Critical Angle
Angle beyond which total internal reflection occurs.
Formula
Variables
How it works
The critical angle is the smallest angle of incidence in the optically denser medium for which the refracted ray would skim along the boundary, making the angle of refraction 90°. It comes directly from Snell’s law: at that limiting case, n1 sin(θc) = n2 sin(90°) = n2, so sin(θc) = n2/n1, which only makes sense when light goes from higher refractive index to lower refractive index. For any larger incidence angle, Snell’s law would require sin of the refracted angle to be greater than 1, which is impossible, so the light is totally internally reflected.
When to use it
Use this when checking whether total internal reflection will occur at a boundary, such as in optical fibres, prism reflectors, or light escaping from water or glass into air. It is especially useful in advanced optics problems involving ray paths, guiding, and angle thresholds between two media.
Worked example, step by step
A light ray travels from crown glass (n1 = 1.52) into water (n2 = 1.33). Step 1: Confirm the higher index is first: 1.52 > 1.33, so a critical angle exists. Step 2: Substitute into the formula: θc = sin^-1(n2/n1) = sin^-1(1.33/1.52). Step 3: Calculate the ratio: 1.33/1.52 = 0.875. Step 4: Take the inverse sine: θc = sin^-1(0.875) ≈ 61.0°. Final answer: The critical angle is about 61.0°, so any incidence angle greater than 61.0° in the glass will produce total internal reflection at the glass-water boundary.
Common mistakes to avoid
- Using the formula when light goes from lower refractive index to higher refractive index, where no critical angle exists
- Swapping n1 and n2, which gives an incorrect angle and can even make the ratio greater than 1
- Forgetting that the angle is measured from the normal, not from the surface
Did You Know?
Total internal reflection past the critical angle keeps light bouncing down optical fibres across oceans.
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