## Sunday, February 24, 2008

### Basic Lens Equation

I find myself using the basic lens equation quite a bit so I would like to derive some important formulas. Consider a lens of focal length f. The object is at distance I from the lens, while the image is formed at distance I'. The size of the object is s, the size of the image is s'. See the diagram here.

The basic lens equation is: 1/f = 1/I + 1/I' (1)

The magnification by definition is M = S'/S = I'/ I (2)

If we use equation (2) to solve for either I or I' and substitute it in equation (1), we obtain these two useful formulas:

M = f/x (3) and M = x'/f (4)

From (3) and (4) we can write (1) as: f**2 = x x'

If the subject is far away from the lens (low magnification) then I >> f and I = x, I' = f, so the magnification is approximately equal to M = f/I. This is the low magnification approximation.

At high magnifications I gets close to f, and I' gets very large, so I' = x' and M = I'/f. this is the high magnification approximation.

An interesting situation occurs at M = 1, then x = x' = f, and the subject is at distance 2f from the lens and the image is formed at distance 2f from the lens. In this case the total distance from the object to the film plane is the smallest possible (4f).