Find the refractive index of the second medium with respect to the first medium, if light moves through the first medium with velocity 2 × 10⁸ ms⁻¹, which changes to 1.25 × 10⁸ ms⁻¹ in the second medi

Question

Find the refractive index of the second medium with respect to the first medium, if light moves through the first medium with velocity 2 × 10⁸ ms⁻¹, which changes to 1.25 × 10⁸ ms⁻¹ in the second medium.

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Correct — option (3), 1.6. The refractive index of a second medium with respect to a first is the ratio of the speed of light in the first medium to the speed of light in the second — the speeds go the opposite way round from the media, which is the single point on which this calculation turns. Writing n₂₁ for the refractive index of medium 2 with respect to medium 1, n₂₁ = v₁ / v₂. Substituting the values printed in the stem, v₁ = 2 × 10⁸ m s⁻¹ and v₂ = 1.25 × 10⁸ m s⁻¹, gives n₂₁ = 2 × 10⁸ ÷ (1.25 × 10⁸). The powers of ten cancel, leaving 2 ÷ 1.25, which is 1.6. Like relative density, refractive index is a ratio of two like quantities and therefore carries no unit. There is a check on the answer that costs nothing and catches most of the ways this question is lost. Light slows down when it enters the second medium — from 2 × 10⁸ to 1.25 × 10⁸ metres per second — which means the second medium is optically denser than the first, and the refractive index of a denser medium with respect to a rarer one must be greater than 1. Any answer below 1 is therefore wrong before it is checked arithmetically. A second and more satisfying cross-check comes from the absolute indices. The absolute refractive index of a medium is the speed of light in vacuum divided by the speed in that medium, so with c taken as 3 × 10⁸ m s⁻¹, the first medium has an absolute index of 3 ÷ 2 = 1.5, which is about the value for ordinary glass, and the second has 3 ÷ 1.25 = 2.4, which is about the value for diamond. The relative index then follows as n₂₁ = n₂ / n₁ = 2.4 ÷ 1.5 = 1.6, the same answer by a different route, and the two media turn out to be a realistic pair rather than arbitrary numbers. So the value asked for is 1.6, which is option (3).

Why the other options are wrong — (a) This is the product of the two speeds' coefficients rather than their quotient: 2 multiplied by 1.25 gives 2.5. It is an operation error rather than an arithmetic one, and it is what a candidate produces who remembers that refractive index involves two speeds without remembering how they are combined. The definition is a ratio, and the same relation can be reached from Snell's law, which states that n₁ sin i = n₂ sin r and therefore that the refractive index of the second medium with respect to the first equals sin i divided by sin r — a ratio again, never a product. There is also a quick way to see that 2.5 cannot be right even if the operation is forgotten. A refractive index of 2.5 would make the second medium denser than diamond, whose index is about 2.42, relative to a first medium that is itself denser than vacuum; the physical world offers nothing in that range for a pair of ordinary transparent media, and the numbers in the stem are far too gentle to produce it.  (b) This option is not derivable from the two speeds in the stem at all: no combination of 2 × 10⁸ and 1.25 × 10⁸ produces 1.33. It is there because 1.33 is the most memorised number in this part of the syllabus — the refractive index of water with respect to air, four-thirds — and a candidate who recognises the shape of a refractive-index question before reading its numbers will reach for the value they know. That is a distinct and very common failure: recall substituting for calculation. Whenever a numerical question supplies its own data, the answer must come out of that data, and a familiar constant appearing among the options is a warning rather than a help. The standard values are worth holding — air about 1.0003, water about 1.33, ordinary glass about 1.5, diamond about 2.42 — but they belong to questions that ask for them, and here the stem asks for a relative index between two media whose speeds it has already given.  (d) This is the ratio taken the wrong way round: 1.25 divided by 2 is 0.625. The value is not meaningless — it is the refractive index of the FIRST medium with respect to the second, n₁₂, and the two are reciprocals, so n₂₁ × n₁₂ = 1 and 1.6 × 0.625 is indeed 1. What the option gets wrong is which of the two the question asked for, and the stem is explicit: the refractive index of the second medium with respect to the first. This is the most instructive of the three wrong answers, because it fails on the direction of a comparison rather than on any physics, and directions are what such questions are usually built on. The physical check disposes of it immediately as well. Light slowed down on entering the second medium, so the second medium is optically denser and the index of the second with respect to the first has to exceed 1; a value of 0.625 asserts the opposite of what the stem's own numbers say.

Concept: Light travels fastest in vacuum, at about 3 × 10⁸ metres per second, and more slowly in any material medium. The absolute refractive index of a medium is defined as the speed of light in vacuum divided by the speed of light in that medium, so it is always greater than 1 and is larger for media in which light is slower. When light passes from one medium to another it changes direction at the boundary, and the relation governing that bending is Snell's law: n₁ sin i = n₂ sin r, where i is the angle of incidence and r the angle of refraction, both measured from the normal. Rearranged, this gives the refractive index of the second medium with respect to the first as n₂₁ = n₂ / n₁ = sin i / sin r = v₁ / v₂, three equivalent expressions of the same ratio. Light bends towards the normal when it enters a denser medium and away from the normal when it enters a rarer one. That second case has a limit: beyond a certain angle of incidence, called the critical angle and given by sin C = 1 / n, light travelling from a denser medium into a rarer one is not refracted out at all but reflected entirely back, which is total internal reflection. It is what makes optical fibres carry signals, what makes a diamond sparkle — diamond's very high index of about 2.42 gives it a critical angle of only about 24 degrees — and what produces mirages over a hot road. Refractive index also varies slightly with the wavelength of light, which is dispersion, and is why a prism spreads white light into a spectrum and why a rainbow has colours.

State PCS 2021 Q53 — Science & Technology PYQ with Answer

State PCS 2021 Q53 — Science & Technology PYQ with Answer