Lens formula, magnification and power of a lens
The lens formula 1/f = 1/v − 1/u links object distance, image distance and focal length. Power P = 1/f (f in metres) is measured in dioptre.
Cartesian sign convention
- All distances are measured from the optic centre (O) of the lens.
- Distances measured in the direction of the incident light (usually to the right) are positive. Distances against it are negative.
- So the object distance u is negative when the object is on the left of the lens.
- Heights above the principal axis are positive; heights below it are negative.
- The focal length of a convex lens is positive; that of a concave lens is negative.
Lens formula, magnification and lens maker's formula
The lens formula is 1/f = 1/v − 1/u. Here u is the object distance, v the image distance and f the focal length. Magnification (m) tells how many times bigger the image is: m = h′/h = v/u. Here h is the height of the object and h′ the height of the image. A negative m means a real, inverted image. A positive m means a virtual, erect image. If the size of m is more than 1, the image is magnified. Lens makers use the lens maker's formula to make a lens of a chosen focal length: 1/f = (μ − 1)(1/R₁ − 1/R₂). Here μ is the refractive index of the lens material. R₁ and R₂ are the radii of curvature of its two faces.
Power of a lens
The power (P) of a lens is its ability to converge or diverge light. It is the reciprocal of the focal length in metres: P = 1/f. The SI unit is the dioptre (D). 1 dioptre is the power of a lens of focal length 1 metre. A convex lens has positive power; a concave lens has negative power. A lens with a shorter focal length is more powerful.
Image formed by a convex lens
An object 2 cm tall is placed 30 cm in front of a convex lens of focal length 20 cm. Find the position, size and nature of the image.
- Given: u = −30 cm, f = +20 cm, h = 2 cm.
- Lens formula: 1/f = 1/v − 1/u, so 1/v = 1/f + 1/u.
- 1/v = 1/20 + 1/(−30) = 3/60 − 2/60 = 1/60.
- v = +60 cm, on the other side of the lens.
- Magnification: m = v/u = 60/(−30) = −2.
- Image height: h′ = m × h = −2 × 2 cm = −4 cm.
The image is 60 cm behind the lens and 4 cm tall. It is real and inverted (m is negative) and magnified two times.
Power of two lenses
Find the power of (i) a convex lens of focal length 25 cm (ii) a concave lens of focal length 50 cm.
- (i) f = +25 cm = +0.25 m. P = 1/f = 1/0.25 = +4 D.
- (ii) f = −50 cm = −0.5 m. P = 1/f = 1/(−0.5) = −2 D.
The convex lens has power +4 D (converging); the concave lens has power −2 D (diverging).
A spectacle prescription such as −1.5 D or +2.0 D gives the power of each lens. A minus sign means a concave lens (for myopia); a plus sign means a convex lens (for hypermetropia).
1 marks: “What is the SI unit of the power of a lens?” Dioptre (D).
2 marks: “A concave lens has a focal length of 20 cm. Find its power.” f = −20 cm = −0.2 m; P = 1/f = 1/(−0.2) = −5 D.
4 marks: “An object is placed 15 cm from a convex lens of focal length 10 cm. Find the image distance and the magnification.” 1/v = 1/10 + 1/(−15) = 3/30 − 2/30 = 1/30, so v = +30 cm; m = v/u = 30/(−15) = −2: real, inverted, twice the size.
Power is one over f in metres: change centimetres to metres first, or your dioptres will be wrong by 100 times.