A convex lens of focal length `f_(1)` is kept in contact with another lens of focal length/2. Find an expression for the focal length of the combination.

Explanation

Consider two thin lenses A and B of focal lengths `f_(1)` and `f_(2)` placed in contact. Let a point object be placed at O, beyond the focus of first lens A. Lens A forms a real image at `I_(1)`. This image serves as a virtual object for second lens B and the final real image is formed at I.

For image `I_(1)` formed by first lens, we have
`1/v_(1) -1/u =1/f_(1)`
and for the image I formed by the second lens, we have
`1/v - 1/v_(1) = 1/f_(2)`...(ii)
Adding (i) and (ii), we have
`1/v - 1/u =1/f_(1) + 1/f_(2)`...(iii)
If the two lens system is considered as equivalent to a single lens of focal length f, then
`1/v -1/u = 1/f`
Comparing (iii) and (iv), we find that : `1/f = 1/f_(1) + 1/f_(2)`