If the refractive index of the medium is more than the refractive index of double concave or double convex lens, then the focal length of the lens becomes ________.

Question

UnlockIAS · Accepted Answer

Correct — option (4), "negative". Everything here follows from the lensmaker's formula written for a lens sitting in a medium rather than in air: 1/f = (n_lens/n_medium − 1) × (1/R₁ − 1/R₂) The second bracket is fixed by the shape of the lens and does not care what surrounds it. The first bracket is the one the medium controls. When the lens is denser than its surroundings — glass in air, the ordinary case — n_lens/n_medium is greater than one, the bracket is positive, and the lens behaves as we expect: a double convex lens converges and has a positive focal length. Now make the medium denser than the lens, as the stem does. Then n_lens/n_medium is less than one, the bracket (n_lens/n_medium − 1) turns negative, and multiplying by a negative number flips the sign of 1/f. The optical character of the lens is reversed. A double convex lens, which converged in air with a positive focal length, now diverges, and its focal length is negative — which is the Commission's credited answer, option (4). The classroom demonstration is a convex lens of crown glass, refractive index about 1.5, immersed in carbon disulphide, refractive index about 1.63. Parallel light entering the arrangement spreads out instead of coming to a focus. The everyday version is an air bubble trapped in water: the bubble is convex in shape but is optically rarer than the water around it, and it behaves as a diverging lens, which is why bubbles in a glass of water look bright-rimmed and never focus light to a point. Why the sign flips at all is worth understanding physically rather than algebraically. Refraction at each surface bends light towards the normal when it enters a denser medium and away from the normal when it enters a rarer one. In air, light entering the curved face of a convex glass lens is entering a denser medium and is bent inwards, towards the axis — convergence. Surround the same lens with something denser than the glass and light crossing that face is now entering a rarer medium; it bends the other way at every surface, and the same shape that gathered rays now spreads them. A second consequence, less often asked but worth having: the magnitude of the focal length also grows. The factor (n_lens/n_medium − 1) is smaller in size than (n_lens − 1) was in air, so 1/f is smaller, f is longer, and the power of the lens P = 1/f falls. A lens immersed in a medium of similar optical density is a weak lens as well as a reversed one, and in the limiting case where the two refractive indices are equal the bracket vanishes altogether, 1/f becomes zero and the focal length becomes infinite — the lens stops bending light and optically disappears. That limiting case is what option (1) is doing on the page: it is a real result, but for the condition n_lens = n_medium, not for the condition the stem states. One note for the careful reader. The stem names the double concave lens alongside the double convex one, and the reversal principle applies to both — worked through the same formula, each type has its converging or diverging character inverted when the surrounding medium is the denser one. The physical rule to carry away from this question is therefore the reversal itself: in a medium denser than the lens material, a converging lens diverges and a diverging lens converges. On this paper the Commission's final key is option (4), which is the result for the double convex case, and it is the final key that governs the mark.

Why the other options are wrong — (a) Option (1) states a real result attached to the wrong condition. The focal length becomes infinite when the refractive index of the medium EQUALS that of the lens: the bracket (n_lens/n_medium − 1) then vanishes, 1/f is zero, and the lens neither converges nor diverges — it becomes optically invisible, which is why a glass rod in a liquid of matched refractive index seems to disappear. The stem specifies a medium denser than the lens, not one of equal density, so the bracket is negative rather than zero.  (b) Option (2) is impossible for any lens under any condition. A focal length of zero would mean 1/f infinite — a lens of unbounded power, bringing parallel light to a focus at the lens itself. Nothing in the lensmaker's formula can produce it: neither bracket can become infinite for a lens of finite curvature made of a material of finite refractive index. This is the option offered to a candidate who has correctly grasped that something goes to a limiting value but has confused which side of the reciprocal it lies on.  (c) Option (3) fails to register that anything has changed. A positive focal length is what a double convex lens has in air, where it converges; the whole point of the stem's condition is that immersing the lens in a denser medium reverses that character, so the sign of the focal length cannot stay as it was. A candidate marking this has read the stem as a question about magnitude — the focal length does indeed grow longer in a dense medium — and missed that the question asks about sign.

Concept: The lensmaker's formula, 1/f = (n_lens/n_medium − 1)(1/R₁ − 1/R₂), separates the two things that determine a lens's behaviour: its geometry, carried by the radii of curvature, and the optical contrast between the lens material and its surroundings, carried by the ratio of refractive indices. Geometry alone does not decide whether a lens converges. What converges light is a difference in refractive index across a curved boundary, and the direction of that difference matters as much as its size. A shape that gathers rays when it is optically denser than its surroundings will spread them when it is optically rarer, because refraction at each surface bends the other way. This is why the words converging and diverging are properly attached to a lens together with the medium it is used in, and why a laboratory lens described as 'convex' is really shorthand for 'convex and used in air'. The formula also carries the sign convention that gives the answer its form: by the Cartesian convention a positive focal length denotes a converging system and a negative one a diverging system, so a reversal of character shows up as a reversal of the sign of f.

State PCS 2026 Q54 — Science & Technology PYQ with Answer

State PCS 2026 Q54 — Science & Technology PYQ with Answer