# 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.
The text passage contains a series of questions related to the Stefan–Boltzmann constant and the Planck theory of thermal radiation. It includes a mathematical problem and asks for the evaluation of an integral and the computation of a physical constant.
multiple_questions
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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.
user
The text passage contains a series of questions related to the Stefan–Boltzmann constant and the Planck theory of thermal radiation. It includes a mathematical problem and asks for the evaluation of an integral and the computation of a physical constant.

TYPE: MULTIPLE_QUESTIONS