Describe internal reflection and total internal reflection using both experimental and everyday examples

3.2.2 Refraction of Light

Key Definitions

• Normal – an imaginary line drawn perpendicular to the surface at the point of incidence.
• Angle of incidence (θ_i) – the angle between the incident ray and the normal.
• Angle of refraction (θ_r) – the angle between the refracted ray and the normal.
• Refractive index (n) – a measure of how much a medium slows light compared with vacuum.

  For two media 1 and 2 it is given by

  \$n = \frac{\sin\thetai}{\sin\thetar} = \frac{c}{v}\$

  where \(c\) is the speed of light in vacuum and \(v\) the speed in the medium.

Snell’s Law (Law of Refraction)

When a light ray passes from medium 1 (refractive index \(n1\)) into medium 2 (refractive index \(n2\)) the relationship between the angles is

\[

n1\sin\thetai = n2\sin\thetar

\]

This explains why the ray bends towards the normal when it enters a denser medium (\(n2>n1\)) and bends away when it leaves a denser medium.

Passage of Light Through a Transparent Material (Two‑Medium Boundaries)

In the Cambridge syllabus the ray is considered to encounter only two distinct media at each surface. The same law applies at every boundary, irrespective of the shape of the transparent object.

1. First boundary (air → glass) – the ray bends towards the normal because \(n{\text{glass}} > n{\text{air}}\).
2. Second boundary (glass → air) – the ray bends away from the normal. If the angle of incidence at this second boundary exceeds the critical angle, the ray is reflected back into the glass instead of emerging.

Internal Reflection and Total Internal Reflection (TIR)

• Internal reflection – any reflection that occurs inside the original (denser) medium when light travels from a higher‑index medium to a lower‑index medium. It follows the law of reflection: θ_i = θ_r (reflected).
• Total internal reflection – a special case of internal reflection that happens only when:

  • the light is incident from the denser medium (\(n1>n2\)), and
  • the angle of incidence exceeds the critical angle θ_c.

  In TIR the refracted ray disappears; all the light is reflected back into the original medium.

Critical Angle – Meaning and Formulae

The critical angle θ_c is the smallest angle of incidence (measured in the denser medium) for which the refracted ray in the rarer medium travels exactly along the interface (i.e. at 90° to the normal). For any larger angle the refracted ray cannot exist and total internal reflection occurs.

\[

\thetac = \sin^{-1}\!\left(\frac{n2}{n1}\right)\qquad (n1>n_2)

\]

When the second medium is air (\(n_2 = 1.00\)) the formula can be written as

\[

n1 = \frac{1}{\sin\thetac}

\]

Thus the critical angle depends only on the ratio of the two refractive indices; the shape of the denser medium (block, prism, fibre) does not affect its value.

Experimental Demonstration

Apparatus

• Ray box with a narrow, collimated beam
• Transparent rectangular glass block (≈ 3 cm × 3 cm × 5 cm)
• Equilateral glass prism (≈ 3 cm sides)
• Rotating stage or protractor with a fine scale
• White screen or tracing paper to record emergent rays
• Ruler and pencil for noting angles

Procedure (Rectangular Block)

1. Place the block on the rotating stage. Align the incident ray to strike the first face near the centre.
2. Set the angle of incidence to a small value (≈ 10°) and mark the position of the emergent ray on the screen.
3. Measure both the incident angle (θ_i) and the refracted angle inside the block (θ_r) using the protractor. Record the data.
4. Increase θ_i in steps of 5° up to about 70°, recording the emergent ray each time.
5. When the ray reaches the second (glass–air) face, continue increasing θ_i. Note the angle at which the emergent ray disappears and a bright reflected ray appears inside the block – this is the experimental critical angle.
6. Repeat the whole set‑up with the triangular prism and compare the measured critical angles and the pattern of the refracted rays.

