Formula & Calculator
Critical Angle (Total Internal Reflection)
Calculates the minimum angle of incidence at which light traveling from a denser to a less dense medium undergoes total internal reflection.
Interpretation
Critical angle: θ_c = arcsin(n₂/n₁), where n₁ > n₂. It is the angle of incidence above which total internal reflection occurs. Example: n₁=1.5 (glass), n₂=1.0 (air) → θ_c = arcsin(0.667) ≈ 41.8°.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| theta_c | Critical angle | degrees |
| n1 | Index of refraction of the denser (incident) medium | |
| n2 | Index of refraction of the less dense (transmitted) medium |
What it means
The critical angle is the angle of incidence in a denser medium (higher n) for which the refracted ray in the rarer medium grazes the interface (angle of refraction = 90°). For any incidence angle greater than θ_c, total internal reflection occurs, meaning all light is reflected back into the denser medium. This phenomenon is the basis for optical fibres, where light is guided along the core by repeated total internal reflection. It is also used in prisms, endoscopes, and in diamond cutting to exploit its brilliance. The formula θ_c = arcsin(n₂/n₁) is derived from Snell’s law. Understanding the critical angle is essential for designing fibre optics and for studying light propagation in media.
Worked example
Critical Angle – Two Examples
Real‑World| Parameter | Value |
|---|---|
| n₁ | 1.5 |
| n₂ | 1.0 |
| Parameter | Value |
|---|---|
| n₁ | 1.33 |
| n₂ | 1.0 |
Common mistakes
- Critical angle: Only defined when n₁ > n₂ (light going from denser to rarer medium).
- Arcsine: θ_c = arcsin(n₂/n₁) – ensure the ratio ≤ 1; if >1, no total internal reflection.
- Units: Result in degrees or radians – be consistent.
- Total internal reflection: Occurs when angle of incidence > θ_c.
- Media: n₁ is the incident medium, n₂ is the refracting medium.
Applications
The critical angle for total internal reflection, θ_c = arcsin(n₂/n₁), determines the angle above which light is totally reflected within a medium. This principle is the basis for optical fibres, which rely on total internal reflection to guide light with minimal loss. Engineers use it to design fibre optic cables, endoscopes, and telecommunications systems. It is also applied in prism spectrometers, in refractive index measurement, and in light‑guiding displays. By understanding the critical angle, professionals can ensure efficient light transmission and create innovative optical devices that exploit this effect.
- Design of optical fibre communication networks
- Endoscopes and medical imaging devices
- Prism spectrometers and refractometers
- Optical waveguides and couplers
- Light‑guiding displays and signage
Frequently Asked Questions
The critical angle θ_c is the angle of incidence in the denser medium (n₁) for which the refracted ray in the less dense medium (n₂) travels along the interface. It is given by θ_c = arcsin(n₂/n₁), where n₁ > n₂.
Applying it when light travels from a less dense to a denser medium (n₂ > n₁). In that case, total internal reflection is not possible because the sine of the critical angle would be > 1.
All light is reflected back into the denser medium; no light transmits into the second medium. This is the principle behind optical fibres.
For water (n₁=1.333) to air (n₂=1.000), θ_c = arcsin(1/1.333) ≈ 48.6°.
For typical glass (n₁=1.5) to air, θ_c ≈ 41.8°. This is why glass prisms can reflect light internally.
Light is confined within the fibre core by repeated total internal reflection at the core‑cladding interface. The cladding has a lower refractive index than the core.
The acceptance angle is determined by the numerical aperture: NA = n₀·sinθ_a = √(n₁² – n₂²).
In seismology, the critical angle is used to analyse the refraction of seismic waves at layer boundaries, helping to determine Earth's internal structure.