Refraction at a spherical surface
Take a curved glass surface of radius R separating medium n₁ (where the object is) from medium n₂. A point object O on the axis sends one ray along the axis and one ray to a point N on the surface. For rays close to the axis (paraxial), angles are small, so tan θ ≈ θ ≈ sin θ.
Step 1: angles
Let α, β, γ be the angles that ON, NC and NI make with the axis (C is the centre of curvature, I the image). From the triangles: i = α + γ and r = γ − β.
Step 2: Snell's law for small angles
n₁ i = n₂ r, so n₁(α + γ) = n₂(γ − β).
Step 3: put in distances
α ≈ MN/OM, γ ≈ MN/MC, β ≈ MN/MI. Cancel MN, use signs (OM = −u, MC = +R, MI = +v):
n₂/v − n₁/u = (n₂ − n₁)/R
This one formula covers convex and concave surfaces, real and virtual images, if signs are used correctly.
Lens maker's formula
A thin lens (index n) in air has two surfaces with radii R₁ and R₂.
Step 1: first surface
Light goes from air (1) into glass (n). It would form an image I₁ at v₁: n/v₁ − 1/u = (n − 1)/R₁.
Step 2: second surface
I₁ acts as the object for the second surface, where light goes from glass (n) into air (1). For a thin lens, the thickness is ignored: 1/v − n/v₁ = (1 − n)/R₂.
Step 3: add
1/v − 1/u = (n − 1)(1/R₁ − 1/R₂).
Step 4: object at infinity
If u = ∞, the image is at the focus, v = f:
1/f = (n − 1)(1/R₁ − 1/R₂)
For a double convex lens R₁ is + and R₂ is −, so f is +. If the lens sits in a liquid of index n_m, replace n by n/n_m. If n = n_m, the lens stops bending light at all.
Thin lens formula
Putting the lens maker's result into Step 3 above gives the thin lens formula:
1/v − 1/u = 1/f
Use the same Cartesian sign convention as for mirrors, with distances measured from the optical centre O. A convex (converging) lens has f positive; a concave (diverging) lens has f negative. A real image forms on the far side (v positive); a virtual image forms on the object side (v negative).
Magnification by a lens
m = h′/h = v/u (no minus sign for lenses, unlike mirrors).
- m negative → real and inverted.
- m positive → virtual and erect.
Other forms: m = f/(f + u) = (f − v)/f.
Power of a lens
The power P of a lens tells how strongly it bends light: P = 1/f, with f in metres. Its unit is the dioptre (D): 1 D = 1 m⁻¹. A convex lens has positive power; a concave lens has negative power. A lens of focal length 25 cm has P = 1/0.25 = +4 D.
Using the lens maker's formula, P = (n − 1)(1/R₁ − 1/R₂): more curved faces and a higher n give more power.
Thin lenses in contact
Put lens A (f₁) touching lens B (f₂). Lens A alone forms an image at v₁: 1/v₁ − 1/u = 1/f₁. This image is the object for lens B: 1/v − 1/v₁ = 1/f₂. Add the two:
1/v − 1/u = 1/f₁ + 1/f₂, so the pair acts as one lens with 1/F = 1/f₁ + 1/f₂ and P = P₁ + P₂.
The total magnification is the product: m = m₁ × m₂. Camera and microscope lenses combine lenses this way to get the power they need and to reduce colour blurring.
Try it: measure f with the Sun
Take a magnifying glass outdoors. Hold a sheet of paper under it and move the lens up and down until the Sun's spot is smallest and brightest. (Do not look at the Sun and do not leave it on the paper.) The distance from lens to paper is f. Work out its power in dioptres: P = 1/f (in m). Then hold the lens close to printed text: the letters look bigger and upright — the object is inside F. In the 3D, check this with the slider.
Key formulas and definitions
- n₂/v − n₁/u = (n₂ − n₁)/R
- 1/f = (n − 1)(1/R₁ − 1/R₂)
- 1/v − 1/u = 1/f
- m = h′/h = v/u
- P = 1/f (f in metres), unit dioptre D
- Lenses in contact: 1/F = 1/f₁ + 1/f₂, P = P₁ + P₂, m = m₁m₂
Worked examples
1. A double convex lens (n = 1.5) has both radii 20 cm. Find its focal length.
Step 1: R₁ = +20, R₂ = −20. Step 2: 1/f = (1.5 − 1)(1/20 + 1/20) = 0.5 × 0.1 = 0.05. Step 3: f = 20 cm.
2. An object is 30 cm from a convex lens of focal length 20 cm. Find the image position, magnification and nature.
Step 1: u = −30, f = +20. Step 2: 1/v = 1/f + 1/u = 1/20 − 1/30 = 1/60. Step 3: v = +60 cm. Step 4: m = v/u = 60/−30 = −2. Real, inverted, twice as big, 60 cm on the other side.
3. An object is 10 cm from a convex lens of f = 15 cm. Find the image.
Step 1: u = −10, f = +15. Step 2: 1/v = 1/15 − 1/10 = −1/30. Step 3: v = −30 cm. Step 4: m = −30/−10 = +3. Virtual, erect, 3× bigger, on the object side.
4. A concave lens has f = 20 cm. An object is 20 cm away. Find the image and power.
Step 1: u = −20, f = −20. Step 2: 1/v = −1/20 − 1/20 = −1/10, v = −10 cm. Step 3: m = −10/−20 = +0.5. Step 4: P = 1/(−0.2) = −5 D. Virtual, erect, half size.
5. Find the power of a lens of focal length −50 cm, and say what kind of lens it is.
Step 1: f = −0.5 m. Step 2: P = 1/f = −2 D. Step 3: negative power → concave (diverging) lens.
6. Two thin lenses of powers +5 D and −2 D are in contact. Find the focal length of the pair.
Step 1: P = P₁ + P₂ = +3 D. Step 2: F = 1/P = 0.333 m ≈ 33.3 cm. It behaves as a converging lens.
7. A glass lens (n = 1.5) has f = 20 cm in air. What is its focal length in water (n = 4/3)?
Step 1: in air 1/20 = (1.5 − 1) K, so K = (1/R₁ − 1/R₂) = 0.1. Step 2: in water use n_rel = 1.5/(4/3) = 1.125: 1/f′ = 0.125 × 0.1 = 0.0125. Step 3: f′ = 80 cm. The lens is 4 times weaker in water.
8. A convex surface of glass (n = 1.5, R = 10 cm) faces a point object in air 30 cm away. Where is the image inside the glass?
Step 1: n₁ = 1, n₂ = 1.5, u = −30, R = +10. Step 2: 1.5/v − 1/(−30) = 0.5/10. Step 3: 1.5/v = 0.05 − 0.0333 = 0.0167. Step 4: v = 90 cm inside the glass (real).
9. A plano-convex lens (n = 1.5) must have power +2 D. What radius should the curved face have?
Step 1: flat face R₂ = ∞, so 1/f = (n − 1)/R₁. Step 2: 2 = 0.5/R₁. Step 3: R₁ = 0.25 m = 25 cm.
Common mistakes
- Using the mirror formula 1/v + 1/u = 1/f for a lens. For lenses it is 1/v − 1/u = 1/f.
- Putting f in cm into P = 1/f. Power in dioptres needs f in metres (or use P = 100/f with f in cm).
- Taking both radii as positive for a double convex lens. With the sign convention R₁ = +R, R₂ = −R.
- Adding focal lengths of lenses in contact. You add powers (or 1/f), not f.