Derive the expression for effective focal length of two thin lenses kept in contact.

Explanation

(a) Thin lenses in contact:

A thin lens is that lens whose thickness can he neglected when compared to its radius of curvature. By applying universal lens formula of refraction through `L_1` ,
`1/(f_1)=1/(-u) + 1/v` , ....(1)
Image at lc serves as a virtual object for the thin lens `L_2`. The virtual object distance is taken as +ve in the direction of incident ray.
`therefore 1/(f_2)=1/(-v.) + 1/v` ...(2)
Adding (1) and (2) we get
`1/(f_1) +1/(f_2) =1/(-u)+1/v=1/f`
where f is known as the focal length of equivalent lens.
An equivalent lens is that single lens that produces the same effect as those of combined lenses in contact.
In general,
`1/f=1/(f_1) + 1/(f_2) + 1/(f_3)` + ...... + `1/(f_n)`
Since, `P=1/f` =power of the lens
`P=P_1+P_2+P_3`+.....+`P_n`
`P_1,P_2` may be ve or - ve depending upon the type of lenses.
(b)`1/f=1/(f_1) + 1/(-f_2)`
where .f. is in metre
i.e. `P=(P_1-P_2)` and .P. is in dioptre.