# When Two Thin Lenses Are Kept in Contact, Prove that Their Combined Or Effective Focal Length F is Given by : 1/F = 1/F_1 + 1/F_2 Where the Terms Have Their Usual Meaning. - Physics (Theory)

When two thin lenses are kept in contact, prove that their combined or effective focal length F is
given by :

1/F = 1/f_1  + 1/f_2

where the terms have their usual meaning.

#### Solution

1) The image is formed in two steps :

In the first step, the lens L1 produces the image I' of the object A. It is a real image at a distance v' from L1

Then for the lens L1 , we have,

1/(v') + 1/(-u) = 1/f_1

:. 1/(v') - 1/u = 1/f_1   ... (1)

2) In the second step, the image I' acts as an object for L2 and the final image I is formed. The image I' is not observed, hence it acts a virtual object for lens L2 it is assumed to be kept on left, we can write for L2.

1/v + 1/(-v') = 1/f_2

:. 1/v - 1/(v') = 1/f_2    ...(2)

3) Adding equation (1) and (2), we get

1/(v') - 1/u + 1/v - 1/(v') = 1/f_1  + 1/f_2

:. 1/v - 1/u = 1/f_1  + 1/f_2   ....(3)

4) If a single lens of focal length f of equivalent lens is used for object A and corresponding image I is formed then we have,

1/v + /(-u) = /1/f     ...(4)

5) From equations (3) and (4),

1/f = 1/f_1  + 1/f_2    ...(5)

Concept: Refraction at Spherical Surfaces and by Lenses - Combination of Thin Lenses in Contact
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