Snell's Law and Refractive Index

Snell’s law describes the relationship between the angle of incidence and the angle of refraction when light passes between two media. The refractive index quantifies how much a medium slows light down.

Refractive index

$n = \frac{\text{speed of light in vacuum}}{\text{speed of light in the medium}}$

A higher refractive index means the medium slows light more. Glass has $n \approx 1.5$; water has $n \approx 1.33$; air has $n \approx 1.00$.

Snell’s law

For light passing from air (or vacuum) into a medium of refractive index $n$:

$n = \frac{\sin i}{\sin r}$

where $i$ is the angle of incidence (in air) and $r$ is the angle of refraction (in the medium).

Critical angle and refractive index

$n = \frac{1}{\sin c}$

where $c$ is the critical angle for the medium.

Worked example: finding the angle of refraction

Light enters a glass block ($n = 1.5$) at an angle of incidence of 45$^\circ$. Find the angle of refraction.

$\sin r = \frac{\sin i}{n} = \frac{\sin 45}{1.5} = \frac{0.707}{1.5} = 0.471$

$r = \sin^{-1}(0.471) = 28^\circ$

Worked example: finding the critical angle

Find the critical angle for glass with $n = 1.5$.

$\sin c = \frac{1}{n} = \frac{1}{1.5} = 0.667$

$c = \sin^{-1}(0.667) = 42^\circ$

Common errors and how to correct them

Putting the angles in the wrong positions in Snell’s law. The angle of incidence $i$ is measured in the less dense medium (usually air). The angle of refraction $r$ is measured in the denser medium. $n = \sin i / \sin r$, not $\sin r / \sin i$.

Using degrees in a calculator set to radians. Ensure your calculator is in degree mode when calculating $\sin$ and $\sin^{-1}$.

How to apply this in an exam

State Snell’s law, substitute the known values, and solve for the unknown angle. For the critical angle, use $n = 1 / \sin c$. Show all working including the $\sin$ and $\sin^{-1}$ steps.