Snell's Law

Physical Law

Consider a ray of light crossing the threshold between two media.

Let its velocity:

• in medium 1 be $v_1$,
• in medium 2 be $v_2$.

Let it meet the threshold at:

• an angle $\alpha_1$ from the vertical in medium 1,
• an angle $\alpha_2$ from the vertical in medium 2.

Then Snell's law states that:

$\displaystyle \frac {\sin \alpha_1} {v_1} = \frac {\sin \alpha_2} {v_2}$

Proof

Snell's law can be derived from Fermat's Principle as follows:

Let it travel from $A$ to $P$ in the medium 1.

Then let it travel from $P$ to $B$ in medium 2.

The total time $T$ required for that journey is:

$\displaystyle T = \frac {\sqrt{a^2 + x^2}} {v_1} + \frac {\sqrt{b^2 + \left({c - x}\right)^2}} {v_2}$

from the geometry of the above diagram.

From Fermat's Principle, this time will be a minimum.

From Derivative at Maximum or Minimum, we need $\dfrac{\mathrm{d}{T}}{\mathrm{d}{x}} = 0$.

So:

$\displaystyle \frac x {v_1 \sqrt{a^2 + x^2}} = \frac {c - x} {v_2 \sqrt{b^2 + \left({c - x}\right)^2}}$

which leads directly to:

$\displaystyle \frac {\sin \alpha_1} {v_1} = \frac {\sin \alpha_2} {v_2}$

by definition of sine.

$\blacksquare$

Source of Name

This entry was named for Willebrord Snell.

However, it had previously been discovered by several other scientists, including Ibn Sahl in 984 and Thomas Harriot in 1602.

It was also discovered independently by René Descartes in 1637. In France this law is called la loi de Descartes or loi de Snell-Descartes.

Sources

• George F. Simmons: Differential Equations (1972): $\S 6$