Stefan Boltzmann Law- Formula, Derivation

Ludwig Boltzmann and Josef Stefan are the two scientists who introduced the Stefan-Boltzmann law. As per the Stefan Boltzmann law, the amount of heat reflected or radiated from the black body is proportional to the power of four of the black body’s absolute temperature or thermodynamic temperature. The constant that shows its proportionality and describes the law equation is called the Stefan Boltzmann constant value. Stefan Boltzmann’s law demonstrates the connection between the black body and its temperature. It is used to study more about the black body. It depicts the amount of heat reflected from the black body in different situations. 

Stefan Boltzmann law

Stefan Boltzmann’s law demonstrates that the amount of heat or energy radiated from the per-unit total area of black body along all its wavelength per unit radiant emittance is proportional to the power of four of the absolute and thermodynamic temperature (T) of the black body. 

Stefan- Boltzmann constant value

Stefan- Boltzmann’s constant value is the proportionality constant which satisfies the equation and formula of Stefan Boltzmann’s law. The symbol of Stefan Boltzmann’s constant value is σ.

  σ =  2 π5k4/15c2h4

Stefan- Boltzmann constant value,

σ = 5.670367 × 10-8 W⋅m-2⋅K-4

In the above equation, 

W = Watt

m = metre

K = Kelvin 

The dimensional formula of Stefan Boltzmann’s constant value is [M1L2T-2K-1]. 

As per the Committee on Data for Science and Technology (CODATA), the Stefan Boltzmann constant value becomes, 

 σ = 2π5 R4 / 15 h3 c2 NA4

In the above equation, 

NA = The value of Avogadro’s constant, which is equal to 6.02214086 × 1023 mol-1

R = The value of the universal gas constant, which is equal to 8.314 J.mol-1.K-1

Stefan- Boltzmann constant value, as per the CODATA. 

σ = (32 π5 h R4 R∞4) / (15 Ar(e)4 Mu4 c6 α8)

Here, 

R∞ = Value of Rydberg constant. 

Ar(e) = Value of Electron’s relative atomic mass 

Mu = Value of molar mass constant 

α = Value of structure constant. 

Stefan Boltzmann Law Formula

The Stefan Boltzmann law is the total amount reflected per unit area along all wavelengths. It can be obtained after interconnecting Plank’s radiation formula.

The formula of Stefan Boltzmann law:

u/A =  σT4

Stefan Boltzmann’s law formula for the body is not the black body. 

u =  eσAT4

Here, e = emissivity of the body, which is merely equal to the absorptive power of the body. 

Stefan Boltzmann Law Derivation 

The Stefan Boltzmann law is the total amount reflected per unit area along all wavelengths. It can be obtained after interconnecting Plank’s radiation formula. 

Then, it becomes:

dP/dλ.1/A = 2πhc2/λ5(hc/eλkT – 1)

In the above equation, 

P = Radiated power 

A = Total area of black body

λ = wavelength

h = Planck’s constant

c = light velocity 

k = Boltzmann constant value 

T = thermodynamic temperature 

After computing and integrating above equation with respect to λ and separating the value of Constant,

Put x = hc/eλk

Then, 

dx = hc/eλk dλ

We know that, 

h = xλkT/c and c = xλkT/h

Therefore, 

dλ = – λ2kT/hc dx

Putting all these values in the above equation and compare the above equation with standard form of integration, 

Putting the above value in the equation, 

= P/A = 2π(kT)4/h3c2). π4/15

On computing further, the results come, 

P/A = σT4

Putting, P/A = E 

E = σT4

σ = Stefan Boltzmann constant value 

Applications of Stefan Boltzmann Law

Stefan Boltzmann’s law is widely used in various domains of different fields like astronomy, physics, and mathematics. Let’s understand some of its Applications. 

Measurement of Star Radius 

When we measure the star’s radius, the star’s luminosity is considered. The luminosity is the total amount of heat radiated by the star in the universe. This luminosity basically depends upon two different factors, which are surface area and temperature. The Stefan Boltzmann law demonstrates the relation of the amount of heat and temperature; that’s why it is used to measure the radius of stars. 

Heating of Metal Road 

When we heat any metal road, the heat from the hotter side transfers to the colder side, but it is a misconception. The truth is that both the colder and hotter regions radiate the atmosphere’s heat. Only the difference lies in their amount, as the hotter region radiates more heat and the colder region radiates less. To calculate the amount of heat, we need to consider the hotter region’s area and the colder region’s area. This can be done by using Stefan Boltzmann’s law, where surface area is proportional to the four power of temperature. 

Conclusion 

Stefan Boltzmann’s law formula demonstrates a relationship between the temperature and the surface area of the black body. Although it also demonstrates the relation between temperature and area of the normal body, but mainly used for the black body. It is also used to measure the quantity of heat released by the black body or normal body. The Stefan Boltzmann constant value is widely used in the conversions of units. Along with the scientific uses, Stefan Boltzmann’s law is used in our day-to-day life, like in hot metal rods and any other hotter surfaces.