The Stefan-Boltzmann constant The Planck theory of thermal radiation tells us that in the (angular) frequency interval \(\omega\) to \(\omega+d \omega\), a black body of unit area radiates electromagnetically an amount of thermal energy per second equal to \(I(\omega)\) d \(\omega\), where $$ I(\omega)=\frac{\hbar}{4 \pi^{2} c^{2}} \frac{\omega^{3}}{\left(\mathrm{e}^{\hbar \omega / k_{a} T}-1\right)} $$ Here \(h\) is Planck's constant over \(2 \pi, c\) is the speed of \(h\) ght, and \(k_{B}\) is Boltzmann's constant. a) Show that the total rate at which energy is radiated by a black body per unit area, over all frequencies, is $$ W=\frac{k_{B}^{4} T^{4}}{4 \pi^{2} c^{2} h^{3}} \int_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} \mathrm{~d} x . $$ b) Write a program to evaluate the integral in this expression. Explain what method you used, and how accurate you think your answer is. c) Even before Planck gave his theory of thermal radiation around the turn of the 20 th century, it was known that the total energy \(W\) given off by a black body per unit area per second followed Stefan's law: \(W=\sigma T^{4}\), where \(\sigma\) is the StefanBoltzmann constant. Use your value for the integral above to compute a value for the Stefan-Boltzmann 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.

Step by step solution

01

- Define the given expression

The power radiated per unit area by a black body across all frequencies is given by the integral: \[ W = \frac{k_{B}^{4} T^{4}}{4 \pi^{2} c^{2} h^{3}} \int_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} \mathrm{~d} x \]

02

- Substitution

To simplify the integral, we use the substitution \( x = \frac{\hbar \omega}{k_{B} T} \). This leads to \[ x = \frac{\hbar \omega}{k_{B} T} \] which further implies that \( \omega = \frac{k_{B} T x}{\hbar} \) and \( d\omega = \frac{k_{B} T}{\hbar} dx \).

03

- Transform the integral

Substituting these values into the integral, we get: \[ W = \frac{1}{4 \pi^{2} c^{2} h^{3}} \int_{0}^{\infty} \frac{(\frac{k_{B} T x}{\hbar})^{3} \cdot \frac{k_{B} T}{\hbar} \mathrm{d} x}{\mathrm{e}^{x}-1} \]

04

- Simplification

Simplify the integral: \[ W = \frac{k_{B}^{4} T^{4}}{4 \pi^{2} c^{2} h^{3}} \int_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} \mathrm{~d} x \]

05

- Write a Program to Evaluate the Integral

Write a simple Python program using numerical integration methods like Simpson's rule or the trapezoidal rule to evaluate the integral \( \int_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} \mathrm{~d} x \). Ensure the chosen method provides high accuracy. Here's an example of using the scipy library:```pythonimport scipy.integrate as integrateintegral_value, error = integrate.quad(lambda x: (x**3)/(np.exp(x)-1), 0, np.inf)print('Integral Value:', integral_value)```

06

- Compute the Stefan-Boltzmann Constant

Using the calculated integral value, we can compute the Stefan-Boltzmann constant \( \sigma \) as follows: \[ \sigma = \frac{k_{B}^{4}}{4 \pi^{2} c^{2} h^{3}} \int_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} \mathrm{~d} x \] Substitute the known values and the computed integral value to obtain: \[ \sigma \approx 5.67 \times 10^{-8} \text{ W/m}^2 \text{K}^4 \]

07

- Compare with Known Value

Finally, compare the computed value of \( \sigma \) with the known value \( 5.67 \times 10^{-8} \text{ W/m}^2 \text{K}^4 \) to validate our result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stefan-Boltzmann Constant

The Stefan-Boltzmann constant is denoted by \( \sigma \). It describes the power radiated per unit area of a black body as a function of its temperature. This concept is crucial in understanding thermal radiation.
According to Stefan's law, the total energy \( W \) emitted per unit area of a black body is proportional to the fourth power of its absolute temperature. The law is expressed as: \[ W = \sigma T^4 \] Here, we calculated the value of \( \sigma \) using Planck's law by numerically evaluating an integral. Given known constants and integration, we confirmed that \( \sigma \) is approximately \( 5.67 \times 10^{-8} \text{ W/m}^2 \text{K}^4 \). This constant helps in predicting thermal radiation accurately.

Numerical Integration

Numerical integration is a method used to approximate the value of an integral when an analytical solution is difficult or impossible to find. In this exercise, we evaluated the integral \( \int_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} \mathrm{~d} x \) using numerical integration techniques.
Some common methods of numerical integration include:

• Simpson's Rule
• Trapezoidal Rule
• Midpoint Rule

We used Python's SciPy library for our calculations, a widely used tool for scientific computing. The function `quad` from SciPy allows us to perform numerical integration efficiently and accurately. The result of our integral provided a value that helped us compute the Stefan-Boltzmann constant precisely. Here's a brief snippet of the code used:
```python
import scipy.integrate as integrate
import numpy as np
integral_value, error = integrate.quad(lambda x: (x**3)/(np.exp(x)-1), 0, np.inf)
print('Integral Value:', integral_value)
```
This integral plays a vital role in deriving physical constants and understanding energy radiation.

Planck's Law

Planck's Law explains the electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. It was a groundbreaking discovery by Max Planck that resolved issues with previous models of radiation.
The law states that the spectral radiance of a black body is a function of frequency \( \omega \) and temperature \( T \). The formula is given by:
\[ I(\omega) = \frac{\hbar}{4 \pi^{2} c^{2}} \frac{\omega^{3}}{\left(\mathrm{e}^{\hbar \omega / k_{B} T} - 1\right)} \]
Here,

• \( \hbar \) is Planck's constant divided by \( 2 \pi \)
• \( c \) is the speed of light
• \( k_{B} \) is Boltzmann's constant

Planck's law is fundamental in quantum mechanics and provides the foundation for understanding black body radiation, including the derivation of Wien's Law and the Stefan-Boltzmann Law. It highlights the quantized nature of energy emission and absorption, marking a departure from classical physics predictions.

Thermal Radiation

Thermal radiation refers to the emission of electromagnetic waves from all matter that has a temperature greater than absolute zero. This phenomenon occurs due to the thermal motion of particles within matter.
Some key points about thermal radiation:

• It is one of the modes of heat transfer
• Emission depends on the temperature and nature of the material
• A black body is an ideal emitter that completely absorbs and emits radiation uniformly

Key principles in thermal radiation include:

• Stefan-Boltzmann Law: Total energy emitted is proportional to the fourth power of temperature
• Planck's Law: Describes spectral radiance of radiation as dependent on temperature and frequency
• Wien's Displacement Law: Peak wavelength of emission shifts inversely with temperature

Understanding thermal radiation is essential in fields like astronomy, climate science, and engineering, where heat and energy transfer play significant roles. It helps us predict and manipulate thermal properties in various technological and natural processes.