How f-numbers works?

The actual f-numbers themselves denote the number of times the effective diameter of the aperture divides into the lens focal length. So f/2 means setting an aperture diameter one-half the focal length; f/4 is one-quarter, and so on. The system works because each f-number takes into account two main factors which control how bright an image is formed:

Distance between lens and image. Doubling the distance of a surface from a light-source quarters the light it receives. And since distant subjects are focused one focal length from the lens, a lens of, say, 100 mm focal length basically forms an image only one-quarter as bright as a lens of 50 mm.
Diameter of the light beam. Doubling the diameter of a circle increases its area four times. So if the diaphragm of the first lens passes a beam of light 12 mm wide and the second only 6 mm wide, the first image is four times as bright as the second.

Now if you express (2) as a fraction of (1), you find that both lenses are working at near-enough relative apertures of f/8 (100 ÷ 12, and 50 ÷ 6), which is correct since their images match in brightness. So: f-number = lens focal length ÷ effective aperture diameter

In practice the f-number relationship to brightness breaks down when working very close up, because the lens-to-image distance will then differ greatly from one focal length. The f-number settings are also occasionally referred to as 'stops'. In early photography, long before iris diaphragms, each stop was a thin piece of metal punched with a hole the required size which you slipped into a slot in the lens barrel. Hence photographers speak of 'stopping down' (changing to a smaller opening, higher f-number). The opposite action is 'opening up'.

You will find in practice that upper and lower limits of the f-number scale vary with different lenses. Most small-format camera lenses stop down to f/16 or f/22. Larger, sheet-film camera lenses are designed to continue down to f/32 or f/45. Smaller apertures are useful for extra depth of field, see below, but if taken to extremes diffraction starts to destroy image detail. This is why no lens will stop down literally to pinhole size.

At the other end of the f-number scale, limits are set by price and the current state of technology. The wider the maximum relative aperture setting the more difficult it is for the manufacturer to suppress aberrations. The lens must also be bigger, and costs more. But then a wide-aperture lens passes more light (it is 'fast') and this is handy in dim conditions – for photo-journalism for example. Apertures of f/1.4 are quite common on standard non-zoom lenses for small-format cameras. Often a lens design may produce acceptable image quality up to a relative aperture wider than, say, f/2 but not as wide as the next f-number, f/1.4. A maximum setting of f/1.8 or some other irregular f-number will then appear as the limiting factor on its scale. Most large-format camera lenses only open to about f/4 at the most. In fact, the 'best' aperture with most lenses is around f/8, being a compromise between the opposite influences of lens aberrations and diffraction.

The f-number of your lens' maximum aperture, together with its focal length, name and individual reference number, are engraved on the lens rim. You may find that, of two lenses identical in make and focal length, one is almost twice the cost of the other because it has a maximum aperture one stop wider. This can be a high price to pay for the ability to shoot in poorer light or use faster shutter speeds, particularly when you can buy excellent ultra-fast film.

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