CDS (II) 2020 General Knowledge · Question 101 of 120

Correct — B, Lens with power +2·5 D. The power of a lens is defined as the reciprocal of its focal length in metres, P = 1/f, and it is precisely a measure of how sharply the lens bends light. A large power means a short focal length, and a short focal length means a ray parallel to the axis is turned through a large angle before it reaches the focus. The sign only records the direction of the bending — plus for a converging lens that pulls rays together, minus for a diverging lens that pushes them apart — so the amount of bending is set by the magnitude alone. The four magnitudes here are 2·0, 2·5, 1·5 and 2·0 dioptres, and 2·5 is the largest. Its focal length is 1/2·5 = 0·4 m, or 40 cm, the shortest of the four, so it deviates the light most.

• (a)Lens with power +2·0 D — Converging, but weaker than the answer. Its focal length is 1/2·0 = 0·5 m, or 50 cm, longer than the 40 cm of the +2·5 D lens, so it bends a parallel ray through a smaller angle.
• (c)Lens with power –1·5 D — The weakest lens on the list in magnitude. Its focal length is 1/1·5 = 0·67 m, about 67 cm — the longest focal length here and therefore the least bending, whatever the sign.
• (d)Lens with power –2·0 D — The trap for anyone who reads the minus sign as making the lens stronger, or who ranks the numbers as signed values and picks the most negative. Its magnitude is 2·0 D, the same as option (a); it bends light by the same amount as that lens, just in the opposite sense, and both are beaten by 2·5 D.

Power measures the converging or diverging ability of a lens and is the reciprocal of the focal length expressed in metres, so its unit, the dioptre, is one per metre. A convex lens has positive power and a concave lens negative power. Powers add when thin lenses are placed in contact, which is why an optician can specify a correction as a single number and why spectacle prescriptions are written in dioptres.

The item hinges on separating a vector-like sign from a scalar magnitude. 'Bends through the largest angle' asks how much, not which way, so the ranking has to be done on the absolute values: 2·5 beats 2·0, which ties with 2·0, which beats 1·5. A student who instead sorts the signed numbers from largest to smallest would still land on +2·5 D here and get the mark without the reasoning; the same student meets a paper where the strongest lens is the negative one and loses it. Note also that stating the answer as a focal length makes it vivid — 40 cm against 50, 50 and 67 cm, the shortest focal length doing the sharpest bending.

• Power P = 1/f with f in metres; the unit is the dioptre, D, equal to one per metre.
• A convex or converging lens has positive power; a concave or diverging lens has negative power.
• The magnitude of the power fixes how strongly light is bent; the sign fixes only the direction.
• A +2·5 D lens has a focal length of 0·4 m, a +2·0 D lens 0·5 m and a −1·5 D lens 0·67 m.
• Powers of thin lenses in contact add algebraically: P = P₁ + P₂.

• Ranking signed powers instead of magnitudes when the question asks how much a lens bends light.
• Substituting the focal length in centimetres into P = 1/f, which gives an answer a hundred times too large.
• Assuming a negative power means a weaker lens; it means a diverging one.

As a direct power-to-focal-length conversion, as a comparison of several lenses, or as a spectacle-prescription item asking which lens corrects a given defect.

• practice — not a real PYQ

  The focal length of a lens of power −4 D is

  • (a)−25 cm
  • (b)−4 cm
  • (c)−40 cm
  • (d)−0·25 cm

  Answer(a) −25 cm — since P = 1/f, f = 1/(−4) = −0·25 m, which is −25 cm. The negative sign marks a diverging lens.

• practice — not a real PYQ

  Two thin lenses of powers +3 D and −1 D are placed in contact. The power of the combination is

  • (a)+4 D
  • (b)+2 D
  • (c)−3 D
  • (d)+1·5 D

  Answer(b) +2 D — powers of thin lenses in contact add algebraically, so +3 D combined with −1 D gives +2 D.