If a star has a surface temperature of 21,000K, at what electromagnetic wavelength does the star emit its maximum amount of energy?

Question

BlackTom AI · Accepted Answer

To determine where a hot star emits its maximum energy, we apply Wien's displacement law, which links the peak wavelength lambda_max to the temperature T via lambda_max = b / T, where b ≈ 2.898 × 10^-3 m·K.

Option 1: 0.1 micrometers (0.1 μm = 1.0 × 10^-7 m). Using T = b / lambda_max, this gives T ≈ (2.898 × 10^-3) / (1.0 × 10^-7) ≈ 29,000 K. At a temperature of 21,000 K, 0.1 μm is in the right ballpark for a very blue, high-energy peak; the calculation is consistent with a peak near this small wavelength.

Option 2: 1.0 micrometers (1.0 μm = 1.0 × 10^-6 m). If lambda_max were 1.0 μm, then T ≈ (2.898 × 10^-3) / (1.0 × 10^-6) ≈ 2,898 K, which is far cooler than 21,000 K. This would place the peak in the infrared, not in the ultraviolet, so this is inconsistent with the given temperature.

Option 3: 0.5 micrometers (0.5 μm = 5.0 × 10^-7 m). Here, T ≈ (2.898 × 10^-3) / (5.0 × 10^-7) ≈ 5,796 K. This is still much cooler than 21,000 K, indicating a peak at a longer wavelength than the star’s actual maximum emission.

Option 4: 10 micrometers (10 μm = 1.0 × 10^-5 m). Then T ≈ (2.898 × 10^-3) / (1.0 × 10^-5) ≈ 290 K, which corresponds to a very cool object emitting in the far-infrared. This is far from the 21,000 K stated and clearly not the peak for such a hot star.

Synthesis: Since hotter objects peak at shorter wavelengths, the 21,000 K star should have its peak in the ultraviolet region, near a wavelength on the order of 0.1 μm. Among the options, 0.1 μm best matches the high temperature and Wien’s law relationship, while the other choices place the peak at infrared or misrepresent the temperature scale.

Additional note: Real stars emit across a range of wavelengths around the peak, but Wien’s law provides the most direct estimate for the peak emission wavelength given the temperature. The mismatch in the other options arises because they correspond to temperatures far from 21,000 K, hence the incorrect predictions for lambda_max.

If a star has a surface temperature of 21,000K, at what electromagnetic wavelength does the star emit its maximum amount of energy?

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