Refraction of light and Snell's law

Refractive index

• Light travels fastest in a vacuum, at $c = 3.0 \times 10^{8}$ m s⁻¹, and more slowly in every material.
• The refractive index $n$ of a material measures how much it slows light down:

$$n = \frac{c}{v}$$

• $n$ — refractive index (no units — it is a ratio)

• $c$ — speed of light in a vacuum ($3.0 \times 10^{8}$ m s⁻¹)

• $v$ — speed of light in the material (m s⁻¹)

• Because light always slows, $n$ is always greater than 1.

• Typical values worth recognising:

  • vacuum: exactly $1$; air: $1.00$ (treated as $1$),
  • water: $1.33$,
  • glass: about $1.5$,
  • diamond: $2.42$.

• A larger $n$ means a slower speed, a shorter wavelength inside the material, and more bending at the surface. The material is described as optically denser.

  • "Optically denser" is not the same as "heavier". It refers only to how much the material slows light.

Snell's law

$$n_1 \sin\theta_1 = n_2 \sin\theta_2$$

• $n_1$ — refractive index of the medium the light starts in

• $\theta_1$ — angle of incidence, measured from the normal

• $n_2$ — refractive index of the medium the light enters

• $\theta_2$ — angle of refraction, measured from the normal

• The equation is symmetrical: light travelling the other way along the same path refracts by the same angles. Light paths are reversible.

• The direction of bending follows directly:

  • entering a denser medium ($n_2 > n_1$): $\theta_2 < \theta_1$, so the ray bends toward the normal,
  • entering a less dense medium ($n_2 < n_1$): $\theta_2 > \theta_1$, so the ray bends away from the normal,
  • at $\theta_1 = 0°$ (along the normal): $\sin 0° = 0$, so $\theta_2 = 0°$ — no bending, though the speed still changes.

Change the angle of incidence and the refractive index below, and watch the refracted ray respond:

Light through a parallel-sided block

• A ray entering a rectangular glass block bends toward the normal.
• At the far surface it bends away from the normal by exactly the same amount.
• The emerging ray is therefore parallel to the original ray, but displaced sideways.
• The bigger the refractive index or the angle of incidence, the bigger the sideways shift.

Applying Snell's law

• Draw the normal at the surface and label the two angles.
• Identify which medium is 1 (where the light starts) and which is 2.
• Substitute into $n_1\sin\theta_1 = n_2\sin\theta_2$.
• Rearrange for the unknown, then take $\sin^{-1}$ if you are solving for an angle.
• Check the direction of bending is consistent with which medium is denser.