Refracted Angle (θ₂)
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Critical Angle
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What is Refraction?

When light passes from one transparent medium to another (e.g., air to glass), it changes speed. If the light enters at an angle, this speed change causes the light to bend — this is refraction. The direction of bending depends on whether light is speeding up or slowing down.

Key Rule: Light bends towards the normal when entering a denser medium (slower), and away from the normal when entering a less dense medium (faster).

Snell's Law

n₁ sin(θ₁) = n₂ sin(θ₂)

Where:

  • n₁, n₂ = refractive indices of medium 1 and 2
  • θ₁ = angle of incidence (measured from normal)
  • θ₂ = angle of refraction (measured from normal)

The refractive index n = c/v, where c is the speed of light in vacuum and v is the speed in the medium. Higher n means light travels slower.

Critical Angle & Total Internal Reflection

θc = arcsin(n₂/n₁)

When light travels from a denser medium to a less dense medium (n₁ > n₂), there's a special angle called the critical angle. At this angle, the refracted ray travels along the boundary (θ₂ = 90°).

For angles greater than the critical angle, total internal reflection (TIR) occurs — all light is reflected back. This is the principle behind:

  • Fiber optics — light trapped inside glass fibers
  • Diamonds — brilliant sparkle due to TIR
  • Prisms — used in binoculars and periscopes

Refractive Index Table

Material n Critical Angle (to air)
Vacuum/Air 1.00 —
Water 1.33 48.8°
Glass (Crown) 1.52 41.1°
Flint Glass 1.66 37.0°
Diamond 2.42 24.4°

Worked Examples

✅ Example 1: Light Entering Glass

Light travelling in air hits a glass surface (n = 1.52) at 45°. Find the angle of refraction.

Given: n₁ = 1.00, n₂ = 1.52, θ₁ = 45°
Formula: sin(θ₂) = n₁ sin(θ₁) / n₂
Solution: sin(θ₂) = (1.00 × sin 45°) / 1.52 = 0.707 / 1.52 = 0.465
θ₂ = arcsin(0.465) = 27.7°

✅ Example 2: Critical Angle of Glass

Calculate the critical angle for light going from glass (n = 1.52) to air.

Given: n₁ = 1.52, n₂ = 1.00
Formula: θc = arcsin(n₂/n₁)
Solution: θc = arcsin(1.00/1.52) = arcsin(0.658) = 41.1°

✅ Example 3: Water to Air - TIR Check

A diver shines a light upward at 55° from underwater. Does the light escape or reflect?

Step 1: Find critical angle for water (n = 1.33) to air
θc = arcsin(1/1.33) = arcsin(0.752) = 48.8°
Step 2: Compare: 55° > 48.8°
Answer: Since angle > critical angle, Total Internal Reflection occurs! Light is reflected back into the water.

✅ Example 4: Finding Unknown Refractive Index

Light enters an unknown medium from air at 40° and refracts to 25°. What is the medium's refractive index?

Given: n₁ = 1.00, θ₁ = 40°, θ₂ = 25°
Formula: n₂ = n₁ sin(θ₁) / sin(θ₂)
Solution: n₂ = (1.00 × sin 40°) / sin 25° = 0.643 / 0.423 = 1.52 (likely glass)

✅ Example 5: Diamond's Sparkle

Why does diamond sparkle more than glass? Calculate both critical angles.

Glass: θc = arcsin(1/1.52) = 41.1°
Diamond: θc = arcsin(1/2.42) = 24.4°
Answer: Diamond's lower critical angle means more light rays undergo TIR and eventually exit the top, creating more sparkle. The high n = 2.42 also increases light "bending" for striking rainbow effects.
💡 Tip: Critical angle only exists when going from denser to less dense medium (n₁ > n₂). If n₁ < n₂, there's no critical angle — light always refracts!