Before we dive into the math of radiometry, let's get comfortable with some of the common light sources we might observe and that will help us understand why we're doing all of this. Probably the deemed most important light source to know is something called the blackbody source. It seems very weird to call a light source black, and the reason is when you take any object and you put her to uniform temperature, it radiates ideally with a spectrum that depends on that temperature, and you can tell it's pretty darn warm, about 700 degrees K, all of that radiation is off in the infrared, below what the human eye can see. So it's radiating, but it appears black. So that's the origin of the term. The sun is a pretty close approximation of a blackbody radiator. It's darn warm, around 6000 degrees K at the surface. That radiates out in the yellow, we happen to know, and that gives you your first signpost of how warm you have to be. If you have an electric burner on your stove or you heat metal up and it starts to glow red and then blue, that's the blackbody spectrum happening. Humans, at body temperature, are also black and that is they don't radiate in the visible because that would look funny and we would know that. They radiate at around 10 microns and that's relevant, for example, if you wanted to make a passive night viewing system to see humans, you'd need lenses and detectors that could work into that infrared part of the spectrum. We'll define this term here in a minute, but they are also ideal in the sense that the light coming off a blackbody source, and I'm speaking vaguely here because we're about to be more precise in this soon, but the radiance does not depend on angle. You already saw this with your white piece of paper; when you tilted that paper and looked at it sideways, it didn't look massively brighter. That paper was as a matter of fact not a bad blackbody source, and indeed you saw that the radiation didn't seem to depend too much on angle. We'll come back and define that here in a minute. I've given you kind of some of the standard equations that are nice to know for blackbody radiation. One wouldn't be expected to carry these around in your head but they're nice to have. The most important is what is the radiance (watts per meter squared per steradian) as a function of the temperature and the wavelength. So if you can put any particular temperature in here, then you can trace out what the spectrum looks like, and I plotted that over here for a handful of interesting temperatures from about 4000 to 7000 degrees K. You see in general that as you increase the temperature, the spectrum moves out in frequency or shorter in wavelength. That moves it from the infrared down into the visible, and it also radiates more. It gets brighter and that makes sense because there's more thermal energy there. The deviation of observation from this equation is actually part of what led to quantum mechanics because a fully classical description of a blackbody radiator, it blows up. There's an infinity predicted in the spectrum, and this was one of the little niggling details that was being patched up at the end of the 19th century to finish classical physics and it didn't work. Planck found that if he discretized the radiators, he made quantum radiators, then he could get an equation which described the real world. That was one of the several initial hints that classical physics wasn't quite good enough. Besides just knowing what the shape of the spectrum is, it's nice to know where the maximum is because often that describes the color, let's say of the sun. So this is just taking the derivative of that equation and setting it equal to zero. You can find where the maximum temperature is and for example here is the sun at 5500 degrees K radiating in the green and that makes sense. As a matter of fact, you will hear the word color temperature if you work around light much. A color temperature is if you take any light source, an incandescent light bulb, and you describe generally what color it appears to peek at, or what color it appears to be visually, you can then turn that into a temperature through this law right here. So it's common to describe light sources as equivalent to a blackbody of a particular temperature and that's the color temperature of a light source.
