Planck's Theory of Blackbody Radiation

Black body radiation refers to the electromagnetic radiation emitted by a perfect absorber and emitter of radiation, known as a black body. The concept originated in the late 19th century when scientists observed that heated objects emit light across a continuous spectrum. This phenomenon challenged classical physics and led to the development of quantum mechanics. The basic concept revolves around the idea that a black body absorbs all incident radiation and emits radiation across all wavelengths, with the intensity of emission depending on the temperature of the body.

The electromagnetic radiation that a black body in thermal equilibrium at a specific temperature emits is successfully described by Planck's law. Max Planck, who first suggested the law in 1900, is honored by the name of the law. It is an innovative outcome of quantum theory and contemporary physics.

Planck introduced an entirely new idea about blackbody radiation. He mentioned that the emission of blackbody radiation is not continuous, but in the form of energy packets, called ‘quanta’ where each quantum carrying energy , where  is the frequency of radiation and h is the Planck’s constant.

Spectral Radiance

A body's spectral radiance, or , is a measure of the energy it emits as various frequencies of radiation. Spectral radiance can be measured by the amount of power emitted per unit area of the object, per unit solid angle that the radiation is measured over, per unit frequency. Planck demonstrated that a body's spectral brightness or radiance at an absolute temperature of T is equal to:

Here,

Alternatively, the law can be expressed for the spectral energy density at given temperature is given as:

Rather than measuring the spectrum brightness per unit frequency, it is also possible to do so per unit wavelength. In this case, it's provided by:

Alternatively, the law can be expressed for the wavelength of spectral energy density at given temperature is given as:

In 1900, Max Planck formulated the law based solely on empirically known constants. Subsequently, he demonstrated that the law, when stated as an energy distribution, represents the only stable distribution for radiation in thermodynamic equilibrium. It belongs to a family of thermal equilibrium distributions as an energy distribution, which also includes the Bose–Einstein distribution, the Fermi-Dirac Distribution and the Maxwell – Boltzmann distribution.

Experimental Verification

The Planck’s formula is found to be in complete agreement with experiment for the entire wavelength range at all temperatures.

Case 1: When  is very small, then , so that Planck’s formula given by the equation

Here, implement the above condition , the above equation can be written as:

Now, put  and , we have:

The above expression is of Wein’s Law which agrees with experiment at short wavelength.

Case 2: When  is very large, then , so that Planck’s formula for the wavelength of spectral energy density at given temperature can be written as:

This is Rayleigh – Jeans law which agrees with the experiment at long wavelengths.