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

Explanation

Thin lenses in contact:-

A thin lens is that lens whose thickness can be 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 I. 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