Refraction at a Curved Surface
The single curved boundary that lenses, eyes and fish bowls are built from.
Lesson at a glanceShow the key points
- Same coordinate sign convention: origin where the axis meets the surface, positive along the incident light.
- From the triangle angles and small-angle Snell's law μ₁i = μ₂r: μ₂/v − μ₁/u = (μ₂ − μ₁)/R (18.12).
- A ray toward the centre of curvature passes undeviated, which locates the image of an extended object.
- Lateral magnification m = μ₁v/(μ₂u) (18.13).
Before lenses, master one curved boundary. The front of your eye, the side of a goldfish bowl and each face of a lens are all single spherical surfaces between two media.
Setting up
Two transparent media, of refractive indices μ₁ and μ₂, are separated by a spherical surface with centre of curvature C. A point object O in medium 1 lies on the line through C, which meets the surface at P. We use the same sign convention as for mirrors: P is the origin, the axis is OPC, positive in the direction of the incident light; u, v and R are the x-coordinates of object, image and centre.
Deriving the formula
The ray OP hits the surface normally and goes straight on. Another ray OD makes a small angle α with the axis and refracts along DI, where the image I forms. CD is the normal at D. With β the angle DCP and γ the angle DIC, the triangles give
For small angles Snell’s law becomes μ₁i = μ₂r, and α ≈ DP/PO, β = DP/PC, γ ≈ DP/PI. Putting these together:
With PO = −u, PI = +v, PC = +R:
An object in air is 20 cm from a convex glass surface (μ = 1.5) of radius 40 cm. Locate the image.
Extended objects: magnification
A ray from the tip Q of an extended object heading for C meets the surface normally and goes straight on, so the image of Q lies on the line QC. Similar triangles and the formula above give the lateral magnification:
A small 2.0 cm fish is 30 cm from the near side of a spherical bowl of water (μ = 1.33), seen by someone outside. The bowl’s surface has radius 15 cm. Ignore the thin glass. Where does the fish appear, and how big?
- 1μ₂/v − μ₁/u = (μ₂ − μ₁)/R, using the coordinate sign convention.
- 2A ray aimed at the centre of curvature passes undeviated.
- 3Lateral magnification m = μ₁v/(μ₂u).
- 4R → ∞ recovers refraction at a flat surface.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.