Black Body and Stefan-Boltzmann Law

Black Body and Black Body Radiations

A perfect black body is one which absorbs heat radiation of all wavelengths that fall upon it.

When a black body is heated it also reflects the heat radiations absorbed by it. In this universe, no any black body is found to be 100%. The scientist designed a scale body which absorb approximately 98% heat radiations incident on it.

Ferry’s Black Body

The simplest and most commonly used black body was designed by Fery shown in the figure. It consists of a hollow doubled walled metallic sphere having small opening O on one side and a conical Projection P just opposite to it. Inside the sphere is coated with lamp black. A heating coil HH is placed between the walls to heat the body. Any radiation entering the hole is reflected many times at the inner walls of the sphere and after few reflections, almost of the radiation is absorbed.

When the body is heated, it emits radiation through the hole O and so, O acts as a black body radiator. Radiation coming from the hole is called cavity radiation.

Emissive Power

Emissive power of a body at a particular temperature is the total energy of all wavelengths radiated per second per unit area of the body. Since the wavelength of radiation emitted ranges from the zero to infinity

$$\text {Emmisive power},e=\int_0^\infty e_ \lambda d\lambda$$

Its S.I. unit is Jm^-2s^-1 or Wm^-2.

Emissivity (e)

The ratio of heat energy radiated per second per unit area by the given body to the total amount of heat energy radiated per second per unit area by the perfectly black body is known as emissivity. It is denoted by ‘e’ for a perfectly black body its value is 1.

$$emissivity, \epsilon =\frac {e}{E}$$

$$\text {or}, e = \epsilon E$$

Stefan’s Law

It states that, “The total amount of heat energy radiated per second per unit area by the perfectly black body is directly proportional to the fourth power of the absolute temperature of its surface.”

It ‘E’ be the amount of heat energy radiated per second per unit area by the perfectly black body having an absolute temperature of its surface is ‘T’. Then we can write,

$$E \propto T^4$$

$$\therefore E =\sigma T^4$$

Where \(‘\sigma’\) is constant and is shown as Stefan’s constant. It’s value in CGS system is \(5.67 \times 10^{-5} erg s^{-1}cm{-2}k^{-4}\) and in SI system \(5.677 \times 10^{-8} watt m^{-2}k^{-4}\).

In 1879, Stefan's verified this law experimentally and five years later in 1884, Boltzmann theoretically verified these result. Therefore, this law is known as Stefan's Boltzmann law.

This law account only for the emission of energy. According to this law, "When a black body of absolute temperature 'T' is surrounded by another black body of absolute temperature T₀, then the net energy emitted per second per unit area by the former one is given by

$$E = \sigma T^4 - \sigma T_0^4$$

$$E = \sigma (T^4 - T_0^4)$$

If the body is not perfectly black having emissive power 'e'. Then net energy emitted per second per unit area is given by

$$E =e \sigma (T^4 - T_0^4)$$