MPSC 2026 Prelims Paper-I · Question 54 of 100

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.

• (1)infinite — 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.
• (2)zero — 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.
• (3)positive — 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.

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.

Two limiting cases sit on either side of the situation the stem describes, and MPSC has used both. When the medium's refractive index is greater than the lens material's, the bracket is negative and the lens's character reverses — the case here, illustrated by a crown glass convex lens in carbon disulphide, or by an air bubble in water. When the two indices are exactly equal, the bracket vanishes, the focal length becomes infinite and the lens ceases to act on light at all; this is the principle behind the classroom trick of making a glass rod vanish inside a beaker of liquid whose refractive index matches the glass, and it is also why the crystalline lens of the eye, which sits between the aqueous and vitreous humours rather than in air, has far less refracting power than its shape alone would suggest — most of the eye's refraction is done at the air-cornea boundary, where the index contrast is largest. Everyday optical instruments exploit the same arithmetic in reverse: oil immersion microscope objectives use a liquid of refractive index close to that of glass precisely to control refraction at the boundary.

• Lensmaker's formula in a medium: 1/f = (n_lens/n_medium − 1)(1/R₁ − 1/R₂). The medium enters only through the first bracket, and the shape of the lens only through the second.
• If the surrounding medium is optically denser than the lens material, the first bracket becomes negative and the sign of 1/f is reversed — the lens's converging or diverging character is inverted.
• If the refractive index of the medium equals that of the lens, the bracket is zero, 1/f is zero and the focal length is infinite: the lens neither converges nor diverges and effectively disappears optically.
• Standard illustrations: a crown glass convex lens (n about 1.5) immersed in carbon disulphide (n about 1.63) acts as a diverging lens; an air bubble in water is convex in shape but optically rarer than its surroundings and likewise diverges.
• By the Cartesian sign convention a positive focal length denotes a converging lens and a negative focal length a diverging lens; power is P = 1/f in dioptres when f is in metres, so a reversal of sign is also a reversal of the sign of the power.

• Believing a convex lens always converges. Convergence depends on the optical contrast with the surrounding medium, not on shape alone; in a denser medium a convex lens diverges.
• Attaching the infinite-focal-length result to the wrong condition. The focal length becomes infinite when the medium and the lens have EQUAL refractive indices, not when the medium is denser.
• Reading a sign question as a magnitude question. Immersing a lens in a dense medium both reverses the sign of f and increases its magnitude — this stem asks about the sign.
• Remembering the reversal rule for only one lens type. The principle is a reversal of character, so it runs in both directions: converging becomes diverging and diverging becomes converging.

Ray optics is a fixture of MPSC's general science section, and the lens-in-a-medium result is one of its most reliable items because it can be asked as a single sentence and cannot be answered by rote recall of a formula alone. Expect three shapes. The first is this one — a condition on refractive indices with a one-word conclusion about the focal length. The second is the disappearing-lens question, asking what happens when the two indices are equal. The third is applied: why an air bubble in water diverges light, or why a swimmer's vision is blurred underwater but clear behind goggles, which is the same arithmetic applied to the cornea. All three come out of a single relation, so the efficient preparation is to memorise the lensmaker's formula in its medium form and then reason from it rather than to memorise the three results separately.

No directly related past PYQ was found.

• practice — not a real PYQ

  A glass lens becomes invisible when immersed in a certain liquid. This happens because :

  • (a)the liquid absorbs all the light falling on the lens
  • (b)the refractive index of the liquid equals that of the glass
  • (c)the liquid is denser than the glass
  • (d)total internal reflection occurs at every surface of the lens

  Answer(b) the refractive index of the liquid equals that of the glass. With no index contrast at the boundary there is no refraction and no partial reflection at the surfaces, so nothing marks out the lens from its surroundings; in the lensmaker's formula the bracket (n_lens/n_medium − 1) becomes zero, 1/f is zero and the focal length is infinite. If the liquid were merely denser than the glass the lens would still refract light, only in the reversed sense.

• practice — not a real PYQ

  An air bubble inside a glass of water behaves optically as a :

  • (a)converging lens, because it is convex in shape
  • (b)diverging lens, because it is optically rarer than the water around it
  • (c)plane refracting surface
  • (d)perfectly reflecting sphere

  Answer(b) diverging lens, because it is optically rarer than the water around it. The bubble is convex in shape, but shape alone does not decide the behaviour of a lens — the contrast in refractive index does. With air inside and water outside, the surrounding medium is the denser one, the bracket in the lensmaker's formula is negative, and the convex shape spreads light instead of gathering it.