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6. Stars emit radiation with a range of wavelengths - Scottish Highers Physics - Question 6 - 2019

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6. Stars emit radiation with a range of wavelengths. The peak wavelength of the radiation depends on the surface temperature of the star. (a) The graph shows how th... show full transcript

Worked Solution & Example Answer:6. Stars emit radiation with a range of wavelengths - Scottish Highers Physics - Question 6 - 2019

Step 1

Add a line for the second star's radiation

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Answer

To add a line for the second star with a surface temperature of 6000 K on the existing graph, we need to reference the Stefan-Boltzmann Law and Wien's Displacement Law. According to Wien's Law, the peak wavelength (λpeak\lambda_{peak}) is inversely proportional to temperature (TT):

λpeak=bT\lambda_{peak} = \frac{b}{T}

where b≈2.9×10−3 m Kb \approx 2.9 \times 10^{-3} \text{ m K}. For a temperature of 6000 K, the calculation is:

λpeak=2.9×10−36000≈4.83×10−7 m\lambda_{peak} = \frac{2.9 \times 10^{-3}}{6000} \approx 4.83 \times 10^{-7} \text{ m}

Now, plot this value on the graph. The energy emitted should peak at this wavelength, and thus the new line should reach a higher energy value than that of the 5000 K star.

Step 2

Use the data to show the relationship

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For part (b), we can calculate the product TrT_r and λpeak\lambda_{peak} for each star:

  1. For T=7700T = 7700 K: Trλpeak=7700×3.76×10−7≈2.89×10−3T_r \lambda_{peak} = 7700 \times 3.76 \times 10^{-7} \approx 2.89 \times 10^{-3}
  2. For T=8500T = 8500 K: Trλpeak=8500×3.42×10−7≈2.91×10−3T_r \lambda_{peak} = 8500 \times 3.42 \times 10^{-7} \approx 2.91 \times 10^{-3}
  3. For T=9600T = 9600 K: Trλpeak=9600×3.10×10−7≈2.98×10−3T_r \lambda_{peak} = 9600 \times 3.10 \times 10^{-7} \approx 2.98 \times 10^{-3}
  4. For T=12000T = 12000 K: Trλpeak=12000×2.42×10−7≈2.90×10−3T_r \lambda_{peak} = 12000 \times 2.42 \times 10^{-7} \approx 2.90 \times 10^{-3}

From the calculations, we observe that all values are consistent with the formula Trλpeak=2.9×10−3T_r \lambda_{peak} = 2.9 \times 10^{-3}, which confirms the relationship.

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