NDA (I) 2017 General Ability Test · Question 106 of 149

Correct — A, +7.5 D. Use the lens maker's formula, 1/f = (n − 1)(1/R1 − 1/R2). In a double convex lens the first face bulges towards the incoming light and the second away from it, so with the usual sign convention R1 = +10 cm and R2 = −20 cm. That gives 1/f = (1.5 − 1) × (1/10 + 1/20) = 0.5 × 3/20 = 3/40 per centimetre, so f = 40/3 cm, which is 0.1333 metre. Power is the reciprocal of the focal length in metres, so P = 1/0.1333 = +7.5 dioptre. The sign is positive because a double convex lens of glass in air converges light.

• (b)−7·5 D — The size of the number is right but the sign is not. A negative power belongs to a diverging lens — a concave one, or a convex one made of a medium optically rarer than its surroundings. Glass in air with both faces bulging outward converges light, so its power must come out positive.
• (c)+2·5 D — This is what you get by writing 1/10 − 1/20 = 1/20 instead of 1/10 + 1/20, that is by giving both radii the same sign. Then 1/f = 0.5 × 1/20 = 1/40 per centimetre, f = 40 cm = 0.4 m and P = 2.5 D. The second surface of a double convex lens has its centre of curvature on the incoming side, so its radius carries the opposite sign, and missing that is the commonest slip in this question.
• (d)+5·0 D — This follows from using only the first face, as though the lens were plano-convex with R = 10 cm: 1/f = 0.5 × 1/10 per centimetre gives f = 20 cm and P = 5 D. Both curved faces refract the light, so both radii must enter the formula.

The lens maker's formula ties the focal length of a thin lens to two things — how much the material bends light, through (n − 1), and how sharply the two faces are curved, through 1/R1 − 1/R2. Power is simply 1/f with f measured in metres, and its unit, the dioptre, is therefore per metre. A converging lens has a positive power, a diverging lens a negative one, and when thin lenses are placed in contact their powers add directly, which is why opticians work in dioptres rather than centimetres.

Everything in this question turns on the sign convention. Light is taken to travel left to right, distances are measured from the optical centre, and a radius is positive when the centre of curvature lies on the outgoing side. For a double convex lens that makes the first radius positive and the second negative, so the two fractions add rather than subtract — and the moment they add, the arithmetic is a few seconds' work. Notice also that the answer is bigger than either single-surface value, which is the physical signal that both faces are doing the converging.

• The lens maker's formula is 1/f = (n − 1)(1/R1 − 1/R2) for a thin lens in air.
• Power in dioptre is the reciprocal of the focal length in metres, so a focal length of 40/3 cm gives 7.5 D.
• A converging lens carries positive power and a diverging lens negative power.
• For thin lenses in contact the powers add, P = P1 + P2, which is why spectacle prescriptions are written in dioptres.

• Giving both radii the same sign and losing the addition of the two fractions.
• Reporting the focal length in centimetres and then calling its reciprocal the power — the metre is the unit that defines a dioptre.
• Assuming a convex lens always converges; a glass convex lens immersed in a denser liquid diverges light.

NDA asks for the power of a lens of given focal length, or the focal length of a stated power, and occasionally runs the lens maker's formula as here.

• practice — not a real PYQ

  The power of a convex lens of focal length 20 cm is

  • (a)+2 D
  • (b)+5 D
  • (c)−5 D
  • (d)+20 D

  Answer(b) +5 D — 20 cm is 0.2 m, and the reciprocal of 0.2 is 5, positive because the lens converges.

• practice — not a real PYQ

  A convex lens of power +3 D is placed in contact with a concave lens of power −5 D. The focal length of the combination is

  • (a)+0.5 m
  • (b)−0.5 m
  • (c)+2 m
  • (d)−2 m

  Answer(b) −0.5 m — the powers add to −2 D, and the focal length is the reciprocal of that, so the pair behaves as a diverging lens.