How to understand Black-Body Curves and get useful information from them?

Question

Here I have got a black-body curve. It shows the amount of radiation released as the wavelength varies for different temperatures. But I don't seem to understand it much.

1. How do we properly study a black-body curve (like this one) and draw different conclusions about the body ?
2. Why after some wavelength, radiation decreases slowly (the low slope area) and why at lower wavelength the radiation decreases quickly (the high slope area) ?
3. Also, how the shape of the graph would change as the temperature changes ?

Answer 1

1. I'm sorry, but you interpret these graphs like any other graphs: first make sure you know what 'spectral radiance' and 'wavelength' mean, and what these terms are being applied to: I'd recommend that you take a black body to be a small hole in the wall of a cavity with walls at uniform temperature. So the radiation emerging from the hole is a sample of the radiation in the cavity.

   Now consider the shapes of the graphs, noting features such as the peaks, and the shifting of the peaks to lower wavelengths at higher temperatures, and the steep rise in the heights of the graphs with increasing temperature. Visualise what these features mean in terms of the radiation emitted at different wavelengths. As an example, note that there is less blue than red light emitted at 3000 K, but that the balance is much more even at 5000 K.

   You will probably be curious to know the exact rules governing these features. I recommend that you look up the Stefan-Boltzmann law that tells you how the area under the graph (representing total radiance over all wavelengths) varies with temperature, and Wien's displacement law, telling you how the wavelength of the peak of the curve depends on temperature.

   Both these relationships (and others) can be deduced from the Planck radiation law. This is the equation for the curve itself.

2. To understand why the curve has the shape it has, you need to understand the derivation of the Planck law. It cannot be explained both quickly and clearly without some knowledge of statistical mechanics, but here is a hand-waving attempt at the impossible... The cavity I've mentioned will be full of e-m radiation of many wavelengths. The shorter the wavelength the more modes of standing wave there will be per unit wavelength interval. But shorter wavelengths have a greater minimum energy per mode, without which that mode can't exist (Planck's quantum hypothesis). So although the number of modes per unit wavelength interval increases indefinitely with decreasing wavelength, it becomes increasingly unlikely that any such mode will get any energy, because each such mode would take such a big fraction of the available energy. That's why the graph goes down on the left. It goes down on the right because there are increasingly few long wavelength modes of standing wave.

3. You need to refer to the three laws that I have already named.

Answer 2

1. The main thing to notice is that the peak moves to the left as the temperature increases, the peak of the $3000K$ curve is above 0.9 micrometers, but the peak of the $5000K$ curve is above about 0.6 micrometers, so by looking at the position of the peak the temperature of the body can be deduced.

2. You should look up the formula for the 'Planck black body distribution', but the derivation involves quantum theory and is complicated.

3. The shape is similar as the temperature changes, but the change mentioned in 1. happens and also the total power emitted is higher as temperature increases, so the peak (and the whole curve) becomes higher.