# Compound lenses and virtual images

Ok so I have this problem where I have a system of two lenses. All I know is that the distance between the object and the first lens is 30cm, the distance between the object and the final image is 70cm, the focal distance of the first lens is 20 and the total magnification is -0.666. I need to find the focal distance of the second lens and the distance between lenses 1 and 2.

Now I've tried calculating the parameters of the first lens' image (it should be 60cm past the first lens and the magnification should be -2). My problem is that if I now use this image as an object for the second lens and solve the equations, I obtain absurd results (values should put the lens between the first lens and the final image but they don't).

How should I go about solving such a problem?

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Most likely, your problem is the sign of the distance from the image of the object through the first lens, to the second lens. When you use the formula for the ideal thin lens you must be really careful about the convention used.

For example, I usually use $\frac{1}{f} = \frac{1}{o} + \frac{1}{i}$, where $f$ is the lens' focal distance, $o$ is the distance from the object to the lens, and $i$ is the distance from the lens to the image. The sign convention for this formula is

letter   positive           negative
o        left of the lens   right of the lens
i        right of the lens  left of the lens
f        converging         diverging


This means that the reference system is different for $o$ and $i$, which can be confusing.

Other common error for this kind of problems is that $o$ and $i$ is the distance to the lens, so you can't use the number you get for $i$ from the first lens as the $o$ for he second lens. You need to take into account the distance between the lenses.

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A little late considering this was homework but I figured it out in time and you are spot on. Thank you very much kind sir! – pwny Nov 20 '11 at 3:08