The Stefan-Boltzmann Law

The Austrian physicist Ludwig Boltzmann used the rules of thermodynamics to understand how surfaces release heat energy. The Stefan-Boltzmann law states that the total illuminated heat energy released from a surface is equal to the fourth power of its absolute temperature. 

If E is the illuminated heat energy released from a designated area in 1 second and T is its absolute temperature in kelvins, then E=σT⁴. Here, σ represents the proportionality factor of the blackbody radiation, commonly known as the Stefan-Boltzmann constant. The value of the Stefan-Boltzmann constant is 5.6703710⁻⁸ Wm-²K-⁴.

This law is only applicable to blackbodies and other surfaces that absorb radiation.

Blackbody Radiation

An imaginary object which is perfectly effective at sucking up and emitting radiation is called a blackbody, and its emissions are called blackbody radiation. At higher temperatures, these emissions can glow at visible wavelengths. 

Absolute Temperature

• It is the temperature of a material which is measured in relation to its absolute zero temperature. 
• It is also known as thermodynamic temperature. 
• Absolute zero is the state in which systems are usually at their lowest energy. Electric devices don’t operate at this temperature, and the kinetic energy is also nil.

Stefan-Boltzmann Constant

• The Stefan-Boltzmann constant, derived by the scientist Josef Stefan, is signified with the Greek letter σ. 
• It is a physical quantity related to the emissions given out by blackbodies. 
• The quantity determines how much energy a blackbody can give out per unit area, as per the rules of thermodynamics. 
• The Stefan-Boltzmann Constant value is nearly 5.67 x 10⁻⁸ Wm-2K-⁴.

Stefan’s Law Formula

The formula for the Stefan-Boltzmann law is:

P=εσT4A

Here, 

• P: Radiated energy
• σ: Stefan-Boltzmann constant
• T: Absolute temperature
• ε: Emissivity of the material
• A: Area of the emitting body 

Examples

1. An object has an emissivity factor of 0.2 and its surface area is 300 m2. Find out the rate at which the object emits energy if its temperature is 700 K.

Ans: P=εσT⁴A

P = 0.2 x (5.67 x 10⁻⁸ Wm–²K–⁴) x (700 K)4 x 300 m²

P=8.1710⁵W

2. A steel ball with a radius of 2 cm is heated to the temperature of 4000°C. If its emission rate is 0.8, at what rate does it radiate energy?

Ans: The temperature of the steel ball in Kelvin is (4000°C – 273°C) K/°C = 3727 K.

The surface area of the sphere is 4πr2 = 4π(0.02m)2 = 0.005 m2.

So the energy radiated by the ball is:

P=εσT4A

P = 0.8 x 5.67 x 10−8 Wm-2K-4 x (3727 K)4 x 0.005 m2

P = 4.38 x 104 W

3. Assuming that the Sun is a blackbody and its temperature is exactly 6000 K, how much energy does it radiate per square metre? The emissivity of the Sun is approximately 1.

Ans: The energy emitted per square metre is :

P=εσT⁴A

P = 1 x 5.67 x 10⁻⁸ Wm–²K–⁴ x (6000 K)4 x 1 m²

P = 7.35 x 107 W

Usage of the Stefan-Boltzmann Law

The temperature of the Sun

Using the Stefan-Boltzmann law, Stefan determined the approximate temperature of the Sun. 

• He took data from the previous scientists who postulated that the energy flux density of the sun was 29 times larger than the flux density of a warmed lamella (a metal plate). 
•  Stefan also theorised that the Earth’s upper layers absorb half of the energy given out from the Sun.
• Using this information, he concluded that the Sun’s temperature was about 5700 K.

The temperature of Earth and Stars

With this method, we can also calculate the effective temperatures of the stars and other planets, like the Earth, by treating the energy emitted by them as blackbody emissions.

Applications of Stefan-Boltzmann Law

• It is used to derive the laws of fluid dynamics regarding energy and mass.
• It is applicable in physical cosmology, which includes the Big Bang theory, the formation of dark matter and the production of light elements.
• It is helpful in the fields of general relativity and astronomy.
• It is used to find the energy emitted from a black body.

Conclusion

We can conclude that the Stefan-Boltzmann law was a necessary and efficient discovery that further helped us to better understand the laws of thermodynamics. The Stefan-Boltzmann constant is critical for determining the temperature of blackbodies and similar objects. It also gives us the approximate temperature of far-away objects, like the Sun and stars. This law has also been applied in various fields such as astronomy and fluid dynamics.