Nigel Wood Photography - cobwood studio

Magnification vs Focal Length

This article discusses the magnification, i.e. relative size, of an image to the subject and the implications of using a 'cropped' sensor.

magnification

The diagram above depicts a tower of height 'm' at distance 's' from the lens, which has a focal length of 'f'. An image of height 'n' forms on the sensor at a distance 'v' behind the lens.

From the thin lens equation,

equation 1

By similar triangles,

equation 2

Substituting for 1/v from (1) into (2) and simplifying,

equation 4

Hence,

equation 5

The term on the left (n/m) is the ratio of the size of the image to the size of the tower and is therefore the magnification. Now, for normal photography well outside the macro range, the distance to the subject 's' is very much bigger than the focal length, i.e. s>>f. So in this range, the magnification of the image is much less than 1 and the magnification is very nearly proportional to the focal length.

Effect of reducing sensor size

Palma Castle
Cropped sensor overlaid on full-frame sensor - image taken at 30 mm

The image above of Palma Castle was taken with a full-frame DSLR with a sensor of 36 by 24 mm. Imagine that we are looking at the image on the sensor itself. For comparison, the white rectangle represents the size of a typical APS-C sensor of 24 by 16 mm.

If we were to photograph the same scene from the same distance with the same lens but with an APS-C camera, the sensor would record the area inside the white rectangle and produce the image below:

cropped
Scene as taken by an APS-C camera also at 30 mm

Crop factor

A sensor smaller than the 'full-frame' standard of 36 by 24 mm, such as the APS-C sensor in this example, is called a 'cropped sensor'. The 'crop factor' is then the ratio of the size of the sensor to the size of a full-frame standard sensor. In the case of this APS-C sensor, its width is 24 mm compared with the full-frame width of 36 mm, i.e. a ratio of 1 to 1.5 and so its crop factor is 1.5

When comparing lenses used with cropped sensors, we often want to refer to the equivalent lens on full-frame; equivalent meaning having the same field of view. So if we are discussing a particular lens on an APS-C camera, the question arises, "what lens would have the same field of view on a full-frame camera?"

Looking back at the images above, the top one was taken with a 30 mm lens on a full-frame camera, while the bottom picture shows the scene as taken by an APS-C camera with the same lens. The question is then, what lens on the full-frame camera would reproduce the scene in the lower picture?

Looking at the upper picture we can see that the image needs to be magnified by a factor of 1.5 so that the area in the white rectangle expands to cover the whole sensor. That factor is of course the same 'crop factor' we have just described above.

In the mathematics above we concluded that at these distances magnification is effectively proportional to the focal length. So to magnify the upper image by 1.5, we need to zoom the lens in by a similar ratio, i.e. from 30 mm to a focal length of 45 mm.

In general we can conclude that if we have a camera with a cropped sensor and are using a lens of focal length 'f', we will have a field of view equivalent to using a lens of 'f' multiplied by the 'crop factor' on a full-frame camera.

For example, the table below shows lenses for the Micro Four Thirds system (crop factor = 2) and the APS-C cameras that would be equivalent to the 'standard' 50 mm lens on a full-frame camera.

Micro 4/3
(crop factor 2)
APS-C (Nikon)
(crop factor 1.5)
Full frame
25 mm 33 mm 50 mm

Note that Canon APS-C cameras have a slightly smaller sensor than Nikon's and have a crop factor of 1.6

For further discussion of equivalence of sensor formats see equivalence and for the effect of sensor formats on depth of field see: depth of field