How to determine the focal length of a combination of two convex lenses separated by a distance $d$ experimentally without formula?

Forget—for a moment—that the system under discussion is two lens. Just focus on the question

How do I make an experimental measurement of the focal length of [an optical element]?

For instance, how would you measure the focal length of a single convex lens? (The problem is considerably harder if the optical element is diverging, BTW, so let's focus on the converging part of the problem to start.)

Now, this is a very basic problem and there is a well known answer: you measure the image distance when projecting a very distant object. That is, you hold the lens up near a screen and move it around until you get a crisp image of the landscape seen through the laboratory window, and the distance from the lens to the screen is is the focal length. (You may want to darken the room before starting.)

To see why this works look at the thin lens formula:¹ $$ frac{1}{f} = frac{1}{d_o} + frac{1}{d_i} ;, tag{1}$$ and notice that is $d_o gg f$ and $d_o gg d_i$ then the fraction $1/d_o$ is much smaller than either of the other fractions and we can simplify (1) to $$ frac{1}{f} approx frac{1}{d_i} $$ or even $$ f approx d_i ;.$$.

Finally, we need to pay attention to the fact that our optical element is a compound system and probably isn't "thin". As it turns out the only adjustment we need to make is defining²

• object distance as the distance from the subject to the leading element of the compound system.
• image distance as the distance from the trailing element of the compound system to the image.

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If your two-element system is diverging, then the measurement requires you to combine it with a third element that is converging. But the pattern "figuring out how to do the measurement without worrying too much about the nature of the system being measured" remains the same.

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¹ With a two (or more) element system the word "thin" probably doesn't apply anymore, but we can fudge it by choosing the right definitions for object distance and image distance.

² Question for the student: how do the rules for treating multi-element systems imply that these are the natural definitions of $d_o$ and $d_i$?