06 · 20 min
Light
Reflection, refraction, the critical angle, and the thin-lens equation.
n = sin i / sin r
The idea
Angles of rays are measured from the normal, the line at 90° to the surface, not from the surface itself. The law of reflection: angle of incidence equals angle of reflection.
When a ray enters a slower medium such as glass it bends towards the normal. For a ray from air, Snell’s law is n = sin i / sin r. The general form is n1 sin θ1 = n2 sin θ2. The refractive index of a material compared with air is also the ratio of speeds, n = c / v.
Going from glass towards air, the ray bends away from the normal. Past the critical angle it does not leave at all: it reflects inside. That is total internal reflection, and sin c = 1/n for a boundary with air. For a converging lens, this course uses the real-is-positive convention: 1/f = 1/u + 1/v.
Formulas
Snell’s law from air
n = sin i / sin r
The incident ray is in air. i and r are from the normal, in degrees in this solver.
- n
- refractive index
- —
- i
- angle of incidence
- °
- r
- angle of refraction
- °
Critical angle
sin c = 1 / n
From the denser medium into air. There is no critical angle the other way.
- c
- critical angle
- °
- n
- refractive index
- —
Thin converging lens
1/f = 1/u + 1/v
Real-is-positive. A negative v from the solver means a virtual image. Object must be beyond the focus for a real image.
- f
- focal length
- cm
- u
- object distance
- cm
- v
- image distance
- cm
Worked examples
A ray into glass
A ray in air hits glass of refractive index 1.50 at 30° to the normal. Find the angle of refraction.
- i
- 30°
- n
- 1.50
Try it on paper first. Then open the working one step at a time.
Critical angle of glass
Glass has refractive index 1.50. Find the critical angle for a ray going from glass into air.
- n
- 1.50
Try it on paper first. Then open the working one step at a time.
Checks
A ray in air strikes glass (n = 1.50) at 60° to the normal. What is the angle of refraction, to the nearest degree?
A medium has refractive index 2.0 for a boundary with air. What is the critical angle, in degrees?
An object is 60 cm from a converging lens of focal length 20 cm. Use 1/f = 1/u + 1/v. What is the image distance, in cm?
Total internal reflection happens when a ray
Easy places to lose a mark
- Measuring angles from the glass surface instead of from the normal.
- Using sin c = n instead of sin c = 1/n.
- In the lens equation, subtracting the reciprocals in the wrong order. 1/v = 1/f − 1/u.
Practical
Ray box
Log angles and see that sin i / sin r stays equal to n.