Refraction and Total Internal Reflection

When light passes from one medium to another, it changes speed. If the light enters at an angle to the boundary, this change in speed causes a change in direction — this is refraction. Under certain conditions, light can be completely reflected back into the original medium — this is total internal reflection.

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Refraction

When a wave crosses a boundary between two media in which it travels at different speeds, several things can happen:

• The wave changes speed.
• The wave changes wavelength (but the frequency stays the same).
• If the wave enters at an angle other than 90° to the boundary, it changes direction.

Why Does the Frequency Stay the Same?

At the boundary, the number of wavefronts arriving per second must equal the number leaving per second (otherwise wavefronts would pile up or disappear at the boundary). Since frequency is the number of wavefronts per second, it remains constant. Since v = fλ and f is constant, a decrease in speed means a decrease in wavelength, and vice versa.

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Refractive Index

The refractive index of a medium, n, is defined as:

n = c/v

where c is the speed of light in a vacuum (3.00 × 10⁸ m s⁻¹) and v is the speed of light in the medium.

Medium	Approximate Refractive Index
Vacuum	1.000 (exactly)
Air	1.000 (approximately)
Water	1.33
Glass (crown)	1.52
Glass (flint)	1.65
Diamond	2.42
Optical fibre (core)	~1.46–1.62

A higher refractive index means the light travels more slowly in that medium.

Exam Tip: The refractive index of a medium is always ≥ 1 (since v ≤ c). Air has n ≈ 1.00, so for most purposes, air and vacuum are treated as equivalent.

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Snell's Law

When light passes from medium 1 (refractive index n₁) to medium 2 (refractive index n₂), the relationship between the angles of incidence and refraction is given by Snell's law:

n₁ sin θ₁ = n₂ sin θ₂

where θ₁ is the angle of incidence (measured from the normal) and θ₂ is the angle of refraction (measured from the normal).

Key Observations

• Light passing from a less dense to a more dense medium (n₁ < n₂) bends towards the normal (θ₂ < θ₁).
• Light passing from a more dense to a less dense medium (n₁ > n₂) bends away from the normal (θ₂ > θ₁).
• Light entering along the normal (θ₁ = 0°) passes straight through without changing direction (θ₂ = 0°), but still changes speed.

Worked Example 1 — Light travels from air (n = 1.00) into crown glass (n = 1.52) at an angle of incidence of 40°. Calculate the angle of refraction.

n₁ sin θ₁ = n₂ sin θ₂

1.00 × sin 40° = 1.52 × sin θ₂

sin θ₂ = sin 40°/1.52 = 0.6428/1.52 = 0.4229

θ₂ = arcsin(0.4229) = 25.0°

The light bends towards the normal as it enters the denser medium.

Worked Example 2 — Light passes from water (n = 1.33) into air (n = 1.00) at an angle of incidence of 30°. Calculate the angle of refraction.

1.33 × sin 30° = 1.00 × sin θ₂

sin θ₂ = 1.33 × 0.500 = 0.665

θ₂ = arcsin(0.665) = 41.7°

The light bends away from the normal as it enters the less dense medium.

Worked Example 3 — Light of wavelength 589 nm in vacuum enters glass of refractive index 1.52. Calculate the speed and wavelength of the light in the glass.

Speed: v = c/n = (3.00 × 10⁸)/1.52 = 1.97 × 10⁸ m s⁻¹

Wavelength in glass: λ_glass = λ_vacuum/n = 589/1.52 = 388 nm

The frequency remains the same: f = c/λ_vacuum = (3.00 × 10⁸)/(589 × 10⁻⁹) = 5.09 × 10¹⁴ Hz

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Total Internal Reflection (TIR)

When light travels from a more dense medium (higher n) to a less dense medium (lower n), the refracted ray bends away from the normal. As the angle of incidence increases, the angle of refraction increases until it reaches 90° — the refracted ray travels along the boundary.