I can't copy and paste and still keep the formulae from a word document; but my take on "useful formulae" can be downloaded
here (2 page pdf). Pasting has this effect (so if you think it might be helpful, download and you'll see the equations that I can't paste in).
3. Useful formulae for close up photography
There are two basic formulae used with lenses:
where f = the focal length
u = the distance the subject is in front of the lens
v = the distance of the plane of focus behind the lens.
Equation 1 Basic lens equation
Since lenses used in photography are thick lenses with several elements, the actual plane within the lens used for these measurements are the nodal planes, and these won’t be marked for you. But, as we’ll see, this doesn’t matter a great deal. The other basic formula relates the magnification of the subject to the distances in front of and behind the lens:
where M = the magnification and u and v are as above
Equation 2 Magnification as a function of subject and image distances
Using simple algebraic substitutions in these formulae, we can get equations to answer the most common questions in close up photography
The distance the subject will be in front of the nodal plane of the lens at a given magnification is given by
When working at 1:1, M = 1 and u = 2f
When working at 1:2, M = ½ and u = 3f
Equation 3 Subject distance as a function of magnification
We saw in chapter 9 that this formula is actually an approximation that works reasonably well until you get really close to the subject, but is increasingly in error after (say) 1:1. The exact formula should substitute
for M, where P is the pupil magnification which can be a significant size, as we saw in chapter 9 when we looked at reversing a lens.
Many lens makers specify the closest focusing distance in terms of distance from the focal plane of the camera, rather than the front of lens; this formula will give you a reasonable idea of what your working distance will really be.
If you want to calculate more exactly how far in front of the lens the subject will be, or if the camera maker only tells you how far the subject is from the lens and you want to work out what the value of M is (some compact cameras specify the closest focusing distance in this way, and leave you wondering what that means in terms of magnification) then you can use the following formula:
Equation 4 Subject distance as a function of magnification
If you know the closest focusing distance from the front of the lens, then you also know that if you add the distance from the focal plane to the front of the camera to this distance, the result will be approximately u + v (or exactly if the front and rear nodal planes of the lens coincide) which is
And knowing f, you can work out the magnification.
Exposure increases when using tubes or bellows can be worked out either from the length of the extra extension E and the focal length of the lens F using the formula
Exposure increase =
Equation 5 Exposure increase for a given extension
Or using the magnification M and the marked aperture f you can calculate the effective aperture F
Equation 6 Effective f stop and magnification