Question

A convex lens and a convex mirror of radius of curvature `20 cm` are placed co-axially with the convex mirror placed at a distance of `30 cm` from the lens. For a point object at a distance of `25 cm` from the lens, the final image due to this combination coincides with the object itself. What is the focal length of convex lens ?

Answer 1

Fig. shows the given set up.
Here, `R = PC = 20 cm`
`C_1 P = 30 cm, OC_1 = 25 cm`
As the final image due to the combination coincides with the object itself, the rays reflected from convex mirror must be retracting their path. This can happen only when rays refracted from convex lens tend to meet at centre of curvature of convex mirror. Therefore, for convex lens,
`v = C_1 C = C_1 P + PC = 30 + 20 = 50 cm`,
when `u = OC_1 = -25 cm`
From `(1)/(f)=(1)/(v)-(1)/(u)=(1)/(50)+(1)/(25)=(3 + 6)/(150)=(9)/(150)`
`f = 150//9 = 16.67 cm`.
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