Why F Number?

I feel myself lucky to have friends who are genuinely interested in photography. Today Sudeep (Sudeep Sarkar) asked me a very interesting question. About which I had never given thought. Why F numbers are used instead of aperture diameter to standardise measurement of aperture?

A very interesting question. So, after giving some thought, I derived the solution and here it is.

Suppose there is an object of intensity L at a certain distance R from you. Say at night you see a train with it's headlight on. As the train approaches you or you approach the train, the brightness increases. Similarly, if we move further away from the train, the brightness decreases.

This intensity that we feel is inversely proportional to the square of the distance R. This is as per inverse square law.

L∝1/ R²

Now focal length acts as this distance R. When focal length is increased light has to travel more and it's intensity decreases.

L∝ A

Where A is the area of the aperture. Obviously, larger the area, more will be the light passing through and hence, higher intensity.

So, we can safely derive that

L∝ {A/R²}

or

L = K × {A/R²} where K is a constant

Exposure Value E is a function of ISO, shutter speed and amount of light entering.

Earlier we derived that amount of light entering is dependent on two factors. The area of the aperture and the distance the light has to travel.

Therefore, the area of the aperture alone cannot define the amount of light passing through. And this is exactly the reason why area or radius or diameter (since area, radius and diameter are directly dependent on each other) alone cannot be used as a standard as far as calculation of amount of light passing through is concerned.

This is where F number comes to rescue. F number is the ratio of the focal length to the aperture diameter.

A particular F number will allow similar amount of light to pass through itself irrespective of focal length and the area (or radius or diameter) of the aperture.

We will give examples for both in order to provide further clarity.

For our understanding, we will maintain the following assumptions.

Two lenses of focal length 50 mm and 200 mm.

Aperture diameter of 25 mm.

ISO is kept constant as I

Shutter speed is kept constant as S

Exposure E is a function of ISO, Shutter speed and amount of light i.e. the intensity of light as a function of aperture diameter.

So, if can be written as

E = f (I,S,L)

Since I and S are constant in both the cases, we can rewrite the equation as

E= _f _(L)

For 200 mm lens

E = f {(K* Π 2525)/(4200200)}

Since K and Π are constant

E = P * 0.0039 where P is some constant derived from the equation consisting of I,S,K and Π

For 50 mm lens

E= P*0.0625 where value of P is same as in the previous equation.

Clearly, both the values are different.

However, if we use F number instead of aperture diameter the results will be interesting.

Let us assume F number is F 2.

This means, for the 200 mm lens,

F= focal length/diameter of aperture

or diameter of the aperture = 200/2 = 100 mm

for 50 mm lens with F number F 2

diameter of the aperture = 50/2 = 25 mm

Putting these values in the equation of Exposure, we can find

E= _f{(_KΠ100100)/(4200*200)}

E= P * 0.0625 for 200 mm lens

&

E= P * 0.0625 for 50 mm lens

Both the exposure values are same!

Clearly F number is a better and simpler standardisation!