Problem

The thermal radiation from the sun peaks in the visible part of the spectrum. Estimate the temperature of the sun using Wien’s displacement law.

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Data

• Wien’s constant: $b = 0.2898 \times 10^{-2} \, \mathrm{m \cdot K}$
• Wavelength at peak intensity: $\lambda_{\max} = 0.5 \, \mu \mathrm{m} = 0.5 \times 10^{-6} \, \mathrm{m}$

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Prerequisite Concepts

Wien’s displacement law relates the temperature $T$ of a blackbody to the wavelength $\lambda_{\max}$ at which its radiation is at peak intensity:

$$\lambda_{\max} \cdot T = b$$

where $b$ is Wien’s constant. Rearranging for temperature:

$$T = \frac{b}{\lambda_{\max}}$$

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Solution

Using the given values:

$$T = \frac{0.2898 \times 10^{-2} \, \mathrm{m \cdot K}}{0.5 \times 10^{-6} \, \mathrm{m}}$$

Simplify the calculation:

$$T = \frac{0.2898 \times 10^{-2}}{0.5 \times 10^{-6}}$$ $$T = 0.5796 \times 10^{4} \, \mathrm{K}$$ $$T = 5796 \, \mathrm{K} \approx 5800 \, \mathrm{K}$$

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Answer

The estimated temperature of the sun is approximately $5800 \, \mathrm{K}$.