Refraction of light through a prism

A prism is a piece of transparent optical material, usually glass, bounded by two plane surfaces inclined to each other. The angle between these two refracting surfaces is called the angle of the prism, $A$. The third side, opposite the edge where the two refracting surfaces meet (the apex), is called the base of the prism — and it is the direction "towards the base" that keeps coming up as the key qualitative result of this topic.

• Two surfaces, not parallel. A rectangular slab has two refracting surfaces that are parallel to each other. A prism's two refracting surfaces are tilted with respect to one another, meeting at the apex angle $A$. This single geometric difference is the reason a prism behaves so differently from a slab.
• Refraction happens twice. Every ray that emerges from a prism has crossed two surfaces, each one bending the ray according to the same two laws of refraction learned for the slab: (i) the incident ray, the refracted ray and the normal at the point of incidence all lie in the same plane, and (ii) the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for the pair of media, given by Snell's law,

where: $i$ is the angle of incidence and $r$ is the angle of refraction, both measured from the normal at that surface, and $n_{21}$ is the refractive index of the second medium with respect to the first. At the first surface light goes from air into glass ($n_{21} > 1$), so the ray bends towards the normal; at the second surface it goes from glass back into air, so it bends away from the normal.

• The bends add up instead of cancelling. In a slab, the two normals are parallel, so the "towards" bend at entry and the "away" bend at exit are equal and opposite — the ray exits parallel to how it entered, just laterally shifted. In a prism, the two normals are not parallel (because the surfaces meet at angle $A$), so the two bends do not cancel. The net result is a genuine change of direction of the ray, called the angle of deviation, $\delta$ — the angle between the original incident-ray direction and the final emergent-ray direction.
• The direction is always towards the base. Whichever surface of a symmetric prism the ray enters, the cumulative effect of the two refractions always turns the ray towards the thicker part of the prism — the base — and away from the thin apex.

Trace the two bends in order and the direction of the final ray follows without needing any formula. At the first surface, the ray travels from air (rarer) into glass (denser). By the laws of refraction, it bends towards the normal at that surface — it is pulled closer to the perpendicular line as it slows down entering the denser medium. Inside the glass, the ray travels in a straight line until it reaches the second surface. There it travels from glass (denser) back out into air (rarer), so by the same laws it bends away from the normal — it speeds up again on leaving the denser medium and swings further from the perpendicular. The reason these two bends do not simply cancel, as they do for a slab, is that the normal at the first surface and the normal at the second surface point in different directions — because the two surfaces themselves are inclined to each other by the prism angle $A$. A bend "towards" one normal and a bend "away from" a differently-tilted normal do not add up to zero rotation; they add up to a net rotation of the ray in one consistent sense. Working through the geometry surface by surface shows that this net rotation always carries the ray towards the base of the prism, never towards the apex — the apex is the thin end where the two surfaces converge, and the ray is always turned away from it. This is also why a triangular prism is the standard tool for splitting white light into its colours in the very next topic of this chapter: since the refractive index of glass is slightly different for each colour of light, each colour bends by a slightly different angle of deviation at the very same two surfaces, fanning the colours apart as they all deviate towards the base by not-quite-equal amounts.

Question: A narrow beam of white light travelling through air strikes one face of a triangular glass prism held with its base pointing downward. Describe, surface by surface, what happens to the beam as it crosses the prism, and state the final direction of the emergent ray relative to the incident ray.

Step 1. The beam meets the first refracting surface. It travels from air (rarer medium) into glass (denser medium), so by the laws of refraction it bends towards the normal drawn at that point of incidence.

Step 2. Inside the prism, the refracted ray travels in a straight line across the glass until it strikes the second refracting surface.

Step 3. At the second surface, the ray travels from glass (denser medium) back out into air (rarer medium), so it bends away from the normal drawn at that point.

Step 4. Because the two surfaces are inclined to each other at the prism angle $A$ (not parallel, as in a slab), the "towards normal" bend at the first surface and the "away from normal" bend at the second surface do not cancel out angularly. They combine into one net rotation of the ray.

Answer: the emergent ray leaves the prism bent towards the base (downward, in this case, since the base points down), and it is no longer parallel to the incident ray — the angle between the two directions is the angle of deviation, $\delta$.

Competitive stretch

If the same prism is turned upside down, so its base now points upward, the reasoning of Steps 1 to 4 is unchanged — only the geometry is flipped. The emergent ray will now bend towards the new position of the base, i.e., upward. The rule "the ray always deviates towards the base" tracks the base wherever it is; it is not tied to any fixed direction like "downward."

• Thinking a prism produces only a lateral shift like a slab — a prism's inclined surfaces make the two refractions add up rather than cancel, producing a real change of direction (deviation), not just a sideways nudge.
• Forgetting the ray bends twice — once entering, once leaving — and mixing up which bend goes towards the normal and which goes away from it.
• Saying the ray bends towards the apex — it is always the base, the thick end, that the emergent ray leans towards.
• Trying to plug numbers into a formula linking $\delta$, $A$ and the refractive index — at this level the treatment is qualitative only; no such formula is required or expected.

Ask the tutor: Want a hand sketching the full ray diagram through a prism step by step, or curious how this same setup splits white light into a rainbow in the next topic?