text: # Integrals
  
  Exercise 5.12: The StefanāBoltzmann constant
  
  The Planck theory of thermal radiation tells us that in the (angular) frequency interval Ļ to Ļ + dĻ, a black body of unit area radiates electromagnetically an amount of thermal energy per second equal to I(Ļ) dĻ, where
  
  $I(\omega) = {\hbar\over4\pi^2c^2} \, {\omega^3\over(e^{\hbar\omega/k_BT}-1)}.$
  
  HereĀ $\hbar$ is Planckās constant over 2Ļ, c is the speed of light, and kB is Boltzmannās constant.
  
  1. a) Ā Show that the total energy per unit area radiated by a black body is
  2. $W = {k_B^4 T^4\over4\pi^2c^2\hbar^3}\int_0^\infty {x^3\over e^x-1} dx.$
  
  3. b) Ā Write a program to evaluate the integral in this expression. Explain what method you used, and how accurate you think your answer is.
  4. c) Ā Even before Planck gave his theory of thermal radiation around the turn of the 20th century, it was known that the total energy W given off by a black body per unit area per second followed Stefanās law: W = ĻT4, where Ļ is the StefanāBoltzmann constant. Use your value for the integral above to compute a value for the StefanāBoltzmann constant (in SI units) to three significant figures. Check your result against the known value, which you can find in books or on-line. You should get good agreement.
explanation: The document is an exercise from a section on integrals, specifically focusing on the Stefan–Boltzmann constant. It contains multiple parts, each of which is a question or a task to be completed. The questions involve showing a mathematical derivation, writing a program to evaluate an integral, and using the result to compute a physical constant.
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
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distilabel_metadata.content: # Integrals
  
  Exercise 5.12: The StefanāBoltzmann constant
  
  The Planck theory of thermal radiation tells us that in the (angular) frequency interval Ļ to Ļ + dĻ, a black body of unit area radiates electromagnetically an amount of thermal energy per second equal to I(Ļ) dĻ, where
  
  $I(\omega) = {\hbar\over4\pi^2c^2} \, {\omega^3\over(e^{\hbar\omega/k_BT}-1)}.$
  
  HereĀ $\hbar$ is Planckās constant over 2Ļ, c is the speed of light, and kB is Boltzmannās constant.
  
  1. a) Ā Show that the total energy per unit area radiated by a black body is
  2. $W = {k_B^4 T^4\over4\pi^2c^2\hbar^3}\int_0^\infty {x^3\over e^x-1} dx.$
  
  3. b) Ā Write a program to evaluate the integral in this expression. Explain what method you used, and how accurate you think your answer is.
  4. c) Ā Even before Planck gave his theory of thermal radiation around the turn of the 20th century, it was known that the total energy W given off by a black body per unit area per second followed Stefanās law: W = ĻT4, where Ļ is the StefanāBoltzmann constant. Use your value for the integral above to compute a value for the StefanāBoltzmann constant (in SI units) to three significant figures. Check your result against the known value, which you can find in books or on-line. You should get good agreement.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is an exercise from a section on integrals, specifically focusing on the Stefan–Boltzmann constant. It contains multiple parts, each of which is a question or a task to be completed. The questions involve showing a mathematical derivation, writing a program to evaluate an integral, and using the result to compute a physical constant.
  
  Label: multiple-question