-9} \times 5800 = 2.9 \times 10^{-3} \, \text{m·K}
\]
3. Use Wien’s constant \(b = 2.9 \times 10^{-3} \, \text{m·K}\) for the core temperature:
\[
\lambda_{\text{max}} = \frac{b}{T_2} = \frac{2.9 \times 10^{-3}}{10^7} = 2.9 \times 10^{-10} \, \text{m}
\]
Therefore, the peak wavelength corresponding to the core of the Sun at \(10^7\) K is approximately \(0.29 \, \text{nm}\).
---
This information and calculation is useful for understanding the thermal radiation behavior of different temperatures and can be applied to various astrophysical and scientific contexts involving black-body radiation
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