Refraction and total internal reflectionEdexcel A-Level Physics: Revision notes

Refraction is the change in direction of a wave when it crosses a boundary between media because its speed changes. The refractive index n of a medium measures how much it slows light:

n = c/v

where c is the speed of light in a vacuum (3.00 × 10⁸ m s⁻¹, and almost the same in air) and v is the speed in the medium. Since v < c, n is always greater than 1 (air is about 1.00, water 1.33, glass about 1.5). A medium with a higher n is optically denser.

When light enters a denser medium it slows down and its wavelength shortens, but its frequency is unchanged.

At a boundary between medium 1 and medium 2:

n₁ sin θ₁ = n₂ sin θ₂

where the angles are measured from the normal. Entering a denser medium the ray bends towards the normal, and leaving it the ray bends away from the normal. A ray along the normal (θ = 0) passes straight through.

From n = c/v it follows that n₁/n₂ = v₂/v₁ = λ₂/λ₁.

When light travels from a denser to a less dense medium, the refracted ray bends away from the normal. As the angle of incidence rises, the angle of refraction reaches 90° at the critical angle C. At larger angles there is no refracted ray and all of the light is reflected: total internal reflection (TIR).

Putting θ₂ = 90° into Snell's law for a boundary with air:

sin C = 1/n

For glass of n = 1.50, C = 42°. For water (n = 1.33), C = 49°. For diamond (n = 2.42), C = 24°.

A higher refractive index gives a smaller critical angle. In general, sin C = n₂/n₁ between two media.

Two conditions must both be met for TIR:

1. The light is travelling towards a medium of lower refractive index (e.g. glass to air).
2. The angle of incidence is greater than the critical angle.

To predict what happens, calculate C and compare it with the angle of incidence. Below C, most of the light is refracted (with some partial reflection). Above C, TIR occurs.

Uses: optical fibres carry light by repeated TIR in a core with a cladding of lower n, and cut diamonds sparkle because the small critical angle gives much TIR inside the stone.

Use a rectangular glass block, a ray box with a single slit, a protractor and paper.

1. Draw round the block and draw a normal at the point where the ray enters.
2. Direct a narrow ray along lines at different angles of incidence (about 10° to 70°) and mark the ray with dots well apart on each side.
3. Join the dots to find the ray inside the block and measure θ₁ and θ₂ from the normal.
4. Plot sin θ₁ against sin θ₂. The graph is a straight line through the origin whose gradient is n.

Using a graph reduces random error and shows up anomalies.

A ray in air hits glass (n = 1.50) at 40°.

sin θ₂ = sin 40° ÷ 1.50 = 0.429, so θ₂ = 25°

Speed in glass: v = c/n = 3.00 × 10⁸ ÷ 1.50 = 2.0 × 10⁸ m s⁻¹

Critical angle for glass to air: sin C = 1/1.50, so C = 42°. A ray inside the glass hitting the surface at 50° is totally internally reflected, because 50° is greater than 42°.