NDA (I) 2017 General Ability Test · Question 142 of 149

Correct — D, 8 : 9. The absolute refractive index of a medium is n = c/v, where c is the speed of light in vacuum and v its speed in the medium, so v = c/n. Speed is therefore inversely proportional to the index. Here v(glass) = c ÷ (3/2) = 2c/3 and v(water) = c ÷ (4/3) = 3c/4. The ratio is (2/3) : (3/4), and multiplying both sides by 12 gives 8 : 9. Note the direction — glass has the larger index, so light travels slower in it, and the first number in the ratio must be the smaller.

• (a)3 : 4 — This is what you get by dividing the numerators of the two indices, 3 and 4, without inverting them. It is a plausible-looking wrong turn because it uses the digits in the question.
• (b)4 : 3 — This is the ratio of the two refractive indices themselves, (3/2) ÷ (4/3) = 9/8 inverted to 4 : 3 by a second slip — in any case it is an index ratio, not a speed ratio.
• (c)8 : 7 — No route through the arithmetic gives 8 : 7. It is placed next to the correct 8 : 9 so that a hurried reader who has done the work correctly may still tick the wrong box.

Light travels fastest in vacuum. In any material medium it slows, and the absolute refractive index is simply the factor by which it slows: n = c/v. A larger index means a slower speed and a greater bending at the surface, which is why a diamond, with an index near 2.42, bends and disperses light far more sharply than window glass at about 1.5 or water at about 1.33.

The single most common error in this item is losing the inversion. Write v = c/n before touching the numbers, and the ratio takes care of itself. A useful sanity check afterwards: glass is optically denser than water, so light must be slower in glass, and the answer must be less than one — 8 : 9 is, and 4 : 3 is not. The same relation is what tells you that the ratio of speeds in two media equals the inverse ratio of their indices.

• Absolute refractive index n = c/v, where c is the speed of light in vacuum.
• Speed in a medium is inversely proportional to its refractive index.
• For n(glass) = 3/2, the speed in glass is two-thirds of the speed in vacuum.
• For n(water) = 4/3, the speed in water is three-quarters of the speed in vacuum.
• Typical values are about 1.33 for water, about 1.5 for ordinary glass and about 2.42 for diamond.

• Reporting the ratio of indices when the question asks for the ratio of speeds.
• Getting the magnitude right but the order wrong, giving 9 : 8 instead of 8 : 9.
• Assuming the denser-looking medium always has the larger index without checking the given values.

NDA sets this relation almost every year — sometimes as a two-medium ratio, sometimes as 'find the speed in a medium of index n', sometimes as an inequality between two named materials.

• practice — not a real PYQ

  The speed of light in a medium of refractive index 1.5, taking the speed in vacuum as 3 × 10⁸ m/s, is

  • (a)4.5 × 10⁸ m/s
  • (b)3 × 10⁸ m/s
  • (c)2 × 10⁸ m/s
  • (d)1.5 × 10⁸ m/s

  Answer(c) 2 × 10⁸ m/s — v = c/n = 3 × 10⁸ ÷ 1.5.

• practice — not a real PYQ

  Of water (n = 1.33), crown glass (n = 1.52) and diamond (n = 2.42), light travels fastest in

  • (a)Water
  • (b)Crown glass
  • (c)Diamond
  • (d)The speed is the same in all three

  Answer(a) Water — the lowest refractive index of the three gives the highest speed.