Question

Two thin lenses of focal length f₁ and f₂ are placed in contact with each other coaxially. Prove that the focal length f of the combination is given by f = `(f_1 f_2)/(f_1 + f_2)`.

Answer 1

For thin lenses, the image formed by the first lens acts as the object for the second lens.

Use the lens formula:

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

Let object distance = u, image formed by first lens = v₁.

`1/v_1 - 1/u = 1/f_1`    ...(i)

Since lenses are in contact, the image of the first lens becomes the object for the second lens. So object distance for the second lens = v₁.

Let the final image distance = v

`1/v - 1/v_1 = 1/f_2`    ...(ii)

By adding equations (i) and (ii), we get,

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

Cancel `1/v_1`:

`1/v - 1/u = 1/f_1 + 1/f_2`

For equivalent single lens:

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

Comparing:

`1/f = 1/f_1 + 1/f_2`

`1/f = (f_1 + f_2)/(f_1 f_2)`

Taking reciprocal:

f = `(f_1 f_2)/(f_1 + f_2)`