NDA (II) 2018 General Ability Test · Question 110 of 146

Correct — A, +2 dioptre. The power of a lens is the reciprocal of its focal length expressed in metres, P = 1/f. Here f = 50 cm = 0·50 m, so P = 1/0·50 = 2 dioptre. The sign follows from the type of lens: a convex lens converges light, its focal length is taken as positive in the Cartesian convention, and its power is therefore positive. Putting the two together gives +2 dioptre. The whole item is really a test of whether the candidate converts centimetres to metres before taking the reciprocal, because that single step separates the correct answer from two of the three distractors.

• (b)+0·02 dioptre — This is 1/50 with the focal length left in centimetres. The dioptre is defined only for a focal length in metres, so the conversion has to be made first; forgetting it shrinks the answer by a factor of a hundred.
• (c)–0·5 dioptre — Wrong on both counts. The magnitude is 0·50 rather than its reciprocal, and the minus sign would make this a diverging lens, whereas the question specifies a convex lens.
• (d)+0·5 dioptre — This is the focal length in metres reported as though it were the power. Power and focal length are reciprocals of one another, so 0·50 m gives 2 dioptre, not 0·5.

The power of a lens measures how strongly it bends light. It is defined as the reciprocal of the focal length in metres, and its unit, the dioptre, is therefore an inverse metre. A short focal length means strong bending and a large power; a long focal length means a weak lens. Converging lenses are given positive powers and diverging lenses negative ones, which is why a spectacle prescription for short sight carries a minus sign and one for long sight a plus.

Two habits settle every question of this type. First, convert the focal length to metres before doing anything else. Second, fix the sign from the type of lens rather than from the arithmetic — convex is positive, concave negative. The distractors here are laid out to catch exactly the candidates who skip one of those two steps, and one of them catches both. A useful extension is that when thin lenses are placed in contact their powers simply add, so two +2 dioptre lenses together give +4 dioptre, that is a combined focal length of 25 cm.

• Power P = 1/f, with f measured in metres; the unit of power is the dioptre, equal to one inverse metre.
• A focal length of 50 cm is 0·50 m, so the power is 1/0·50 = 2 dioptre.
• A convex, converging lens has positive focal length and positive power; a concave, diverging lens has both negative.
• For thin lenses placed in contact the powers simply add, so P = P₁ + P₂ and so on.

• Taking the reciprocal while the focal length is still in centimetres.
• Reporting the focal length in metres as if it were the power, which happens to look plausible whenever f is close to 1 m.
• Attaching a negative sign out of habit from mirror problems; a convex lens always has positive power.

Asked as a single-step numerical where the unit conversion, not the formula, is the real hurdle.

• practice — not a real PYQ

  A lens has a power of –4 dioptre. Its focal length and nature are

  • (a)25 cm, convex
  • (b)25 cm, concave
  • (c)4 m, concave
  • (d)0·25 m, convex

  Answer(b) 25 cm, concave — f = 1/P = 1/(−4) = −0·25 m, and the negative sign marks it as a diverging, concave lens.

• practice — not a real PYQ

  Two thin lenses of powers +3 dioptre and –1 dioptre are placed in contact. The focal length of the combination is

  • (a)0·5 m
  • (b)2 m
  • (c)0·25 m
  • (d)1 m

  Answer(a) 0·5 m — powers in contact add, giving +2 dioptre, and the focal length is the reciprocal, 0·5 m.