CDS (I) 2023 General Knowledge · Question 10 of 120

Correct — C, 10 dioptre. The power of a lens is defined as the reciprocal of its focal length measured in metres, P = 1/f. The trap is entirely in the units: the focal length is given as 10 centimetres, which is 0.10 metre, and one divided by 0.10 is 10. The lens therefore has a power of 10 dioptres. One dioptre is by definition the power of a lens whose focal length is one metre, so a lens that brings rays to a focus in only a tenth of that distance must be ten times as powerful — a check that confirms the answer without any arithmetic.

• (a)0.1 dioptre — The focal length in metres, copied out as if it were the power. It also fails the sanity check — 0.1 dioptre would describe a very weak lens with a focal length of 10 metres, not a strong one that focuses in 10 centimetres.
• (b)1 dioptre — The power of a lens of focal length one metre. Using it here means the 10 centimetres was quietly rounded to a metre, which throws away exactly the factor the question is testing.
• (d)100 dioptre — Comes from dividing one by 0.01 instead of by 0.10 — a decimal slip that treats 10 cm as one centimetre. A lens of 100 dioptres would have a focal length of 1 cm.

The power of a lens measures how strongly it bends light, and it is defined as the reciprocal of the focal length in metres. Its unit, the dioptre, is therefore an inverse metre: a one-dioptre lens has a focal length of one metre. Shorter focal length means stronger bending and higher power. Sign carries meaning too — a converging, convex lens has a positive focal length and positive power, while a diverging, concave lens has both negative. Powers add when thin lenses are placed in contact, which is why spectacle prescriptions are written in dioptres rather than in focal lengths.

Everything in this item turns on converting centimetres to metres before taking the reciprocal, and three of the four options are what happens when that conversion goes wrong by a factor of ten in one direction or another. The habit worth building is to write the focal length in metres on the page before touching the reciprocal. The paper prints no sign on any option, and the stem does not say whether the lens converges or diverges, so what is being asked for is the magnitude; had it specified a concave lens the answer would have been minus 10 dioptres. That the printed choices carry no plus or minus sign is a small looseness of the paper rather than a change to what is being tested.

• Power of a lens P = 1/f, with f measured in metres; the unit is the dioptre, an inverse metre.
• A lens of focal length one metre has a power of one dioptre, so 10 cm gives 10 dioptres.
• A convex, converging lens has positive focal length and positive power; a concave, diverging lens has both negative.
• When thin lenses are placed in contact their powers add, which is why optical prescriptions are written in dioptres.

• Taking the reciprocal of the focal length in centimetres and never converting to metres.
• Ignoring the sign, and quoting a positive power for a diverging lens.
• Confusing power with magnification; the two are different quantities with different units.

As a direct conversion between focal length and power in either direction, or as a combination sum where two lenses in contact have to be added.

• practice — not a real PYQ

  The focal length of a lens whose power is -4 dioptre is

  • (a)-25 cm
  • (b)-4 cm
  • (c)+25 cm
  • (d)-40 cm

  Answer(a) -25 cm — f = 1/P = 1/(-4) = -0.25 m, and the negative sign marks it out as a diverging, concave lens.

• practice — not a real PYQ

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

  • (a)+2 D
  • (b)+4 D
  • (c)-3 D
  • (d)+1.5 D

  Answer(a) +2 D — for thin lenses in contact the powers add algebraically, so +3 D and -1 D give +2 D.