Obtain an expression for the combined focal length for two thin convex lenses kept in contact and hence obtain an expression for the combined power of the combination of the lenses.

Explanation

i) Consider two lenses A and B of focal lengths `f_(1)` and `f_(2)` placed in contact with each other.
ii) Let the object be placed at a point O, the first lens forms the image at `I_(1)`, it is real. It serves as the virtual object of lens B, Producing final image at I.
iii) For the image formed by lens A, we get
`(1)/(v_(1))-(1)/(u)=(1)/(f_(1))" ......................(1)"`
For the image formed by lens B, We get
`(1)/(v)-(1)/(v_(1))=(1)/(f_(2))" ...........................(2)"`
Adding equations (1) and (2)
`(1)/(v)-(1)/(u)=(1)/(f_(1))+(1)/(f_(2))" ............................(3)"`
But `(1)/(v)-(1)/(u)=(1)/(f)`
Where f is focal length of the combination
From eq (3), `(1)/(f)=(1)/(f_(1))+(1)/(f_(2))" .........................(4)"`
If several lenses of focal length `f_(1),f_(2),f_(3)"...................... are in contact"`
`(1)/(f)=(1)/(f_(1))+(1)/(f_(2))+(1)/(f_(3))+....................`
In terms of power, we can write
`P=P_(1)+P_(2)+P_(3)+"..........................(5)" (because P=(1)/(f))`