Question

find out expression for equivalent focal length of thin lenses combination​

Answer 1

consider two lenses A and B of focal length $f_1$ and $f_2$ 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 $I_1$, which serves as a virtual object for the 2nd lens B, producing the final image at I.

for the image formed by the first lens A, at $v_1$

using formula, 1/v - 1/u = 1/f

so, 1/$v_1$ - 1/u = 1/$f_1$ ........(1)

for the image formed by the 2nd lens B, at v,

using formula, 1/v - 1/u = 1/f

so, 1/v - 1/$v_1$ = 1/$f_2$ ........(2)

adding equations (1) and (2),

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

we know, 1/v - 1/u = 1/f

so, 1/f = 1/$f_1$ + 1/$f_2$

for several thin lenses of focal length $f_1,f_2,....$ then, effective focal length is given by,

1/f = 1/$f_1$ + 1/$f_2$ + 1/1/$f_3$ + 1/$f_4$ +....