GATE 2022 ME (ME2) – Question 35
Wien’s law is stated as follows: $\lambda_mT=C$, where C is 2898 µm K and $\lambda_m$ is the wavelength at which the emissive power of a black body is maximum for a given temperature T. The spectral hemispherical emissivity $(\varepsilon_\lambda)$ of a surface is shown in the figure below ($1\ \text{Å}=10^{-10}$ m). The temperature at which the total hemispherical emissivity will be highest is ______ K (round off to the nearest integer).
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Correct answer: 4825 to 4835
Explanation
**Idea.** The total emissivity is highest when the black body's emission peak falls where the surface's spectral emissivity is highest. The emissivity curve peaks at 6000 Å, so use Wien's displacement law at that wavelength.
**Step 1: unit conversion.**
$$6000\text{ Å}=6000\times10^{-10}\text{ m}=0.6\ \mu\text{m}.$$
**Step 2: Wien's law.**
$$T=\frac{C}{\lambda_m}=\frac{2898\ \mu\text{m K}}{0.6\ \mu\text{m}}=\mathbf{4830\ K}.$$
Official GATE 2022 answer key: https://gate2026.iitg.ac.in/doc/download/Answer_keys2022/me-merged_2022.pdf