# What's the difference between these two approximations for thin lenses?

I have seen the thin lens equation, which describes the following relation.

$$\frac{1}{\text{image distance}} + \frac{1}{\text{object distance}} = \frac{1}{\text{focal length}} \tag{1}$$

However, I've also seen the following relation.

$$\frac{\text{object height}}{\text{image height}} = \frac{\text{object distance}}{\text{focal length}} \tag{2}$$

I was wondering how you get to one from the other, and what exactly the difference is, in terms of what they're trying to model?

• Is the second equation even correct? Consider an object at 2f, identical image at 2f, LHS equals 1, RHS equals 2 – DJohnM May 24 '18 at 19:55
• My answer on a similar question might help: physics.stackexchange.com/a/396085/162611 – cms May 24 '18 at 20:21
• Please tell where you saw that second equation. – Pieter May 24 '18 at 20:32

A correct formula, of which #2 might be an incorrect interpretation, would be

(object height)/(image height) = (object distance)/(focal distance)

where focal distance is the distance from the lens to the image.

What's different between Formula 1 and the correct Formula 2 is that Formula 1 calculates the image position from the object position, given the focal length of the lens; while correct Formula 2 calculates the magnification, given the object and image positions. The two (#1 and correct #2) can be combined in various ways to yield, e.g., the necessary focal length and lens position to obtain a given magnification with object and image at given positions.

You don't. They are derived independently, from the geometry of the ray tracing figure. The first one is a relationship between the positions of the image and object. The second one allows to calculate the magnification of the lens. You can use the first one to replace one of the distances in the second formula so you can get the magnification as a function of the position of the object (for example), for a given focal length.

• So does that mean the second equation is “missing” some information? – Carpetfizz May 24 '18 at 17:16
• No, each one contain some information about relationships between various parameters. I don't even see how saying that an equation "is missing information make sense". The equation is either right (withing the domain of conditions used to derive it) or is wrong. These two are both correct. – nasu May 24 '18 at 20:11
• The second equation is not correct. – Pieter May 24 '18 at 20:29
• You are right, Pieter. Either I did not see it or it was changed since the first post. – nasu May 25 '18 at 12:57
• Unless is some special case, with conditions unmentioned in the OP. I hope the OP will provide the origin of the formula. – nasu May 25 '18 at 13:05