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For two thin lenses kept in contact with...

For two thin lenses kept in contact with each other, show that :
`" "1/F=1/f_(1)+1/f_(2)`
where the terms have their usual meaning.

Text Solution

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Combined Focal Length of two Thin Lenses in Contact
(i) Both the Lenses are Convex: Suppose two thin convex lenses `L_(1) and L_(2)` of focal lengths `f_(1) and f_(2)` are placed in contact in air having a common principal axis. A point-object O is placed on the principal axis at a distance u from the first lens `L_(1)`. Its image would be formed by the lens `L_(1)` along at I.. distant v. (say) from `L_(1)`. Then, from the lens formula, we have
`(1)/(v.)- (1)/(u)= (1)/(f_(1))` ...(i)

I. serves as a virtual object for the second lens `L_(2)` which forms a final image I at a distance v (say) from it. Then, we have `(1)/(v)- (1)/(v.)= (1)/(f_(2))` ...(ii)
Adding eq (i) and (ii), we get
`(1)/(v)- (1)/(u)= (1)/(f_(1)) + (1)/(f_(2))` ....(iii)
If we replace these two lenses by a single lens which forms the image of an object distant u from it at a distance v, then the focal length F of this equivalent lens would be given by `(1)/(v)- (1)/(u)= (1)/(F)` ..(iv)
From eq. (iii) and (iv), we get
`(1)/(F )= (1)/(f_(1)) + (1)/(f_(2))`.
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