As the lens was not changed, the F-number of the lens still must be the same. The darker image and the reduced field of view are described by the working F-number, also called effective F-number.

The F-number is only defined for objects at infinite distance.

The sensor has according the definition of the focal length exactly the distance “focal loength” from the principal plane when the lens is focussed to infinity.

When an object comes closer, it’s image moves in the same direction. It moves further away from the principal plane.

To keep the object focussed, the distance of lens and sensor must be increased. This is equivalent with adding distance rings reagrding the infinity position of the lens. As result the images gets darker the field of view gets smaller , the therefore the magnification changes.

It holds:

where

working F-number or effective F-number.

image side numerical apertur,

the magnification for this focus distance and

is the pupil magnification.

Especially:

The effective F-number can also be described by the height “delta” of the inserted distance rings:

and

From (1) we get

with this we get from (2) :

From this we can derive a third way to calculate the working F# :

With the length of the needed distance rings delta

we get

or shorter

q.e.d.

see also:

F-Number

Pupil magnification

Numerical Aperture