Refraction Through a Prism
A triangular prism is a transparent solid bounded by two flat refracting faces meeting at the apex. The angle between these faces, A, is the angle of the prism. A ray that enters one face is refracted, travels through the glass, and refracts again on leaving the second face. The net result is that the emergent ray is bent — or deviated — from its original direction by an angle δ.
The geometry
Let i be the angle of incidence on the first face and e the angle of emergence at the second face. Inside the prism the ray makes refraction angles r₁ and r₂ with the two normals. From the geometry of the triangle formed by the ray and the two normals:
$$ r_1 + r_2 = A \qquad \text{and} \qquad \delta = (i + e) - A $$Snell's law
Applying Snell's law at the two faces (with air on the outside):
$$ \sin i = \mu \sin r_1, \qquad \sin e = \mu \sin r_2 $$Together these four relations let us solve for every internal angle and for the deviation δ given just(i, A, μ).
Minimum deviation
If you plot δ against i (try the Graphs page!) you see a smooth U-shape: δ falls, reaches a minimum, then rises again. At that minimum the path is symmetric — the ray inside the prism runs parallel to the base and i = e, r₁ = r₂ = A/2. This gives the classic prism formula used to measure μ:
$$ \mu = \dfrac{\sin\left(\tfrac{A + \delta_{\min}}{2}\right)}{\sin\left(\tfrac{A}{2}\right)} $$What controls δ?
- Angle of incidence (i) — δ first decreases, hits δmin, then increases.
- Refractive index (μ) — δ grows monotonically with μ.
- Apex angle (A) — δ grows with A.
- Wavelength (λ) — μ rises as λ falls (normal dispersion), so violet deviates more than red. This is what splits white light into a spectrum.
Why toward the base?
At each face the ray bends toward the normal as it enters denser glass, then away from the normal as it leaves. Because the two normals tilt outward from the apex, both refractions push the ray in the same rotational sense — toward the thicker part of the prism, i.e. its base.