Power of a lens: P = 1/f in dioptre
Power P = 1/f, with f in metres, in dioptre (D). A convex lens of 50 cm has P = +2 D, and a concave lens of 20 cm has P = -5 D.
The ray diagram
Drag the object or type values. Signs follow the New Cartesian convention: distances are measured from the pole or optical centre, along the incident light is positive, against it negative, and heights above the axis are positive. So u is negative for a real object.
Convex lens of focal length +50 cm. An object +2 cm tall at u = -100 cm: the image is at v = +100 cm, real, inverted, same size.
Show the working
- f = +50 cm, u = -100 cm (the object is in front, so u is negative), h = +2 cm.
- 1/v = 1/f + 1/u = 1/(+50) + 1/(-100) = 50/5000 = 1/100
- v = +100 cm: positive, so the image is behind the lens and real.
- m = v/u = (+100)/(-100) = -1: negative, so inverted, and the same size.
- h′ = m × h = -1 × +2 = -2 cm
- Power P = 1/f = 1/(+0.5 m) = +2 D
Convert to metres, then take 1/f
- Convex lens, f = +50 cm = +0.5 m.
- P = 1/f = 1/(+0.5 m) = +2 D. Positive: a converging lens.
- Concave lens, f = -20 cm = -0.2 m.
- P = 1/f = 1/(-0.2 m) = -5 D. Negative: a diverging lens.
About power
- 1 dioptre is the power of a lens of focal length 1 metre: 1 D = 1 m⁻¹.
- Convex lenses have positive power and concave lenses negative power.
- A shorter focal length bends light more, so the power is larger.
- For thin lenses in contact the powers add: P = P₁ + P₂ + P₃ and so on. Opticians prescribe spectacles this way.
Quick answers
What is the SI unit of power of a lens?
The dioptre, D. A lens of focal length 1 m has a power of 1 D, so 1 D = 1 m⁻¹.
Why is the power of a concave lens negative?
Its focal length is negative in the sign convention, so P = 1/f is negative. Negative power means the lens spreads light out.
What is the power of a convex lens of focal length 50 cm?
f = +0.5 m, so P = 1/0.5 = +2 D.