Derive the equation for effective focal length for lenses in contact.

Explanation

(i) Consider two lenses (1) and (2) of focal length `f_1` and `f_2` are placed coaxially in contact with each other so that they have a common principal axis.
(ii)For an object placed at O beyond the focus of the first lens (1) on the principal axis, an image is formed by it at I'.
(iii) This image I'acts as an object for the second lens (2) and the final image is formed at I.
(iv) As these two lenses are thin, the measurements are done with respect to the common optical centre P in the middle of he two lenses.
(v) Let, PO be object distance u and PI' be the image distance (v') for the first lens (1) and object distance for the second lens (2) and PI = v be the image distance for the second lens (2). Writing the lens equation for first lens (1),
` (1)/(v') - (1)/(u) = (1)/(f_1) " " ....(1)`
Writing the lens equation for second lens (2),
` 1/v - (1)/(u') = (1)/(f_2) " " ....(2)`
Adding the above two equations (1) and (2),
` 1/v - 1/u = (1)/(f_1) + (1)/(f_2) " " ....(3)`
(vi) If the combination acts as a single lens of focal length f so that for an object at the position O it forms the image at I then,
` 1/v - 1/u = 1/F` ...(4)
Comparing equations (3) and (4) we can write,
` 1/F = (1)/(f_1) + (1)/(f_2)`