The Thin Lens Equation

Our first step is to determine the conditions for which a lens will produce an image, be it real or virtual. For an image to be produced, all rays at the image plane which come from one particular point on the object must pass through one corresponding particular point in the image plane. Two special such rays are shown in figure 3.

Figure 3. Diagram of image formation for the derivation of the thin lens formula. Part (a) is for the most often encountered case, part (b) is the same, but with an incoming converging pencil of light forming an apparent object to the right of the lens. This may occur when a lens (not shown above) might form an image at l in part b. This image formed by the undrawn lens is the "object" for the lens shown in part b to work on. The physics is the same in both cases, the latter diagram is easier for understanding signs of quantities.

We shall now work on these diagrams using the Cartesian Sign Convention.

From similar triangles we see y'/l' = y/l,

so the linear magnification is y'/y = l '/l.

The other pair of similar triangles gives y'/(f '- l') = y /f '.

Combining the last two equations to get rid of y' and y gives us the thin lens equation:-

1/l' - 1/l = 1/f '

Note - due to variations in font renderings, that the first three equations above do not contain the number one. The thin lens equation immediately above is one over image distance minus one over the object distance.

This important equations holds pretty well as long as

• i) the lens is "thin",
• ii) the paraxial approximation is appropriate, and
• iii) the refractive of the medium of the surrounds is the same on both sides of the lens.

Return to Lecture summary page

Press the Back button to return to the last page you were viewing