6.2 Total Internal Reflection
Total internal reflection is an optical phenomenon that occurs when light travels from an optically denser medium to an optically less dense medium at an angle of incidence greater than the critical angle. Instead of passing through the boundary into the second medium, all light is reflected back into the denser medium.
Learning Objectives
- State the conditions for the occurrence of total internal reflection of light.
- Define critical angle, \(c\).
- Relate the critical angle, \(c\), to the refractive index, \(n\), using Snell's Law (\(n = \frac{1}{\sin c}\)).
- Explain natural phenomena involving total internal reflection, such as the formation of a rainbow and a mirage.
- Explain daily life applications of total internal reflection, such as prism periscopes, optical fibres, and cat's eye reflectors.
- Solve numerical and ray-tracing problems involving critical angle and total internal reflection.
The Three Regimes of Light at an Optically Dense-to-Less Dense Boundary
Illustrate clearly the transition of light behaviour as the angle of incidence increases from \(i < c\) to \(i = c\), and finally to \(i > c\) (total internal reflection).
Ray Diagrams
Summary of Behaviour
As the angle of incidence increases from below the critical angle to above it, the refracted ray weakens and eventually disappears. At \(i = c\), the refracted ray travels along the boundary. At \(i > c\), all light is reflected back into the denser medium.
Critical angle and refractive index
\(n = \dfrac{1}{\sin c}\)where \(n\) is the refractive index of the denser medium and \(c\) is the critical angle.
Short Explanation
What is Total Internal Reflection?
Total internal reflection is the complete reflection of a light ray within an optically denser medium at the boundary with an optically less dense medium.
Conditions for Total Internal Reflection
1. Light travels from a medium of high optical density to a medium of low optical density.
2. The angle of incidence, \(i\), is larger than the critical angle, \(c\) (\(i > c\)).
Critical Angle, \(c\)
The critical angle is the angle of incidence in the medium of high optical density when the angle of refraction in the medium of lower optical density is equal to \(90^\circ\).
Relationship Between \(n\) and \(c\)
The higher the refractive index of a medium, the smaller its critical angle. For diamond (\(n = 2.42\)), \(c = 24.4^\circ\). For glass (\(n \approx 1.49\)), \(c \approx 42^\circ\).
Derivation from Snell's Law
\(n_1 \sin \theta_1 = n_2 \sin \theta_2\)For light travelling from a denser medium (\(n_1 = n\), \(\theta_1 = c\)) to air (\(n_2 = 1\), \(\theta_2 = 90^\circ\)):
\(n \sin c = 1 \times \sin 90^\circ\)Since \(\sin 90^\circ = 1\), we get:
\(n = \dfrac{1}{\sin c}\)| Medium | Refractive Index, \(n\) | Critical Angle, \(c\) |
|---|---|---|
| Diamond | 2.42 | 24.4° |
| Glass | 1.49 – 1.50 | 42° – 42.2° |
| Water | 1.33 | 48.8° |
Try to Answer First
Answer in your mind, then press “Check Answer”.
State two conditions required for total internal reflection to occur.
Check Answer
2. The angle of incidence must be greater than the critical angle (\(i > c\)).
The critical angle of a type of glass is \(42^\circ\). Calculate the refractive index of this glass.
Check Answer
\(n = \dfrac{1}{\sin 42^\circ} = \dfrac{1}{0.6691} \approx 1.49\)
In an optical fibre, which part has a higher refractive index: the inner core or the outer cladding? Explain why.
Check Answer
Common Mistakes
- Assuming total internal reflection can happen when light travels from air into glass or water. Light MUST travel from a medium of higher optical density to one of lower optical density.
- Confusing critical angle with angle of refraction: The critical angle is an angle of incidence (inside the denser medium) that produces an angle of refraction of \(90^\circ\) (in the less dense medium), not an angle of refraction itself.
- Calculating refractive index by taking \(\sin c\) directly instead of using the reciprocal \(\frac{1}{\sin c}\). Since refractive index \(n \ge 1\) for all ordinary media relative to vacuum/air, \(n\) cannot be less than 1.
- In ray diagrams for right-angled prisms, incorrectly showing refraction when a ray enters perpendicularly through the surface (\(i = 0^\circ\)). A ray hitting a surface along the normal passes straight through without bending.
Concept Misunderstandings
Light is always partially refracted and partially reflected whenever it hits any boundary.
When \(i > c\) and light travels from a denser to a less dense medium, there is absolutely NO refracted ray; 100% of the light undergoes total internal reflection.
A mirage on a hot road is an illusion caused by water vapour evaporating from the tar.
A mirage is an actual optical image of the sky or clouds produced by the refraction and total internal reflection of light rays through hot, less dense layers of air near the ground.
The critical angle is a fixed universal angle for all materials.
Every material has its own unique critical angle depending on its refractive index. A higher refractive index results in a smaller critical angle (e.g., diamond has \(c = 24.4^\circ\), whereas glass has \(c \approx 42^\circ\)).
Summary
- Total internal reflection is the complete reflection of a light ray back into its original denser medium when travelling toward a less dense medium at an incident angle exceeding the critical angle (\(i > c\)).
- The critical angle \(c\) is the incident angle for which the refracted angle \(r = 90^\circ\), defined by \(n = \dfrac{1}{\sin c}\).
- This optical principle is responsible for natural phenomena including rainbows and desert/road mirages.
- It enables practical instruments such as prism periscopes, cat's eye road studs, and medical/telecommunication optical fibres.
Short Activity
Objective: Identify conditions for total internal reflection, critical angle, and applications.
A. Objective Quiz
1 Which of the following conditions is necessary for total internal reflection to occur?
2 The refractive index of a medium is 2.0. What is the critical angle of this medium?
B. Fill in the Blanks
3 The angle of incidence in an optically denser medium when the angle of refraction in the optically less dense medium is equal to 90° is known as the angle.
4 In an optical fibre, the refractive index of the inner core must be than the refractive index of the outer cladding for total internal reflection to occur.
C. Matching / Drag and Drop
Match each item or phenomenon to its correct optical description. If using a phone, tap the answer first, then tap the matching box.
