The magnitude of focal length of a concave lens is 2 m. What is the power of the lens?

Correct — B, -0.5 dioptre. The power of a lens is the reciprocal of its focal length expressed in metres, and it carries the sign of that focal length. Under the standard sign convention a concave lens diverges light and its focal length is negative, so a magnitude of 2 m means f = -2 m. Then P = 1 ÷ (-2) = -0.5 dioptre. The size of the number, half a dioptre, says the lens bends light only weakly; the minus sign says that it spreads the rays apart instead of bringing them to a focus.

• (a)+0.5 dioptre — Right magnitude, wrong sign. This would be a convex lens of focal length two metres, which converges light instead of diverging it.
• (c)+2.0 dioptre — Reports the focal length as though it were the power, and loses the sign as well. A lens of +2.0 dioptre is convex with a focal length of half a metre.
• (d)-1.0 dioptre — Right sign, wrong magnitude. A power of -1.0 dioptre belongs to a concave lens of focal length one metre, not two.

The power of a lens measures how strongly it bends light, and it is defined as the reciprocal of the focal length in metres. The unit is the dioptre, so a lens of focal length one metre has a power of one dioptre. A short focal length means a large power; the sign is positive for a converging lens and negative for a diverging one.

Two habits make this a one-line calculation. The first is to fix the sign before touching the arithmetic — concave means diverging means negative, without exception — because three of the four options here are separated by sign or by a factor rather than by any real difficulty. The second is to keep the focal length in metres. If it is given in centimetres, the working form is power equal to 100 divided by the focal length in centimetres, which is why a lens of focal length 10 cm has a power of 10 dioptre and not 0.1. The relationship also runs backwards, and questions often ask for it that way: a spectacle prescription of -2.0 dioptre describes a diverging lens of focal length half a metre, prescribed for short sight. When thin lenses are placed in contact their powers simply add, sign and all.

• Power equals one divided by the focal length in metres, and the unit is the dioptre.
• A converging or convex lens has positive power; a diverging or concave lens has negative power.
• With the focal length in centimetres the formula becomes power equal to 100 divided by that length.
• A short focal length means a large power, since the lens bends light more sharply.
• Thin lenses placed in contact have powers that add, which is how combinations are handled.

• Dropping the negative sign because the question gives only the magnitude of the focal length.
• Working in centimetres without converting, which changes the answer by a factor of a hundred.
• Reporting the focal length in place of the power when the two happen to be small numbers.

Asked as a one-step calculation in which the sign convention, not the arithmetic, is what is really being tested.

• practice — not a real PYQ

  The power of a convex lens of focal length 25 cm is

  • (a)+0.25 dioptre
  • (b)+2.5 dioptre
  • (c)+4 dioptre
  • (d)+25 dioptre

  Answer(c) +4 dioptre — 100 ÷ 25 = 4, positive because the lens converges light.

• practice — not a real PYQ

  Two thin lenses of powers +5 dioptre and -2 dioptre are placed in contact. The power of the combination is

  • (a)+7 dioptre
  • (b)+3 dioptre
  • (c)-3 dioptre
  • (d)+10 dioptre

  Answer(b) +3 dioptre — powers of thin lenses in contact add with their signs.