BLACK BODY RADIATION

In 1897, Lummer and Pringsheim measured the intensities of different wavelengths of radiations emitted by black body using Bolometer (a device made from thermocouple) and plotted them as shown in the graph.

Wien derived the equation

 Eλ = ae ^{- b / (λt)} / λ⁵ for the energy density at wavelength λ at time t, a being the proportionality constant. This equation is found to be valid for small wavelengths but fails for the longer wavelengths. Rayleigh and Jeans gave the equation Eλ = 8pKt / λ⁴ which is valid for longer wavelengths but not the smaller.

                        ∞

Here,    ∫  8pKt / λ⁴   = ∞

                        0

This shows that the total energy of radiation is infinite which is absurd. This is known as ultraviolet catastrophe. Thus, all above attempts based on thermodynamics and electromagnetic theories failed to explain the energy distribution curves of black body radiation. Finally, in 1900 A.D., Max Planck explained these experimental results.

Planck’s hypothesis:: -

Planck suggested that the walls of cavity emitting radiations are made of electric dipoles. According to their temperatures, different dipoles oscillate with different frequencies and emit radiations of frequencies same as the frequencies of their oscillations. According to classical physics, an oscillator may possess any amount of energy in a continuously varying energy. But Planck hypothesized that the oscillator can possess only such discrete values of energies depending on its frequency as given by the equation En = nhf,

Where n = 1, 2, 3, … and h = 6.625 ´ 10^{- 34} Joule second is Planck’s universal constant.

Thus, the energy of radiation of frequency f may be considered to be made up of many quanta of energy hf and a radiation may be made of quanta of different frequencies. The quantum of radiation energy is called ‘photon’. The radiation of different frequencies is the collection of photons of various frequencies. If an

oscillator possesses energy 3hf, it may be considered equivalent to a collection of 3 photons.

Based on his hypothesis, Planck derived the equation of spectral emissive power of a perfect black body:

Wf =    [            2 p h (f 3)      ]

                c² exp [hf / kt]^-1

Where c = velocity of light in vacuum, T = absolute temperature of a perfect black body, k = Boltzmann’s constant Energy distribution curves could be explained satisfactorily by the above equation. Even Stefan-Boltzmann’s law can be derived from this equation.