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What is the focal length of a convex lens of focal length 30 cm in contact with a concave lens of focal length 20 cm? Is the system a converging or a diverging lens? Ignore thickness of the lenses.

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#### Solution

Focal length of the convex lens, f_{1} = 30 cm

Focal length of the concave lens, f_{2} = −20 cm

Focal length of the system of lenses = f

The equivalent focal length of a system of two lenses in contact is given as:

`1/"f" = 1/"f"_1 + 1/"f"_2`

`1/"f" = 1/30 - 1/20`

= `(2 - 3)/60`

= `-1/60`

∴ f = −60 cm

Hence, the focal length of the combination of lenses is 60 cm. The negative sign indicates that the system of lenses acts as a diverging lens.

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