Given is a thin convex lens of glass (refractive index $μ$ ) and each side having radius of curvature R. One side is polished for complete reflection. At what distance from the lens, an object be placed on the optic axis so that the image gets formed on the object itself.

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$R / μ$

$R / (2 μ - 3)$

$μ R$

$R / (2 μ - 1)$

answer is D.

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Detailed Solution

The focal length 𝑓 of a thin lens is given by the Lens Maker’s Formula:

$\frac{1}{f} = (μ - 1) (\frac{1}{R} - \frac{1}{- R})$

$f = \frac{R}{2 (μ - 1)}$

For effective System (Lens + Mirror)

$\begin{array}{l} \frac{1}{f_{e q}} = \frac{2}{f_{ℓ}} - \frac{1}{f_{m}} \\ \frac{1}{f_{e q}} = \frac{4 (μ - 1)}{R} - \frac{2}{- R} = \frac{1}{R} (4 μ - 4 + 2) \\ \frac{1}{f_{eq}} = \frac{1}{R} (4 μ - 2) \\ {Object should be kept at 2f}_{e q} distance from the combination . \\ u = 2 f_{eq} = 2 (\frac{R}{4 μ - 2}) = \frac{R}{(2 μ - 1)} \end{array}$