I am currently working on a Mathematics research paper in line with my Semester 2 Mathematics syllabus, and this is a part of a section which is to be discarded. I thought that it would be a waste if it were to just be deleted, so I decided to share it with all of you, because WHY NOT? Please take the article as reference if you so please. Take care and happy new year!
A black body is defined as an ideal system that absorbs all radiation incidents, whereby there is no reflection from its surface. Therefore, blackbody radiation is the electromagnetic radiation emitted by the black body.
The phenomenon physicists wanted to study in the late 19th century was the wavelength distribution of radiation leaving the black body, which was represented as a cavity with a small opening (Serway, 2017, p.1048). Any radiation incident that enters the hole is reflected on the interior walls of the cavity and does not leave. According to Serway, the radiation emitted by oscillators in the cavity walls are reflected to form standing electromagnetic waves, which have many possible standing-wave modes. It is the distribution of energy among the modes that determines the wavelength distribution of the blackbody radiation.
Two consistent experimental findings were of paramount importance for the model to be accurate, in that:
The total power (intensity) of the emitted radiation increases linearly with temperature.
The peak of the wavelength distribution decreases as the temperature increases, which is aptly described using Wein’s displacement law,
Therefore, it can be experimentally shown that the intensity of blackbody radiation varies with both temperature and wavelength. However, early efforts in modelling the distribution using means from classical physics failed, as seen in the Rayleigh-Jeans law, which resulted in the intensity reaching the “Ultraviolet Catastrophe” (Intensity approaching infinity). This was soon resolved when Max Planck, father of Quantum Theory, introduced a different interpretation of energy to explain the wavelength distribution.
In contrast to having a distribution of energy in determining the wavelength distribution (classical interpretation), Planck postulated that energy of an oscillator is discrete and not continuous, , therefore energy is quantized, with each discrete energy value corresponding to a different quantum state (quantum interpretation). Concepts such as atomic energy levels and transitioning energy levels were crucial in modelling the radiation intensity curve, because the intensity at a certain wavelength is weighted on the probability of the wave being emitted (probability of a state being occupied and then grounded to release radiation).
According to the Boltzmann distribution law, the probability of a state being occupied is proportional to the Boltzmann factor: e^-(E/kT). At very low frequencies (high wavelengths), there are smaller gaps between energy states, resulting in higher probability (high Boltzmann factor) of excited states and many low energy emissions due to these states. The low energy in each transition results in low radiation intensity at very high wavelengths. This intensity increases up till a maximum due to increasing probability of emission as required energy for said emission decreases. After the peak, at very high frequencies (low wavelengths), large separations between energy gaps lead to few emissions, resulting in low radiation intensity at very low wavelengths.
By using this alternative approach, with the Boltzmann factor and quantisation of energy in mind, a theoretical expression for the wavelength distribution is derived, as shown below:
When the expression is plotted as a graph, a curve as such is formed.
This took me 3 hours. 3 hours wasted...
-Cheerful :D
References:
Serway, R.A. (2017) 'Free-Electron Theory of Metals', in J. W. Jewett, Jr. (ed.) Physics for Scientists and Engineers with Modern Physics. 10th edn. Boston, MA: Cengage, pp. 1048-1053.
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