Question

Two thin converging lenses of focal lengths f₁ and f₂ are placed coaxially in contact. Derive expression for focal length of combination.

Answer 1

Two thin lenses, A and B (focal lengths f₁, f₂), are placed in contact with a common optical center P.

Lens A forms an image (I₁) of the object O.

This image I₁ acts as a virtual object for the second lens B.

Lens B then forms the final image at I.

Using the lens formula:

`1/v - 1/u = 1/f`

For lens A, the object distance is u, and the image distance is v₁.

`1/v_1 - 1/u = 1/f_1`   ...(1)

For the second lens, the image I₁ produced by the first lens serves as a virtual object. For lens B, the object distance is v_1, and the final image distance is v:

`1/v - 1/v_1 = 1/f_2`   ...(2) 

Adding equation (1) and equation (2):

`(1/v_1 - 1/u) + (1/v - 1/v_1) = 1/f_1 + 1/f_2`

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

If the combination is replaced by a single lens of focal length F hat forms an image of the same object at the same position, then:

`1/v - 1/u = 1/F`

Comparing this with equation (3), we get:

`1/F = 1/f_1 + 1/f_2`

`1/F = (f_1 + f_2)/(f_1f_2)`

F = `(f_1f_2)/(f_1 + f_2)`

In terms of power, the equation can be written as

`P = 1/f_1 + 1/f_2   (∵ P = 1/f)`