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I have been given an equiconvex lens (of given focal length), placed on a plane mirror, with water between the lens and the mirror.

An object has been placed at a certain distance and I have been asked the final image of the object. I calculated the radius of curvature using the lens maker's formula. I proceeded to solve the problem applying the given formula for every interface:

$$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \ .$$

I.e., first using this for the air glass interface, then the glass water interface and finally reversing the sign of the image distance obtained, to account for the image formation by the plane mirror.

However, in the solution, it considers the system as a combination of an equiconvex glass lens and plano concave water lens.

This seemed equally justified. The answers from my approach was significantly different from this approach. What was wrong about my approach?

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Your solution does not involve an equiconvex lens. You've solved the problem for the case of a biconvex lens of differing radii of curvature, such that the focal length of one surface when in contact with air is the same as the focal length of the other surface when in contact with water. An equiconvex lens should have two identical radii of curvature. The focal lengths of the two spherical interfaces are different when such a lens is placed in the context of the original problem.

When a lens is specified by its focal length, the surrounding medium needs to be specified. Commonly, focal lengths are specified for lenses in air or vacuum, and this may be left implicit. This seems to be the case here. The "given focal length" of the lens in your problem is the focal length of the lens in air. The official solution is thus using a different lens geometry than your solution (in particular, the equiconvex lens is actually equiconvex!). The solution introduces a zero-thickness layer of air between the lens and the water is so that the given focal length value for the lens, which applies only when the lens is surrounded by air, can be used directly. Similarly, the second lens formed by the water is analyzed as if it were in air, even though physically it's up against glass.

You can also solve the problem in the spirit of your solution by directly handling the two spherical interfaces. To do it correctly, you have to use the lensmaker's equation where both sides are in air to compute the two (equal!) radii of curvature of the lens from the given focal length value. That is, you must apply the lensmaker's equation to the standard situation for which the lens's focal length is specified, not to the problem system. Then you can compute focal lengths for the interfaces in the problem and proceed from there.

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    $\begingroup$ Thank you so much for taking out your time to answer my question. I assumed the two surfaces have the same focal length(despite the medium) and directly applied lens maker formula and hence assumed equal radius of curvature so the formula I used could not give the correct solution. That is where I went wrong I think. $\endgroup$
    – Emmy Noether
    Commented Aug 9 at 12:14

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