Convex and concave lenses: the basic terms
A lens is a piece of transparent material with at least one curved surface. It works by refraction, not reflection.
- Convex (converging) lens: thicker at the middle; bends parallel rays to meet at a point.
- Concave (diverging) lens: thinner at the middle; bends parallel rays outward so they seem to come from a point.
Terms
- Optical centre (O): the central point of the lens; a ray through it goes straight on.
- Centres of curvature (C₁, C₂): a lens has two curved surfaces, so two centres. Principal axis: the line through both.
- Principal foci (F₁, F₂): one on each side. For a convex lens parallel rays meet at F₂; for a concave lens they seem to come from F₁.
- Focal length (f): distance from O to a principal focus. Aperture: the diameter of the lens.
- The points at twice the focal length are marked 2F₁ and 2F₂ in ray diagrams.
Rules for drawing ray diagrams for lenses
- A ray parallel to the principal axis passes through F₂ after refraction (convex) or seems to come from F₁ (concave).
- A ray through F₁ (convex) or heading toward F₂ (concave) comes out parallel to the axis.
- A ray through the optical centre O goes straight through without bending.
Any two rays are enough. In the 3D bench the orange ray uses rule 1 and the teal ray uses rule 3; dotted lines mean the rays only seem to meet there (virtual image).
Image formation by a convex lens (every object position)
| Object position | Image position | Size | Nature |
|---|---|---|---|
| At infinity | At F₂ | Point-sized, highly diminished | Real, inverted |
| Beyond 2F₁ | Between F₂ and 2F₂ | Diminished | Real, inverted |
| At 2F₁ | At 2F₂ | Same size | Real, inverted |
| Between F₁ and 2F₁ | Beyond 2F₂ | Enlarged | Real, inverted |
| At F₁ | At infinity | Highly enlarged | Real, inverted |
| Between F₁ and O | Same side as the object | Enlarged | Virtual, erect |
Uses of these cases: object far away → camera; object between F₁ and 2F₁ → projector; object inside F₁ → magnifying glass.
Image formation by a concave lens
| Object position | Image position | Size | Nature |
|---|---|---|---|
| At infinity | At F₁ | Point-sized | Virtual, erect |
| Between infinity and O | Between F₁ and O, same side | Diminished | Virtual, erect |
A concave lens can never form a real image of a real object. That is why it is used in peepholes on doors (a wide, small, upright view) and to correct short sight.
Sign convention and the lens formula
Lenses use the same New Cartesian sign convention as mirrors, with the optical centre O as the origin. Light travels left to right, so u is negative. Distances to the right of O are positive.
- Convex lens: f positive. Concave lens: f negative.
- Real image (on the other side): v positive. Virtual image (same side as object): v negative.
Lens formula: 1/v − 1/u = 1/f
Magnification: m = h′/h = v/u
Positive m → virtual and erect; negative m → real and inverted. Notice the lens formula has a minus sign and m = +v/u, while the mirror formula has a plus sign and m = −v/u.
Power of a lens
A lens with a short focal length bends light more strongly. We measure this with power:
P = 1/f (f in metres)
- SI unit: dioptre (D). 1 D is the power of a lens with a focal length of 1 m.
- Convex lens: positive power. Concave lens: negative power.
- If f is in cm, use P = 100/f.
- Lenses in contact: powers simply add, P = P₁ + P₂ + ….
Board pointers: ray diagrams for a convex lens (2–3 marks), a numerical using the lens formula and magnification (3 marks), and power / dioptre questions (1–2 marks) are asked almost every year.
Key formulas and definitions
- Lens formula: 1/v − 1/u = 1/f
- Magnification: m = h′/h = v/u
- Power: P = 1/f (f in metres) = 100/f (f in cm), unit dioptre (D)
- Lenses in contact: P = P₁ + P₂
- Convex lens: f and P positive; concave lens: f and P negative
Worked examples
1. Find the power of a convex lens of focal length 50 cm.
f = +0.5 m. P = 1/f = 1/0.5 = +2 D.
2. An optician prescribes a lens of power −2.5 D. What kind of lens is it and what is its focal length?
Negative power → concave lens. f = 1/P = 1/(−2.5) = −0.4 m = −40 cm.
3. An object is 15 cm from a convex lens of focal length 10 cm. Find the image position and magnification.
u = −15, f = +10. 1/v = 1/f + 1/u = 1/10 − 1/15 = (3 − 2)/30 = 1/30, so v = +30 cm. m = v/u = 30/(−15) = −2. Real, inverted, twice as big, 30 cm on the other side.
4. An object is 10 cm from a convex lens of focal length 15 cm. Describe the image.
u = −10, f = +15. 1/v = 1/15 − 1/10 = (2 − 3)/30 = −1/30, so v = −30 cm. m = v/u = (−30)/(−10) = +3. Virtual, erect, 3 times enlarged, on the same side as the object: a magnifying glass.
5. An object is 30 cm from a concave lens of focal length 15 cm. Find the image.
u = −30, f = −15. 1/v = −1/15 − 1/30 = (−2 − 1)/30 = −3/30, so v = −10 cm. m = (−10)/(−30) = +1/3. Virtual, erect, one-third size, 10 cm from the lens on the object's side.
6. A convex lens forms a real image 3 times the size of the object on a screen 60 cm from the lens. Find the object distance, focal length and power.
Real image: v = +60 cm, m = −3. m = v/u → u = v/m = 60/(−3) = −20 cm. 1/f = 1/v − 1/u = 1/60 + 1/20 = 4/60, so f = +15 cm. P = 100/15 ≈ +6.67 D.
7. Two thin lenses of power +2.5 D and −1.0 D are kept in contact. Find the power and focal length of the combination.
P = P₁ + P₂ = 2.5 − 1.0 = +1.5 D. f = 1/P = 1/1.5 ≈ 0.667 m ≈ 66.7 cm. The combination acts as a convex lens.
Common mistakes
- Using the mirror formula (1/v + 1/u) for a lens. For lenses it is 1/v − 1/u = 1/f.
- Putting f in centimetres in P = 1/f. Use metres, or use P = 100/f with f in cm.
- Writing m = −v/u for lenses. For lenses m = v/u.
- Saying a concave lens can form a real image of a real object. It always forms a virtual, erect, diminished image.