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Depth of Field Focal Length

To study the relationship between depth of field depending on the lens focal length, we must rely on a couple of things we already know, namely

a) The size of circle of confusion depends exclusively on the size of the sensor.
(See the article on the Circle of Confusion)
b) The thin lens equation can be expressed as
q = pf / (p - f).
where p is the distance from the objective focal plane.
f is the lens focal length.
q is the distance the image plane, corresponding to the focal plane, the target.
(See: Thin Lenses)
From the knowledge of these facts, calculate the value of q for two different situations concrete ..
In the first, for a 35-mm focal length, get the value of q for two different focal planes, one located 100 m (100000 mm.) And another located 10 m (10000 mm.).
q1 = (100000. 35) / (100000 - 35) = 35.01
q2 = (10000 .35) / (10000 - 35) = 35.12
In the second, for a focal length of 200 mm, we obtain also the value of q for the previous focus planes.
q3 = (100000. 200) / (100000 - 200) = 200.40
q4 = (10000. 200) / (10000 - 200) = 204.08
What these results mean?
Well, for the first case, all planes of focus located between 100 and 10 m levels result image very focused on a range of 35.12 - 35.01 = 0.11 mm.
In the second case, the same planes of focus locate their image planes within a range of 204.08 - 200.40 = 3.68 mm. That is, if the focal length is small (35mm) image planes are much more concentrated than if the focal length is large (200mm).
We can study what happens if we approach a point target at 22.5 m with an aperture of f5.6 and a circle of confusion of 20 microns in diameter (APS-C format).
When using the lens of 35 mm. upper and lower limits of the target area are 7.37me infinity. When we use the target of 200 mm. the same lower and upper limits are now of 21.17 and 24 m. Or what is the same as the area between 10 and 100 m is within the limits of the depth of field when using a 35 mm lens. and much of it is outside when we use the 200mm lens.
The explanation, as can be seen from the figure that appears at the beginning of the article, is that within the limits of the depth of focus that induces the circle of confusion there are many more planes in the first case in the second.
Therefore we can establish the following rule:
a) If the focal length decreases, increasing the depth of field.
b) If the focal length increases they Profunfidad Country.
(It is interesting to note in the same figure that:
If the focal length increases, the depth of focus)