NDA (I) 2018 General Ability Test · Question 105 of 150

Correct — D, 0·25 m. Powers of thin lenses in contact simply add, so the combination has P = 2 + 2 = 4 dioptre. Power is the reciprocal of the focal length in metres, so f = 1 ÷ 4 = 0·25 m, or 25 cm. The result is positive, confirming that the pair still converges light, and it is shorter than either lens on its own, which is the physical point — putting two converging lenses together makes the system stronger, not weaker, and a stronger lens has a shorter focal length.

• (a)0·10 m — This is the focal length of a 10-dioptre lens. It appears if the two powers are multiplied by five, or if a candidate confuses dioptres with centimetres somewhere in the working. Nothing in the data gives 10 dioptre.
• (b)2 m — This is the trap of copying the number 2 straight out of the stem. Two dioptre corresponds to a focal length of 0·5 m, not 2 m, and in any case the question asks about the combination, not one lens.
• (c)4 m — Here the candidate has correctly reached a combined power of 4 dioptre and then written it down as a focal length instead of taking its reciprocal. Adding lenses can never lengthen the focal length of a converging system to four metres.

The power of a lens is defined as P = 1/f with f measured in metres, and the unit dioptre is therefore an inverse metre. For thin lenses placed in contact the equivalent power is the algebraic sum of the individual powers, which is the same statement as 1/F = 1/f₁ + 1/f₂. Powers are used precisely because they add, which is why an optician prescribes in dioptres rather than in focal lengths.

Note the sign discipline that makes this arithmetic safe. A converging lens has a positive power and a diverging one a negative power, so a 2-dioptre convex lens in contact with a 2-dioptre concave lens gives zero power and no focusing at all, while two convex lenses give 4 dioptre. Candidates lose this mark in one of two ways: by forgetting to invert at the end, or by inverting in centimetres and reporting 1/4 cm. Fix the habit of writing the unit at each step — dioptre going in, metres coming out. The relation is also the reason a magnifying lens of short focal length is described as a powerful lens: strength and focal length run opposite to each other.

• Power P = 1/f with f in metres; the unit of power is the dioptre, equal to one per metre.
• For thin lenses in contact the powers add: P = P₁ + P₂ + …
• Convex lenses carry positive power, concave lenses negative power.
• A 2-dioptre lens on its own has a focal length of 0·5 m; two such lenses together give 4 dioptre and 0·25 m.

• Reporting the combined power as though it were a focal length.
• Inverting a power in centimetres instead of metres.
• Forgetting that a concave lens contributes a negative power to the sum.

NDA sets one power-and-focal-length conversion nearly every paper, sometimes as a single lens and sometimes as two in contact.

• practice — not a real PYQ

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

  • (a)+8 D
  • (b)+2 D
  • (c)−2 D
  • (d)+15 D

  Answer(b) +2 D — powers add with their signs, so 5 + (−3) = 2 dioptre, still converging.

• practice — not a real PYQ

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

  • (a)+0·25 m
  • (b)−0·25 m
  • (c)−4 m
  • (d)−0·4 m

  Answer(b) −0·25 m — the reciprocal of −4, and the negative sign marks it as a diverging lens.