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(A) For a silvered lens,the equivalent power $P$ is given by $P = 2P_L + P_M$.Here,$P_L$ is the power of the lens and $P_M$ is the power of the mirror

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$-\frac{20}{3} \, cm$

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(A) For a silvered lens,the equivalent power $P$ is given by $P = 2P_L + P_M$.Here,$P_L$ is the power of the lens and $P_M$ is the power of the mirror.The power of the lens is $P_L = \frac{(\mu - 1)}{R}$. Since it is a plano-convex lens,the curved surface has radius $R = 20 \, cm$ and the plane surface has $R = \infty$.The power of the mirror is $P_M = \frac{1}{f_M} = \frac{1}{R/2} = \frac{2}{R}$.Thus,$P = 2 \left( \frac{\mu - 1}{R} ight) + \frac{2}{R} = \frac{2\mu - 2 + 2}{R} = \frac{2\mu}{R}$.Since the equivalent system acts as a concave mirror,$P = -\frac{1}{f_{eq}}$.Substituting the values: $P = \frac{2 	imes 1.5}{20} = \frac{3}{20} \, cm^{-1}$.Therefore,$f_{eq} = -\frac{20}{3} \, cm$.

The radius of the curved surface of a plano-convex lens is $20 \, cm$ and the refractive index of the lens material is $1.5$. Calculate the equivalent focal length of the lens if the curved surface is silvered.

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