Derive Stefan-Boltzmann law.
Understand the Problem
The question is asking for a derivation of the Stefan-Boltzmann law, which relates the power radiated by a black body to its temperature. We will outline the physical principles behind the law, including concepts from thermodynamics and electromagnetism to provide a coherent derivation.
Answer
The Stefan-Boltzmann law describes the power radiated by a black body as $P(T) = \sigma T^4$.
Answer for screen readers
The Stefan-Boltzmann law states that the power radiated by a black body is given by:
$$ P(T) = \sigma T^4 $$
Steps to Solve
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Introduce the Concept of Black Body Radiation Black body radiation refers to the electromagnetic radiation emitted by a perfect black body in thermal equilibrium. A black body absorbs all incident electromagnetic radiation and re-emits energy as a function of its temperature.
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Planck's Law of Black Body Radiation Planck's law provides the spectral distribution of radiation emitted by a black body as a function of wavelength $\lambda$ and temperature $T$:
$$ I(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda k T}} - 1} $$
where:
- $I(\lambda, T)$ is the intensity per unit wavelength
- $h$ is Planck's constant
- $c$ is the speed of light
- $k$ is Boltzmann's constant
- $T$ is the absolute temperature
- Calculate the Total Power Radiated To find the total power $P$ radiated per unit area, we integrate Planck's law over all wavelengths:
$$ P(T) = \int_0^{\infty} I(\lambda, T) d\lambda $$
- Perform the Integral The integral can be solved, leading to the Stefan-Boltzmann law:
$$ P(T) = \sigma T^4 $$
where $\sigma$ is the Stefan-Boltzmann constant, approximately equal to $5.67 \times 10^{-8} , \text{W/m}^2\text{K}^4$.
- Interpret the Result The result indicates that the power radiated by a black body is directly proportional to the fourth power of its absolute temperature $T$. This means that if the temperature doubles, the power radiated increases by a factor of $16$.
The Stefan-Boltzmann law states that the power radiated by a black body is given by:
$$ P(T) = \sigma T^4 $$
More Information
The Stefan-Boltzmann law is fundamental in understanding thermal radiation. It plays an important role in various fields such as astrophysics, climate science, and thermodynamics. For instance, it can be used to estimate the power output of stars, including our Sun.
Tips
- Misunderstanding the Relationship: Many might confuse the direct proportionality of power and temperature with linear relationships. It's crucial to remember that power varies with the fourth power of temperature, not linearly.
- Ignoring Units: When calculating the Stefan-Boltzmann constant or deriving the law, neglecting the correct units can lead to incorrect conclusions.