Analysis

• Use Snell’s law to calculate the theoretical refracted angles for each measured θ_i and compare with the observed values.
• Calculate the theoretical critical angle from the known refractive indices (glass ≈ 1.50, air ≈ 1.00) and compare with the experimental value obtained from both the block and the prism.
• Explain why the shape of the transparent material does not change the critical angle, but does alter the internal ray path.

Everyday Examples of Internal Reflection and Total Internal Reflection

• Optical fibres – Light is launched into a glass core (n ≈ 1.48) surrounded by a lower‑index cladding (n ≈ 1.40). At every core–cladding interface the incident angle exceeds the critical angle, so the light is continuously totally internally reflected and guided over kilometres.
• Diamond sparkle – Diamond’s high refractive index (n ≈ 2.42) gives a small critical angle (≈ 24°). Light entering the stone undergoes many internal reflections, producing its characteristic brilliance.
• Mirages – On hot days the air close to the ground is less dense than the cooler air above. Light from the sky strikes the hot layer at an angle greater than the critical angle and is totally internally reflected, creating the illusion of water.
• Endoscopes / medical fibrescopes – Bundles of flexible glass fibres rely on TIR to transmit illumination and images from the tip of the instrument to the surgeon’s eye.
• Rain‑droplet sparkle – Water droplets act as tiny spherical lenses. Light entering a droplet can hit the inner surface at an angle > θ_c, producing TIR that contributes to the bright flashes seen in a spray.
• Glass‑water surface as a mirror – When light travels from water (n ≈ 1.33) to air at an angle larger than the critical angle (≈ 48.8°), the surface behaves like a mirror because the light is totally internally reflected back into the water.

Use of Optical Fibres – Expanded Explanation

In a typical step‑index fibre the core‑cladding interface is cylindrical. As long as the launch angle inside the core satisfies

\[

\theta{\text{core}} > \thetac = \sin^{-1}\!\left(\frac{n{\text{clad}}}{n{\text{core}}}\right)

\]

the light undergoes TIR at every point around the circumference. This “light‑pipe” effect allows high‑bandwidth data transmission with very low loss.

Extension: Refraction in Lenses for Vision Correction

Although the syllabus focuses on a single plane interface, it is useful to see how refraction at two surfaces is exploited in corrective lenses.

• Long‑sightedness (hyperopia) – The eye’s focal point lies behind the retina. A converging (convex) lens with a higher refractive index bends incoming parallel rays towards the centre, moving the focal point forward onto the retina.
• Short‑sightedness (myopia) – The eye’s focal point lies in front of the retina. A diverging (concave) lens spreads the incoming rays, moving the focal point backward onto the retina.

Both types of lens rely on Snell’s law at two curved surfaces; the net effect is a change in the overall focal length of the optical system.

Critical Angle Table for Common Material Pairs

Suggested Diagram for Total Internal Reflection

Key Points to Remember

• Light travels in straight lines in a homogeneous medium.
• The normal is perpendicular to the surface; angles are measured from the normal.
• Snell’s law: \(n1\sin\thetai = n2\sin\thetar\) and \(n = \dfrac{\sin\thetai}{\sin\thetar}\).
• Refractive index \(n = c/v\); for air \(n \approx 1.00\).
• Internal reflection occurs only when light moves from a denser to a rarer medium.
• Total internal reflection requires:

  • \(n1 > n2\), and
  • \(\thetai > \thetac\) where \(\thetac = \sin^{-1}(n2/n1)\) (or \(n1 = 1/\sin\thetac\) when \(n2=1\)).

• The critical angle depends solely on the ratio of refractive indices; the geometry of the denser medium does not affect it.
• Applications of TIR – optical fibres, diamonds, mirages, endoscopes, and the mirror‑like behaviour of a water‑air surface – illustrate how a simple physical principle underpins important technology.
• Corrective lenses use refraction at two curved surfaces to adjust the overall focal length of the eye‑optics system.