Worked example — Chromatic refractive index

Problem

A Crown glass prism has refractive index $n_R = 1{,}514$ for red ($\lambda = 700\;\text{nm}$) and $n_V = 1{,}528$ for violet ($\lambda = 400\;\text{nm}$). Calculate the angle of refraction for a ray incident at $\theta_i = 60^\circ$ on the first face, for the two chromatic extremes.

Solution

Setup. Passage from air ($n\approx1$) to glass. Snell’s law $1\cdot\sin\theta_i = n\,\sin\theta_r$, so $\sin\theta_r = \sin\theta_i / n$. With $\sin 60^\circ = 0{,}866$.

Red ($n_R = 1{,}514$):

$\sin\theta_R = \frac{0{,}866}{1{,}514} = 0{,}572 \quad\Rightarrow\quad \ev{\theta_R \approx 34{,}9^\circ}$

Violet ($n_V = 1{,}528$):

$\sin\theta_V = \frac{0{,}866}{1{,}528} = 0{,}567 \quad\Rightarrow\quad \ev{\theta_V \approx 34{,}5^\circ}$

Comment. The difference is only $\sim 0{,}4^\circ$ at the first face, but crossing both faces of the prism it accumulates to $\sim 1$–$2^\circ$: enough to visibly separate the colours (chromatic dispersion). Violet, with the larger $n$, is deviated more.

Links

Topics: Ottica Concepts: Dispersione cromatica · Rifrazione e legge di Snell