Question

Define power of a lens. Write its units. Deduce the relation `1/f =1/f_1  +1/f_2`for two thin lenses kept in contact coaxially.

Answer 1

Consider two lenses A and B of focal length f₁ and f₂ placed in contact with each other. An object is placed at a point O beyond the focus of the first lens A. The first lens produces an image at I₁ (real image), which serves as a virtual object for the second lens B, producing the final image at I.

Since the lenses are thin, we assume the optical centers (P) of the lenses to be co-incident. For the image formed by the first lens A, we obtain

 `1/v_1 -1/u  =1/f_1    ........ (1)`

For the image formed by the second lens B, we obtain

 `1/v -1/v_1  =1/f_2   ........ (2)`

Adding equations (1) and (2), we obtain

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

If the two lens system is regarded as equivalent to a single lens of focal length f, we have

 `1/v -1/u  =1/f_1   ........ (4)`

From equations (3) and (4), we obtain

`1/f_1  + 1/f_2  =1/f_1 `