Refraction through prisms

We start from pure geometry in the ray diagram — no optics at all in part (a) — and reach the two equations that every prism question is built on.

Setting up. The ray strikes the first face at Q with angle of incidence i_1 and is refracted at r_1. Inside the glass it travels to S on the second face, meets the normal there at r_2, and emerges at i_2. Let N be the point where the two normals (at Q and at S) cross. A is the refracting angle of the prism, at the vertex where the two faces meet.

Part (a): A = r_1 + r_2

Step 1 — look at the quadrilateral AQNS. NQ is the normal at Q, so it is perpendicular to the first face; NS is the normal at S, so it is perpendicular to the second face. Therefore

Step 2 — add up the four angles of the quadrilateral. They total 360^{\circ}:

Step 3 — subtract 180^{\circ} from both sides:

Step 4 — now look at triangle QNS. Its three angles are r_1 at Q, r_2 at S, and \angle QNS at N, and they add to 180^{\circ}:

Step 5 — compare Steps 3 and 4. Both left-hand sides equal 180^{\circ}:

Step 6 — cancel \angle QNS from both sides:

Part (b): \delta = i_1 + i_2 - A

Step 7 — how much the ray turns at the first face. It arrives along a direction making i_1 with the normal and leaves making r_1 with the same normal, so it has been turned through

Step 8 — how much it turns at the second face. Same argument, in the same rotational sense:

Step 9 — add the two turns. Both bend the ray the same way, so the total deviation \delta (the exterior angle of the triangle formed by the incident ray produced and the emergent ray produced backwards) is their sum:

Step 10 — group the i's and the r's:

Step 11 — replace (r_1 + r_2) by A, using Part (a):

Result.

What it means. The first relation is purely geometrical — it holds whatever the glass is made of. The second says the prism's total bending is shared between the two faces. Together they reduce four unknown angles to two, which is why every prism numerical is solvable.

Conditions used. The ray must lie in a plane perpendicular to the refracting edge (the "principal section"), and it must actually emerge at the second face — if r_2 exceeds the critical angle, there is no i_2 and these relations cannot be used.

We start from the shape of the \delta–i_1 graph together with the principle of reversibility, and reach the symmetry condition i_1 = i_2, r_1 = r_2.

Step 1 — read the graph. Draw any horizontal line above the lowest point of the curve. It cuts the curve twice. So for every deviation \delta greater than the minimum there are two different angles of incidence that produce the very same deviation. Call them i and i'.

Step 2 — apply the principle of reversibility. If a ray enters at i_1 and emerges at i_2, then reversing its direction sends a ray in at i_2 which emerges at i_1 — along the same path, so with the same deviation \delta.

Step 3 — identify the two roots. Step 2 gives us two angles of incidence with the same \delta, namely i_1 and i_2. Step 1 says there are only two. So the pair in Step 1 must be

Step 4 — now slide the horizontal line down. As \delta decreases towards \delta_m, the two intersection points move towards each other. At the minimum itself the line touches the curve at exactly one point, so the two roots have merged:

Step 5 — carry this inside the prism. Snell's law at each face gives \sin i_1 = n\sin r_1 and \sin i_2 = n\sin r_2. Equal i's therefore force equal r's:

Result. At minimum deviation the ray passes symmetrically, i_1 = i_2 and r_1 = r_2, and therefore the refracted ray inside the prism runs parallel to the base.

What it means. Symmetry is not an extra assumption we make for convenience — it is forced on us by the shape of the curve. This is why the prism formula of the next section is exact, not approximate.

Conditions used. One prism, one colour, and a ray that actually emerges from the second face.

We start from the two prism relations, add the fact that at minimum deviation the ray passes symmetrically, and reach a formula for n from two measurable angles.

Setting up. At minimum deviation the path through the prism is symmetrical: the ray inside runs parallel to the base, the angle of incidence equals the angle of emergence, and the two internal angles are equal. So

Step 1 — put the equal internal angles into A = r_1 + r_2:

Step 2 — divide both sides by 2:

Step 3 — put the equal external angles into \delta = i_1 + i_2 - A:

Step 4 — add A to both sides:

Step 5 — divide both sides by 2:

Step 6 — apply Snell's law at the first face, where light goes from air into glass:

Step 7 — substitute the expressions for i and r from Steps 5 and 2:

Result.

What it means. This is the prism formula, and it is the standard accurate method for finding n. Everything on the right is an angle, and a spectrometer measures angles to within a few minutes of arc — far more precisely than we could measure a length or a depth. Note also that a larger \delta_m for the same prism means a larger n, which is why violet light (deviated most) has the largest refractive index.

Conditions used. The prism must be set at minimum deviation, so that the path is symmetrical; monochromatic light must be used, since n and \delta_m both depend on colour; and the surrounding medium must be air.

We start from Snell's law at the two faces with all angles small, and reach a deviation formula that no longer contains the angle of incidence at all.

Setting up. The refracting angle A is small (a few degrees) and the ray enters nearly normally. Then A = r_1 + r_2 forces r_1 and r_2 to be small, and Snell's law then makes i_1 and i_2 small too. All four are small angles measured in radians.

Step 1 — Snell's law at the first face:

Step 2 — use the small-angle approximation \sin\theta \approx \theta:

Step 3 — make i_1 the subject:

Step 4 — repeat the same three steps at the second face:

Step 5 — substitute both into \delta = i_1 + i_2 - A:

Step 6 — take out the common factor n:

Step 7 — replace (r_1 + r_2) by A, using the first prism relation:

Step 8 — take out the common factor A:

Result.

What it means. The angle of incidence has vanished from the answer. For a thin prism the deviation is fixed by the prism itself — its angle and its glass — so you do not need to hunt for the minimum-deviation position. And because n depends on colour, so does \delta: this one line is the seed of the whole theory of dispersion.

Conditions used. A must be small (in practice under about 10^{\circ}) and the ray must strike nearly normally, so that \sin\theta \approx \theta is safe in Step 2. For a thick prism or a slanting ray the formula fails and the full prism formula must be used.