Stefan's law, also known as the Stefan-Boltzmann law, is a fundamental principle in physics that describes the total radiant power (total energy radiated as electromagnetic radiation) emitted per unit area by a black body. A black body is an idealized object that absorbs all incident radiation without reflecting or transmitting any of it.

The law is named after the Austrian physicist Josef Stefan, who first derived it in 1879, and it was later refined by the Austrian physicist Ludwig Boltzmann.

ELI5: Imagine you have a very special magic ball that can give off light and heat. The more magic power the ball has, the brighter it shines and the hotter it gets. Now, the Stefan's law tells us that the magic power (light and heat) the ball gives off depends on how hot it is.

When the ball is really hot, it gives off a lot of magic power, and it shines very brightly and feels very warm. But when it's not so hot, it gives off less magic power, so it's not as bright and not as warm.

Stefan's law says that the magic power the ball gives off increases very quickly as it gets hotter. So, when something is really, really hot, it shines and feels very, very bright and warm.

This law helps scientists understand how stars and other things in space give off light and heat. It also helps us know how much heat the Earth receives from the Sun, which is essential for our planet's weather and climate.

Key points about Stefan's law:

Temperature Dependence: The radiant power emitted by a black body is directly proportional to the fourth power of its absolute temperature. This means that as the temperature increases, the amount of radiation emitted by the black body increases significantly.
Universal Law: The Stefan-Boltzmann law is a universal law and is applicable to all objects that behave like black bodies, regardless of their composition. While real objects may not be perfect black bodies, the law still provides a good approximation for objects that emit and absorb radiation.
Applications: Stefan's law has significant applications in astrophysics, astronomy, and climate science. It is used to study the radiation emitted by stars, planets, and other celestial objects. In climate science, the law plays a crucial role in understanding the Earth's energy balance and the greenhouse effect.
Inverse Relationship with Distance: The law also explains the inverse square relationship of brightness with distance for objects like stars. As light travels farther from the source, its intensity decreases because the radiant power is distributed over a larger area.

We have,

which is known as Stefan's law.

From Plank's radiation

Case 1: For low frequency range, h𝛾<<K_BT

This is known as Rayleigh Jean's Law.

Case 2: For high frequency range h𝛾 >> K_BT

Using this condition in equation 1, we get

Pressure exerted by the proton gas is

P = 1/3 U/V

= 1/3 x Energy density

U = 3PV

Thus Stefan's law provides essential insights into the behaviour of thermal radiation and its connection to temperature. It is a foundational principle in the study of thermal radiation and has broad implications in various fields of science and engineering.

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