Critical Angle
When light travels from a denser medium (such as glass or water) into a less dense medium (such as air), it bends away from the normal — the imaginary line perpendicular to the surface. As the angle of incidence increases, so does the angle of refraction. There exists a special angle of incidence, called the critical angle, at which the refracted ray grazes along the interface at exactly 90° to the normal. Beyond this angle, no refraction occurs at all, and all the light is reflected back into the denser medium — a phenomenon known as total internal reflection. The critical angle is not a fixed value; it depends on the refractive indices of the two media involved. A medium with a very high refractive index (like dense flint glass, n ≈ 1.9) has a smaller critical angle than a medium with a lower refractive index (like ordinary glass, n ≈ 1.5). This makes intuitive sense: the more a medium bends light, the sooner — at a shallower angle — total internal reflection kicks in. Water has a critical angle of about 49° relative to air, while diamond's critical angle is only about 24.5°, which is why diamonds sparkle so brilliantly. Understanding the critical angle is essential for explaining a wide range of modern technologies — from optical fibres that carry internet data around the world, to endoscopes used in medical imaging, and even the prismatic reflectors used in binoculars and periscopes. Mastering this concept connects Snell's Law to real-world engineering in a profound and practical way.
- Recall Snell's Law
- Set the Refracted Angle to 90°
- Derive the Formula
- What Happens Beyond θ_c
- Special Case: Medium 2 is Air
Recall Snell's Law
Snell's Law governs refraction at any interface: n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices of the first and second media, and θ₁ and θ₂ are the angles of incidence and refraction measured from the normal. Light always bends toward the normal when entering a denser medium and away from it when entering a less dense medium.
Set the Refracted Angle to 90°
The critical angle θ_c is defined as the angle of incidence (in the denser medium) for which the refracted angle equals exactly 90°. At this point the refracted ray travels along the boundary — it no longer passes into the second medium. Substituting θ₂ = 90° into Snell's Law gives: n₁ sin θ_c = n₂ sin 90° = n₂ × 1 = n₂. Therefore sin θ_c = n₂ / n₁.
Derive the Formula
Rearranging gives the critical angle formula: θ_c = arcsin(n₂ / n₁). This is only physically meaningful when n₁ > n₂ — i.e. light must be travelling from the denser into the less dense medium. If n₁ ≤ n₂, total internal reflection cannot occur, and every angle of incidence produces a refracted ray.
What Happens Beyond θ_c
If the angle of incidence exceeds the critical angle, there is no solution to Snell's Law for a real refracted angle. Instead, all the light energy is reflected back into the denser medium according to the law of reflection (angle of incidence = angle of reflection). This is total internal reflection — 'total' because 100% of the light is reflected, unlike ordinary partial reflection at surfaces, making it more efficient than even the best metallic mirror.
Special Case: Medium 2 is Air
In most textbook problems, the second medium is air (n₂ ≈ 1.00). The formula simplifies to sin θ_c = 1 / n₁, or θ_c = arcsin(1 / n₁). For glass with n = 1.5: θ_c = arcsin(1/1.5) = arcsin(0.667) ≈ 41.8°. Any ray inside the glass hitting the glass–air surface at more than about 42° undergoes total internal reflection.
The Key Equation
- \theta_c — Critical angle (degrees or radians): The minimum angle of incidence (measured from the normal, inside the denser medium) at which total internal reflection occurs.
- n_1 — Refractive index of medium 1 (denser) (dimensionless): The refractive index of the medium in which the incident ray travels. Must be greater than n₂ for total internal reflection to be possible.
- n_2 — Refractive index of medium 2 (less dense) (dimensionless): The refractive index of the medium the light is attempting to enter. Typically air (n ≈ 1.00) in standard problems.
Worked Example
Problem. An optical fibre is made of glass with a refractive index of 1.62. The fibre is surrounded by a cladding material with a refractive index of 1.42. Calculate the critical angle for the glass–cladding interface, giving your answer to one decimal place.
Answer. The critical angle for this optical fibre is approximately 61.2°.
Origins · 1 minWillebrord Snellius and how this was discovered
Willebrord Snellius (also known as Snell) discovered the law of refraction in 1621, which now bears his name — Snell's Law. Although he did not publish his findings during his lifetime, the law was disseminated through the work of Descartes and later verified rigorously by Huygens. The critical angle formula is a direct mathematical consequence of Snell's Law, making Snellius the intellectual father of all total internal reflection applications. Snellius was also a skilled mathematician who improved methods for measuring the radius of the Earth by triangulation — he is considered one of the founders of modern geodesy, the science of measuring Earth's shape.
Related Concepts
- Snell's Law of Refraction — The critical angle formula is derived directly from Snell's Law by setting the refracted angle to 90°; understanding refraction is a prerequisite.
- Total Internal Reflection — The critical angle defines the threshold at which total internal reflection begins — the two concepts are inseparable.
- Refractive Index — The critical angle depends entirely on the refractive indices of the two media, so a solid grasp of refractive index is essential.
- Optical Fibres — Optical fibres work by keeping light above the critical angle at every reflection, making this concept the core principle of fibre-optic technology.