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When two thin lenses of focal lengths fi...

When two thin lenses of focal lengths fi and f, are kept coaxially and in contact, prove that their cobined focal length "f" is given by :
`(1)/(f)=(1)/(f_(1))+(1)/(f_(2))`

Text Solution

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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 real image would be formed by the lens `l_(1)` alone at I., distinct v. (say) from `L_(1)`. Then, from the lens formula, we have
`(1)/(v.)- (1)/(u)= (1)/(f_(1))`

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)
on adding equations (i) and (ii), we get: `(1)/(v)- (1)/(u)= (1)/(f_(1)) + (1)/(f_(2))` ..(iii)
If we replaces these two lenses by a single lens which forms the image of an object placed at distance 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 equations (iii) and (iv) we get
`(1)/(F )= (1)/(f_(1)) + (1)/(f_(2))`
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