What is Planck’s distribution?
Well, Planck’s distribution is a black body’s radiation intensity distribution with wavelength. Okay, so that’s a lot of complicated words in one sentence. So let’s try and understand what exactly that means. First of all, what’s a black body? Well, let’s say that this is a black body. It’s an object that perfectly absorbs all of the radiation falling on it. In other words, if we have a beam of light falling out of his black body, then all of that light gets absorbed by the black body and none of it gets reflected. Now, a black body — when it reaches thermal equilibrium or, in other words, it’s at the same temperature as its surroundings — will also emit some of the energy that it absorbs.
And, of course, because it’s in thermal equilibrium, the amount of energy that it emits will be the same as the amount of energy that it absorbs. So in our diagram here, the light that’s falling on the black body is the energy that’s absorbed by the black body. And the pink arrow show the energy that’s emitted by the black body. Now the fact that a black body emits the same amount of energy that it absorbs does not mean that the energy emitted must be at the same wavelength as the energy absorbed. In fact, a black body show cases are very particular emission spectrum.
So if we were to draw a graph of the intensity of radiation emitted by the black body, that’s the radiation represented by these pink arrows, against the wavelength of the emitted radiation on the horizontal axis, then we’ll see that the spectrum showing the radiation from the black body has a very particular shape. Specifically, it has a peak value. In other words, at this point, the radiation intensity is maximum. And this happens at a very particular wavelength which we’ll call 𝜆 subscript 𝑚, for the maximum intensity wavelength. In other words then, the radiation emitted by a black body is of a very specific kind where it emits most of its radiation at this wavelength and then some radiation at wavelengths either side of this peak wavelength. And we can see that for larger and larger wavelengths, the intensity tends to get smaller and smaller.
So what is the value of 𝜆 subscript 𝑚, the wavelength at which the black body emits most of its radiation? Well, that is very much dependent on the temperature of the black body. In other words, this curve that we’ve drawn is specifically for one temperature that the black body could be yet. However, we can draw another curve that shows the emission of the black body at a temperature slightly lower than the temperature for this curve. In that case, we can notice two things.
Firstly, for the lower-temperature curve, the maximum intensity is lower than for the high-temperature curve. And secondly, the peak wavelength is slightly shifted to the right. In other words, the peak wavelength has increased. So a lower temperature results in a larger peak wavelength but a lower maximum intensity. Conversely, we can draw a radiation spectrum for a very high temperature. In that case, the peak intensity is very, very high. But the wavelength at which this peak intensity occurs is lower.
Now these temperature-dependent spectra that show the radiation intensity emitted by a black body are known as Plank’s distribution. And so, as an answer to our question, we can say that Planck’s distribution is the distribution of radiation intensity with wavelength emitted by a black body.