Have you ever seen a hot iron rod? Or a Blacksmith hammering a piece of hot metal? You must have noticed then that those iron pieces glow or emit light. It was very well established in {some date} that Light is an electromagnetic wave i.e. made up of electric and magnetic field. While we had an idea of what light is made of, physicists did not know why hot objects emitted it back then.
How do hot objects emit light?
When we heat a piece of iron, it first glows red, then orangish-red, to yellow ultimately leading to white. The brightness or intensity as it seemed was directly proportional to the increasing temperature, for simplicity's sake.
However, what they wanted to know was how the intensity varied with the wavelength of the light. What you need to understand is that for an object to emit radiation(visible), it need not be so hot to become luminous. All objects emit radiation in the Infrared spectrum(remember the rbg scale which tells us the temperature of the object), this radiation usually occurs at room temperature and hence is invisible to human eyes.
A Black Body
A body which emits and absorbs all radiation incident upon it is deemed to be a black body. For example, Our Sun is a black body. Confused? Let me explain! Our beloved Sol can emit light of all frequencies in the electromagnetic spectrum and hence is an emitter of radiation while it also reflects a very tiny part of light(not visible) thereby being a non-ideal blackbody, the Solar Radiation Spectrum graph is almost identical to the theoretical blackbody graph, as shown below.
Derivation that leads to disaster.
The Radiation spectrum given off by a blackbody at different temperature ranges is shown below:
The experimental work on the basis on which the theoretical work was established was carried out by Otto Lummer and Ernst Pringsheim. Their experiments gave valuable insights into the problem and helped to map the Theoretical characteristic curves as shown in the diagram above for different temperatures.
The theoretical work to solve this problem was carried out by Lord John Rayleigh and James Jeans.
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Image: Jeans Cube
Fig. 1 |
To explain the theoretical graph, the classical calculation begins by considering a
blackbody as a radiation-filled cavity at the temperature T (Fig. 1). Because the
cavity walls are assumed to be perfect reflectors, the radiation must consist of standing em waves, as in Fig. 2. For a node to be at each wall, the path
length from the wall to wall, in any direction, must be an integral number J of half-wavelengths(intuitively). The cavity is a cube L long on each edge.
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Image: Standing Waves
Fig. 2 |
The number of independent standing waves G(ν)dν in
the frequency interval between ν and dν per unit volume in the cavity turned out
to be $$G(\nu)d\nu=\frac{8\pi\nu^2d\nu}{c}$$According to the
theorem of equipartition of energy, the average energy
per degree of freedom of an entity that is a member of a system of such entities in thermal equilibrium at the temperature T is$$Eav=\frac{1kT}{2}$$ where k=1.38 X 10^-23J/K is the Boltzmann Constant.
In standing waves there are two degrees of freedom, hence the total average energy is just kT.
The total energy per unit volume therefore comes to be: $$u(\nu)d\nu=Eav G(\nu)d\nu=\frac{8\pi kT}{c^3}\nu^2d\nu$$
This relation gives us the energy density for frequencies between v and dv. We came to this conclusion solely relying on the laws of classical physics. But, at a glance, this equation is inconsistent with frequencies of the range of the ultraviolet spectrum. Since ultraviolet frequencies are very high the term v^2 will tend to infinity, hence the energy density will go to infinity when frequency is very high, which is inconsistent with the experimental results. In reality, u(v)dv→0 as v→∞, this discrepancy came to be known as the Ultraviolet Catastrophe.
Birth of Quantum Physics
In 1900 the German physicist Max Planck came up with a formula for the spectral energy density of blackbody radiation:$$u(\nu)d\nu=\frac{8\pi h}{c^3}\frac{\nu^3d\nu}{e^\frac{h \nu}{kT}-1}$$Where h=6.626X10^-34 J.s is regarded as the Plank's Constant.
Now our energy density function works like it's supposed to. If you've noticed when v→∞, u(v)dv→0, therefore for high frequency no more of the Ultraviolet Catastrophe, while for low-frequency values this equation boils down to $$u(\nu)d\nu=\frac{8\pi h}{c^3}\frac{\nu^3d\nu}{e^\frac{h \nu}{kT}-1}≅\frac{8\pi kT}{c^3}\nu^2d\nu$$Which is interestingly the Rayleigh-Jeans formula, which works as expected for larger wavelength values.
How did Planck Solve the problem?
What Planck did was to consider light not as a continuous wave but as discrete packets(now known as quantum)of energy. Whose value was taken to be hv. Therefore the average energy in the standing wave is now not kT as for a continuous system.
Conclusion:
Classical mechanics, although highly successful in explaining many phenomena in the physical universe, encountered a significant limitation in its inability to accurately predict the intensity of radiation emitted by bodies across different wavelengths. This challenge ultimately led to the emergence of quantum mechanics. Thus, while classical mechanics laid the groundwork for our understanding of the physical world, this new branch of quantum physics was going to change our perspective on the universe forever.
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