Relationship between f-number and amount of light being let into the lens system [duplicate]

Question

Why does smaller f-number imply greater aperture diameter and consequently, greater amount of light entering the lens of a camera?

As evidenced from my study across a number of resources, an absolute statement is:

A smaller f-number means greater amount of light being allowed in, and vice-versa.

The f-number is, after all, the ratio of a lens's focal length to the diameter of the aperture. Now, if the aperture diameter is kept constant and the focal length is just decreased, that would, in turn, reduce the f-number too.

Yet, in this case, the lens's f-number is reduced, which as per theory, should allow greater amount of light into the system (as evidenced by all theory I've read stating that smaller f-numbers mean greater light being allowed in).

But, ultimately, I believe the amount of light being let in is controlled by the diameter of the lens aperture that does remain constant.

Can someone explain this concept to me?

Answer 1

The f-number is used precisely because what counts is the f/D ratio. Assume you are shooting a grey wall. If with F=50mm you get all the photons reflected towards your front lens from a circle on the wall which is 20" in diameter, with f=100mm you will get the photons from an area which is 10" in diameter, so you get 4 times less photons. So the light you get on the sensor is proportional to the area of the lens (square of diameter) and inversely proportional to the are you see through the lens focal length (square of focal length). In other words if f²/D² is constant (which is the same as saying that f/D is constant) you get the same amount of light through.

Answer 2

Indeed.

If the aperture opening is constant, and you decrease the focal length, the F-number is reduced, meaning increased exposure.

Where does that exposure come from? The lens with its reduced focal length collects light from a wider angle. Thus, more of the light reaching the lens actually reach the sensor and will not be blocked by the lens barrel inner wall.

The f-number is just a handy way of calculating the total exposure. For some other things like background blur, it may be more beneficial to express the aperture opening in millimeters rather than expressing the f-number.

F-number is also dependent on sensor size: a 1.6x crop sensor requires 1/1.6 times the F-number, and 2.56 times higher ISO for equivalent image (same depth of field, same exposure, same level of noise, same background blur, etc).

Well, some could say use the same F-number and same ISO for crop and full frame. Then, the full frame has less noise, shallower depth of field, and more background blur. So full frame and crop would not be equivalent in this case with same F-number and same ISO.