A biconvex lens is made of a material of refractive index 2 such that the ratio of radii of curvature of its two faces is 1 : 2. A point source in air is placed on the principal axis of this lens at a distance \(u\) from the lens. Which one of the following is correct if the image is formed exactly at the same distance as that of the point source (where \(R\) is the radius of curvature of the first face)?
\(3u = 4R\)
For the biconvex lens, take the first face's radius as \(R_1 = R\) (positive) and the second face's radius as \(R_2 = -2R\) (negative, by sign convention, since the ratio of magnitudes is 1:2).
By the lens maker's formula: \(\dfrac{1}{f} = (n-1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right) = (2-1)\left(\dfrac{1}{R} + \dfrac{1}{2R}\right) = \dfrac{3}{2R}\), so \(f = \dfrac{2R}{3}\).
A real image is formed at the same distance as the object (magnification of magnitude 1) only when the object is placed at \(u = 2f\), giving a real, inverted image at \(v = 2f\) on the other side.
So \(u = 2f = 2 \times \dfrac{2R}{3} = \dfrac{4R}{3}\), i.e. \(3u = 4R\). Hence option (d) is correct.
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