Question

the emitted infrared radiation from a dwarf planet has a wavelength of maximum intensity at $lambda_{max}=74100$ nm. what is the temperature $t$, in kelvins, assuming it follows wiens law?

Explanation:

Step1: Recall Wien's law formula

Wien's law is given by \(\lambda_{max}T = b\), where \(b = 2.898\times10^{-3}\text{ m}\cdot\text{K}\) and \(\lambda_{max}\) is the wavelength of maximum intensity.

Step2: Convert the wavelength unit

Given \(\lambda_{max}=74100\text{ nm}\). Since \(1\text{ nm}=10^{-9}\text{ m}\), then \(\lambda_{max}=74100\times10^{-9}\text{ m}=7.41\times 10^{-5}\text{ m}\).

Step3: Solve for temperature \(T\)

From \(\lambda_{max}T = b\), we can express \(T=\frac{b}{\lambda_{max}}\). Substitute \(b = 2.898\times10^{-3}\text{ m}\cdot\text{K}\) and \(\lambda_{max}=7.41\times 10^{-5}\text{ m}\) into the formula: \(T=\frac{2.898\times 10^{-3}}{7.41\times 10^{-5}}\).

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Answer:

\(39.1\text{ K}\)