Define refractive index, n, as the ratio of the speeds of a wave in two different regions

3.2.2 Refraction of Light

Objective

Define the refractive index, $n$, as the ratio of the speeds of a wave in two different regions and use it to predict the behaviour of light at a boundary.

Key Terminology

• Normal – a line perpendicular to the surface at the point of incidence.
• Angle of incidence, θ₁ – the angle between the incident ray and the normal.
• Angle of refraction, θ₂ – the angle between the refracted ray and the normal (measured in the second medium).
• Refractive index, n – a dimension‑less quantity that expresses how much the speed of a wave changes when it passes from one medium to another.

Definition of Refractive Index

The refractive index of medium 2 relative to medium 1 is

\[ n_{12}= \frac{v_{1}}{v_{2}} \] where \(v_{1}\) and \(v_{2}\) are the speeds of the wave in the first and second medium respectively.

If the first medium is vacuum (or air, which may be treated as vacuum for IGCSE), the absolute refractive index of a medium is written simply as

\[ n = \frac{c}{v}, \] with \(c = 3.00\times10^{8}\ \text{m s}^{-1}\) the speed of light in vacuum and \(v\) the speed in the material.

Typical Values (at 589 nm – sodium D‑line)

Material	Refractive Index $n$
Air (≈ vacuum)	1.0003
Water	1.33
Glass (crown)	1.52
Glass (flint)	1.62
Diamond	2.42

Note: The refractive index varies slightly with wavelength (dispersion). The table shows values measured at the sodium D‑line (589 nm).

Using the Refractive Index

1. Find the speed of light in a medium \[ v = \frac{c}{n} \]
2. Snell’s Law (general form) – derived from Fermat’s principle of least time or from the continuity of wave‑fronts – \[ n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2} \] where the angles are measured from the normal.
3. Refractive‑index ratio form (first medium = air/vacuum) \[ n = \frac{\sin\theta_{1}}{\sin\theta_{2}} \] Useful when only the two angles are known.
4. Predict the direction of bending
   • If \(n_{2}>n_{1}\) (light enters a slower medium) the ray bends **towards** the normal.
   • If \(n_{2}

Critical Angle and Total Internal Reflection (TIR)

• Critical angle, θ_c – the smallest angle of incidence in the denser medium for which refraction can still occur: \[ \theta_{c}= \sin^{-1}\!\left(\frac{n_{2}}{n_{1}}\right)\qquad (n_{1}>n_{2}) \]
• Total internal reflection – when the angle of incidence exceeds θ_c, the light is reflected back into the original (denser) medium rather than refracted. Everyday examples: the sparkle of a diamond (high $n$ gives a small θ_c), light guidance in optical fibres, and the right‑angle prism used in periscopes.

Experimental Demonstration (IGCSE Practical)

Objective: Observe refraction through a transparent rectangular block.

1. Place a rectangular glass or acrylic block on a sheet of white paper and draw its outline.
2. Shine a narrow laser beam (or a ray of light from a torch) so that it strikes one face of the block at a known angle of incidence (measure with a protractor).
3. Mark the incident ray, the refracted ray inside the block, and the emerging ray on the paper.
4. Measure the angles of incidence and refraction, then use Snell’s law to calculate the block’s refractive index. Compare with the tabulated value.

Only the two boundary surfaces are considered; any internal reflections are ignored for the purpose of this experiment.

Worked Example 1 – Speed and Refraction Angle

Light passes from air into water (\(n_{\text{water}}=1.33\)). The angle of incidence in air is \(30^{\circ}\).

1. Speed of light in water: \[ v_{\text{water}}=\frac{c}{n_{\text{water}}}= \frac{3.00\times10^{8}}{1.33}\approx2.26\times10^{8}\ \text{m s}^{-1} \]
2. Angle of refraction using Snell’s law (with \(n_{\text{air}}\approx1.00\)): \[ 1.00\sin30^{\circ}=1.33\sin\theta_{2}\;\Rightarrow\; \sin\theta_{2}=0.376\;\Rightarrow\;\theta_{2}\approx22^{\circ} \] The ray bends towards the normal, as expected.

Worked Example 2 – Using the Ratio Form

A ray in air (\(n\approx1\)) strikes a glass slab (crown glass, \(n\) unknown) at an angle of incidence \(45^{\circ}\). Inside the glass the ray makes an angle of refraction \(30^{\circ}\). Find the refractive index of the glass.

\[ n = \frac{\sin\theta_{1}}{\sin\theta_{2}}= \frac{\sin45^{\circ}}{\sin30^{\circ}}= \frac{0.7071}{0.5000}\approx1.41. \] (The result is close to the tabulated value for crown glass, confirming the measurement.)

Worked Example 3 – Critical Angle and TIR

Light travels in water (\(n_{1}=1.33\)) towards an air (\(n_{2}=1.00\)) interface. Find the critical angle and state what happens for an incident angle of \(50^{\circ}\).

• Critical angle: \[ \theta_{c}= \sin^{-1}\!\left(\frac{1.00}{1.33}\right)=\sin^{-1}(0.752)\approx49^{\circ} \]
• Since \(50^{\circ}>49^{\circ}\), the light undergoes total internal reflection and remains in the water.

Relevance to Technology

• Optical fibres – a glass core (\(n_{\text{core}}\approx1.48\)) is surrounded by cladding with a slightly lower index (\(n_{\text{clad}}\approx1.46\)). Light entering the core at an angle greater than the critical angle is trapped by TIR, allowing signals to travel long distances with minimal loss.
• Prisms and lenses – the amount of bending depends on \(n\); higher‑index glass gives stronger refraction, essential for microscopes, telescopes, cameras and periscopes.
• Diamond sparkle – diamond’s high index (\(n=2.42\)) gives a very small critical angle, so many internal reflections occur before light exits, producing the characteristic sparkle.

Common Misconceptions

• Confusing refractive index with the angle of refraction – they are unrelated quantities.
• Assuming \(n\) is always > 1 – gases at low pressure have indices only slightly above 1.
• Thinking light “stays” slower after leaving a medium – the speed changes only while the light is inside the material.
• Believing total internal reflection can occur when light goes from a slower to a faster medium – TIR only occurs when light attempts to leave a denser (higher‑\(n\)) medium.

Summary

• The refractive index \(n\) is the ratio of the speed of a wave in one medium to its speed in another; it is dimensionless.
• Absolute refractive index: \(n = c/v\).
• Snell’s law links refractive indices to the angles of incidence and refraction: \[ n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2} \] or, when the first medium is air/vacuum, \(n = \sin\theta_{1}/\sin\theta_{2}\).
• Higher \(n\) → slower light → greater bending towards the normal.
• When light moves from a higher‑\(n\) to a lower‑\(n\) medium a critical angle \(\theta_{c}= \sin^{-1}(n_{2}/n_{1})\) exists; incidence above this angle produces total internal reflection.
• These principles underpin technologies such as optical fibres, prisms, lenses and the brilliance of diamonds.