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Prove that when two thin lenses are placed in contact, the focal length is given by…?

(a) Prove that when two thin lenses with focal lengths f1 and f2 are placed in contact, the focal length f of the combination is given by ( 1 / f ) = ( 1 / f1 ) + ( 1 / f2 )

Consider a parallel beam of light parallel to the pricipal axis is incident on the first lens of focal length f1.

The image is at its focal point distant f1 from the first lens.

For the second lens which is in contact with the first lens, the object is at u = – f1,
Minus since the object is at the right side of the second lens.

The image distance is found from
1/f2 = -1/f1 + 1/ v

Solving v = {f1 + f2} /f1f2.

If F is the combined focal length, then

1/F = 1/ u + 1/v = 1/ infinity + {f1 + f2} / f1f2

1/F = 1/f1 + 1/f2
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