For a prism with a small refracting angle θ, the ray is turned through α ≈ (n − 1)θ, whatever the (small) angle of incidence. For n = 1.5 and θ = 18° this is 9°. The exact value for α1 = 10° is 9.173°.
Problem statement
Show that in a prism with a small refracting angle θ the beam of light is deviated through the angle α ≈ (n − 1)θ, whatever the angle of incidence, provided that the angle of incidence is also small.
We assume the prism is in air and made of glass of refractive index n. This means Snell's law reads sin α1 = n sin β1 at the first face.
We assume the light has one colour. This means n has one value and there is no spreading into a spectrum.
We assume the ray crosses the prism from one slanted face to the other. This means it meets the two faces only once each.
Symbols: θ is the refracting (apex) angle. α1 is the angle of incidence on the first face. β1 and β2 are the angles of the ray inside the prism at the first and second faces. α2 is the angle of emergence. α is the total turn of the ray, called the deviation.
The big idea in plain words
Each face of the prism bends the ray toward the base. The two bends add up. For small angles, a bend at a face is a fixed fraction of the angle: the ray turns by (n − 1) times the angle inside. The two angles inside add to the apex angle θ. So the total turn is (n − 1)θ, and the angle of incidence has dropped out.
The solution in each frame
Lab frame
The prism is at rest. The full solution is in the next section.
Other frames
No other frame is needed. Nothing moves, so one frame is enough.
The solution step by step
The normals to the two faces meet at the same angle θ as the faces themselves. In the triangle formed by the two normals and the ray inside the prism, the outside angle θ equals the sum of the two opposite inside angles:
At the first face the ray turns toward the base by α1 − β1. At the second face it turns the same way by α2 − β2. The total turn is the deviation:
This result is exact. Now use Snell's law at each face, sin α1 = n sin β1 and sin α2 = n sin β2. For small angles, in radians, sin x is close to x, so
Put these in the exact result:
The angle of incidence has dropped out, so the deviation is the same for every small angle of incidence. For n = 1.5 and θ = 10° it is 5°. With your current values, (n − 1)θ = 9° and the exact deviation is 9.173°, a difference of 0.17° (1.9 %).
A separate check traces the ray with vector refraction at both faces, using the actual directions of the faces. The angle between the incoming and outgoing directions is 9.17312°. The angle formulas give 9.173°.
Five methods
1. Force method
Not needed. This is a ray-optics problem about directions, not forces.
2. Energy method
Not needed. Only the directions of the ray matter.
3. Graph method
The first graph plots the exact deviation against the angle of incidence. Near small angles it is almost flat, and it sits on the yellow line (n − 1)θ. The second graph shows the exact curve leaving the straight line when θ grows. The third shows α against n.
4. Calculus method
The deviation is smallest when the path is symmetric (α1 = α2). Then the derivative of α with respect to α1 is zero, which is why the deviation changes slowly with the angle of incidence near that path.
5. Check method
Limits: at n = 1 the deviation is zero, as it should be for no glass. At θ = 0 the prism is a flat plate and the deviation is zero. Units: all angles are in the same unit, but the small-angle step needs radians for sin x ≈ x.
Fastest: α = (n − 1)θ. Most reliable: the exact result α = α1 + α2 − θ with Snell's law at both faces.
What each slider changes
| Control | Changes the answer? | What it changes |
|---|---|---|
| Refractive index n | Yes | β1, β2, α2, α, the line (n − 1)θ, the drawing and all graphs |
| Apex angle θ | Yes | The shape of the prism, β2, α2, α and (n − 1)θ |
| Angle of incidence α1 | The exact α only | The path of the ray, α2 and the exact α. It does not change (n − 1)θ. |
| Wanted deviation (given-answer mode) | Sets θ | The model solves the apex angle for the wanted deviation |
Parameters
| Quantity | Symbol | Value used |
|---|---|---|
| Refractive index | n | 1.5 |
| Apex angle | θ | 18° |
| Angle of incidence | α1 | 10° |
| Angles inside | β1, β2 | 6.648°, 11.35° |
| Deviation (exact) | α | 9.173° |
| Deviation (small angles) | (n − 1)θ | 9° |
Exam traps
- Using (n − 1)θ for a thick prism or a large angle of incidence. It holds only when the angles are small.
- Writing the deviation as α1 − α2. It is α1 + α2 − θ.
- Measuring the apex angle θ outside the prism instead of between the two faces.
- Mixing degrees and radians in sin x ≈ x.
- Forgetting that the deviation is toward the base, not toward the apex.