Recall and use the equation n = sin i / sin r

3.2.2 Refraction of Light

Learning objective

Students will be able to:

• Define the key terms (normal, angles of incidence & refraction, refractive index, critical angle).
• Describe a simple experiment that demonstrates refraction.
• State and use the full form of Snell’s law n₁ sin i = n₂ sin r and the simplified air‑→‑medium form n = sin i / sin r.
• Derive the critical‑angle relationship from Snell’s law and explain total internal reflection (TIR).
• Calculate refractive indices, unknown angles and critical angles, and apply the concepts to everyday devices such as optical fibres.
• Recognise the limited wavelength dependence of n (dispersion) and its relevance to prisms.

Key definitions

• Normal – an imaginary line drawn perpendicular to the surface at the point of incidence.
• Angle of incidence (i) – the angle between the incident ray and the normal.
• Angle of refraction (r) – the angle between the refracted ray and the normal.
• Refractive index (n) – a dimensionless number that measures how much a medium slows light compared with vacuum. For air at standard temperature and pressure n ≈ 1.00.
• Critical angle (c) – the smallest angle of incidence in the denser medium for which the refracted ray emerges at 90° to the normal; larger incident angles give total internal reflection.
• Dispersion – the slight variation of n with wavelength (colour). It explains why a prism separates white light into a spectrum.

Quick checklist before any calculation

1. Set the calculator to **degrees** (unless the problem states otherwise).
2. Measure every angle **from the normal**, not from the surface.
3. Identify which medium is denser (higher n) – this determines the direction of TIR.
4. Record all measured angles to the same number of significant figures (usually 2 dp for a protractor).
5. When reporting a calculated n, keep the same number of significant figures as the least‑precise angle used.

Simple experiment to demonstrate refraction

1. Place a rectangular glass slab on a sheet of white paper and draw a straight reference line across the paper.
2. Position a ray‑box so that a narrow beam strikes the upper surface of the slab at a known angle of incidence i.
3. Draw the normal at the point of incidence and use a protractor to measure both i and the angle of refraction r inside the glass.
4. Repeat for several different incident angles and record the data in a table.
5. Calculate the refractive index for each trial with n = sin i / sin r. The results should cluster around the accepted value for glass (≈ 1.5).

Snell’s law

The general relationship between the angles and the refractive indices of the two media is

$$n_{1}\sin i = n_{2}\sin r$$

• Full form – used when both media have known indices (n₁ and n₂).
• Simplified air → medium form – when the ray passes from air (≈ vacuum, n≈1) into another medium: $$n = \frac{\sin i}{\sin r}$$
• Inverse‑Snell form (denser → rarer) – if the ray travels from a medium of index n₂ to air: $$\frac{\sin r}{\sin i}= \frac{1}{n_{2}}$$

Derivation of the critical‑angle formula

1. Start with Snell’s law for light moving from a denser medium (n₂) to a rarer medium (n₁≈1): $$n_{2}\sin i = n_{1}\sin r$$
2. At the critical angle the refracted ray runs along the interface, so r = 90° and sin r = 1.
3. Insert sin r = 1: $$n_{2}\sin c = 1$$
4. Re‑arrange: $$\sin c = \frac{1}{n_{2}} \qquad\text{or}\qquad n_{2}= \frac{1}{\sin c}$$

Total internal reflection (TIR)

If the angle of incidence in the denser medium exceeds the critical angle (i > c), the light is reflected back into the denser medium rather than refracted. This is the principle behind:

• Optical fibres
• The sparkle of a diamond (high ≈2.42 gives a small c)
• MIRAGE formation over hot surfaces

Refractive‑index formulas at a glance

• From Snell’s law (air → medium): n = sin i / sin r
• From the critical angle (medium → air): n = 1 / sin c
• Finding an unknown angle:
  • i = arcsin(n sin r)
  • r = arcsin(sin i / n)
  • Critical angle c = arcsin(1 / n)

How to use the equations (step‑by‑step)

1. Measure the required angle(s) with a protractor (always from the normal).
2. Set the calculator to degrees and find the sine (or inverse‑sine) of each angle.
3. Insert the values into the appropriate formula (see the “formulas at a glance” box).
4. Round the final answer to the same number of significant figures as the measured angles.

Typical refractive indices (20 °C, standard pressure)

Medium	Refractive index n
Air (dry)	1.00
Water	1.33
Glass (crown)	1.50 – 1.55
Diamond	2.42
Plastic (polystyrene)	1.59
Optical‑fibre core (glass)	≈ 1.48
Optical‑fibre cladding	≈ 1.46

Worked examples

Example 1 – finding the refractive index of glass

Given: i = 30°, r = 19° (air → glass).

1. sin 30° = 0.500
2. sin 19° = 0.326
3. n = 0.500 / 0.326 ≈ 1.53 → consistent with crown glass.

Example 2 – critical angle for water

Water: n = 1.33. Find the critical angle for light leaving water into air.

1. sin c = 1 / 1.33 ≈ 0.752
2. c = arcsin 0.752 ≈ 48.8°
3. Any incident angle > 48.8° inside the water will produce total internal reflection.

Example 3 – acceptance angle of an optical fibre

Core: n₁ = 1.48, Cladding: n₂ = 1.46, surrounding medium (air) ≈ 1.00.

1. Maximum angle inside the core for TIR: c₁ = arcsin( n₂ / n₁ ) = arcsin(1.46 / 1.48) ≈ 78.9°.
2. Using the numerical‑aperture formula: NA = √(n₁² − n₂²) ≈ √(1.48² − 1.46²) ≈ 0.24.
3. Maximum acceptance angle to the fibre axis from air: θₘₐₓ = arcsin( NA ) ≈ 13.9°.
4. Thus light entering the fibre at ≤ 13.9° to the axis will be guided by TIR.

Common mistakes to avoid

• Measuring angles from the surface instead of the normal.
• Leaving the calculator in radians when the problem uses degrees (or vice‑versa).
• Applying the simplified air‑→‑medium formula when both media have significant indices; use the full Snell’s law.
• Confusing the critical‑angle equation sin c = 1/n with the ordinary Snell’s‑law ratio.
• For TIR, forgetting that the light must travel from the higher‑n to the lower‑n medium.
• Neglecting significant‑figure rules when reporting experimental values of n.

Practice questions

1. A light ray passes from water (n=1.33) into air. If the angle of incidence in water is 45°, find the angle of refraction in air.
2. Light enters a plastic block (n=1.59) from air at an incidence angle of 20°. Determine the angle of refraction inside the plastic.
3. In an experiment the measured angles are i=25° and r=15°. Calculate the refractive index of the unknown material.
4. Calculate the critical angle for a glass‑to‑air interface (n=1.52). What happens to a ray that strikes the glass at 70° to the normal?
5. Optical fibres rely on total internal reflection. If the core has n=1.48 and the cladding n=1.46, find the maximum acceptance angle (largest angle to the fibre axis at which light can enter from air and still be guided).

Everyday applications – optical fibres

Optical fibres consist of a high‑index glass core surrounded by a slightly lower‑index cladding. The small **refractive‑index contrast** (Δn ≈ 0.02) ensures that light entering the core at an angle less than the acceptance angle is repeatedly totally internally reflected, allowing signals to travel long distances with minimal loss.

Key uses:

• High‑speed internet and telephone communication.
• Medical endoscopes and surgical imaging.
• Illuminated signs, decorative lighting, and data‑centre interconnects.

Suggested diagram