Black body Radiation and the Ultraviolet Catastrophy (1 Viewer)

[shredinator]

Can someone please explain to me these concepts and how E = hf explained the observed black body curve. I am extremely confused.

thanks.

 

[helper]

Think of it in terms of filling containers.

The equation E=hf determines the size of the container.
You are going to more likely be able to fill a series of small containers rather than a few large containers. This means that you don't produce many large containers of energy, so the curve drops at the UV end of the spectrum.

 

[shredinator]

What calculations suggested the ultraviolet catastrophy in the first place?

Why was E= hf necessary to justify the observations?

Thanks

 

[helper]

The original models were based on classical wave theories.
The most common was the Rayleigh-Jeans law. These had the radiation being emitted increasing continuosly as you increase the frequency.
http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html#c4

It wasn't so much the equation E=hf that was important but rather the energy was being emitted in packets. This meant the shape of the curve was dependent on the number of each type of packet.

Because of the probability of filling large packets being unlikely, then there would be few of them. For the extremely large packets there wouldn't be enough energy available so they wouldn't be emitted at all. This explained why the curve dropped at high frequency.

 

[jlnWind]

What calculations suggested the ultraviolet catastrophy in the first place?

Why was E= hf necessary to justify the observations?

Thanks

You do not need to know about the calculations used to justify the ultraviolet catastrophe but you can mention it as the reason why classical physics was inadequate in explaining blackbody radiation.

E=hf is a simplified version of what Planck derived. But the key to E=hf is firstly
the constant which he derived, and secondly the fact that Energy is proportional to Frequency. Classical physics has always assumed that energy was proportional to the intensity, and that energy was emitted/ absorbed (as such in blackbody radiation) continuosly, as one wavelength. However the introduction of E=hf, by Planck implied that energy consisted of "bundles" or "bursts" of energy known as photons/ quanta.

The reason why i say implied is because Planck did not necessarily believe the implications of his equation, he believed that it was a mathematical treatment of blackbody curves which simply worked, so yes it is necessary. However he didn't like the idea of energy being quantised, and it was Einstein who clarified and verified the quantum theory through his work in the Photoelectric Effect.

 

[shredinator]

THanks alot! got it.

 

[jlnWind]

THanks alot! got it.

Glad i could help in some menial way

 

[Felix Jones]

What calculations suggested the ultraviolet catastrophy in the first place?

Why was E= hf necessary to justify the observations?

Thanks

from the wave equations.....intensity in inversely proportion to frequency....this means that as frequency gets very small ( e.g. for ultraviolet light) the intensity would be very very high....as frequency approached 0, intensity would approach infinity.


this clearly violates the law of conservation of energy...at this point Planck comes along......

 

[kooltrainer]

this means that as frequency gets very small ( e.g. for ultraviolet light) the intensity would be very very high....as frequency approached 0, intensity would approach infinity.

as frequency gets very high, wavelength would be small, intensity would approach infinite

 

[Captain Gh3y]

You do not need to know about the calculations used to justify the ultraviolet catastrophe but you can mention it as the reason why classical physics was inadequate in explaining blackbody radiation.

True, but they're interesting.

In high school I never saw any link between the blackbody graph and E = hf, they just told us to believe it. So here's the full story:

Consider an ideal reflecting metal box with a small hole. The inner walls contain charges in constant thermal motion. The oscillators (i.e. electrons) emit radiation at all frequencies. The goal is to calculate the spectral energy density. What is that? Well, the graph you've all seen is just spectral energy density as a function of wavelength and temperature.

To work out the spectral energy density in a blackbody for a frequency range between f and f+df, you need two things:

1. The number of standing waves inside the cavity, dN(f) and
2. E(f), the average energy of a wave of frequency f. The E(f) should have a line over it to indicate it's an average but I can't do that on BoS.

The total energy in a range of frequencies f+df is then given by the product
dU(f) = E(f).dN(f)

Classically and in Planck's theory, dN(f) is worked out exactly the same way from a geometrical argument I won't write down here.

The difference came when they worked out the average energy of a given wave inside the blackbody.

In a system of distinct objects (oscillators in the blackbody walls), the probability that a given oscillator has some energy E is
P(E) = e^(-E/kT)
Where k is Boltzmann's constant, and T is temperature. That's nothing special, it just comes from simple thermal physics (which sadly isn't in the hsc). We get the average energy by multiplying one possible value E times the probability of obtaining that value for E, and then sum over all possible values, and divide by the total probability:

E = sum(E*P(E))/sum(P(E)) = sum(E*e^(-E/kT))/sum(e^(-E/kT)

Understanding exactly what that means (it's just a simple statistical argument) is less important than seeing the difference between the classical result and what Planck did:

The classical formulation finishes by taking the integral (i.e. continuous values from zero to infinity)of the above.
Planck substituted in nhf for each "E" in the formula, then he took the sum for each n from zero to infinity (i.e. discrete values) instead of an integral.

After doing the maths for both cases, classically the average energy of a wave E(f) is just kT. In Planck's formulation you get E(f) = hf/(e^(hf/kT)-1)

Classically, frequency is not a factor; any wave might have any energy in a continuum from zero to infinity. Planck's claim was that electromagnetic radiation of frequency f is allowed only energies given by E = nhf.

So the classical result says that the likelihood of a high frequency wave is the same as that of a low frequency wave. Planck says it becomes increasingly unlikely to find waves of higher frequencies.

That's why classical predictions go off to infinity on the left hand side (high frequency) of the blackbody graph, while Planck's formula gets the correct curve, which falls away to zero as frequency gets big.