The f/number of a lens is a way to help standardize the function of various lenses. The f/number of a lens helps to determine the exposure needed, the depth of field and the potential maximum resolution of a lens. All of these factors tie directly into the pupils of the lens and the focal length.
The f/number of a lens is the focal length divided by the entrance pupil size. the 200 mm lens in the previous posting has an entrance pupil 50 mm in diameter. that makes the lens a maximum of f/4. A 100 mm f/4 lens would have an entrance pupil of 25 mm in diameter.
f/number = f/E (f = focal length,E = entrance pupil diameter)
The f-number of a lens is only applicable at infinity focus. Two f/4 lenses will allow about the same amount of light to hit the detector at the same shutter speed, regardless of the focal length. They will not always be equivalent because two lenses will have a different amount of light loss as the light travels through the lens. Light loss in a lens is dependent upon the number of elements and various lens coatings in each lens.
At infinity focus the detector will be one focal length away from the rear principal plane of the lens. A 200 mm lens will be twice as far away as a 100 mm lens but the entrance pupil is proportionally larger. This will result in the aperture for the two lenses being the same size in relation to the detector. The angle that the edges of the aperture makes with the detector will be the same.
You may have noticed that we haven't talked about the exit pupil of the lens, only the entrance pupil. This is because at infinity focus, the relative size and distance of the exit pupil will be constant regardless of the pupillary magnification and won't effect the numbers. At infinity focus, an f/4 lens at the entrance pupil will be f/4 at the exit pupil also. This will not be true for closer focus. This issue will be taken up separately in the next installment.
If I were to stop down the lens to f/8 on my 200 mm lens the entrance pupil will be 25 mm (=200/8)- half of that at f/4. The area of the pupil will be 1/4 that of f/4 and it will let in 1/4 as much light and will require a shutter speed of 4x as long to get the same amount of light to the detector (for a similar exposure). The stop half-way between f/4 and f/8 will let in half as much light. That would be f/5.6 (not f/6, halfway between the two but the 1 over the square root of 2 between them).
The main stops you will see on a lens will be f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32... Each stop will allow half of the light of the next lower number and will lengthen the shutter speed by 2x to get a similar exposure at the detector.