Refractrion, Reflection and Polarisation

When a ray of light meets the boundary between two transparent media, part of it is reflected and part is transmitted (refracted). The golden rule that earns easy marks: measure every angle from the normal — the line drawn perpendicular to the surface at the point the ray strikes it — never from the surface itself.

The law of reflection has two parts:

• The angle of incidence equals the angle of reflection (both from the normal).
• The incident ray, reflected ray and normal lie in the same plane.

Refraction is the change of direction when a wave crosses a boundary and changes speed. Light travels fastest in a vacuum and slows down in a transparent medium:

• Entering an optically denser medium (e.g. air → glass), light slows down and bends towards the normal.
• Entering a less dense medium (e.g. glass → air), light speeds up and bends away from the normal.

A vital detail examiners test: during refraction the frequency does not change (it is fixed by the source — wavefronts cannot pile up at the boundary). Because $v = f\lambda$ and the speed $v$ changes while $f$ is constant, it is the wavelength that changes — it shortens in the denser medium.

So reflection and refraction always happen together at a boundary; how much light goes each way depends on the angle and the two media. The next two sections quantify refraction (Snell's law) and the special case where refraction stops entirely (total internal reflection).

The refractive index $n$ of a medium quantifies how much it slows light down: $n = \frac{c}{v}$ where $c$ is the speed of light in a vacuum ($3.00\times10^8\ \text{m s}^{-1}$) and $v$ is the speed in the medium. Because light always travels slower in a medium than in a vacuum, $v<c$, so $n$ is always greater than 1 and has no unit. Typical values: glass $\approx 1.5$, water $\approx 1.33$, diamond $\approx 2.42$.

Snell's law (the law of refraction) connects the angles on the two sides of a boundary: $n_1\sin\theta_1 = n_2\sin\theta_2$ with $\theta_1$ the angle of incidence and $\theta_2$ the angle of refraction, both measured from the normal. The medium with the larger refractive index is the optically denser one; light bends towards the normal on entering it.

A reliable method

1. Label the media: $n_1,\theta_1$ on the incident side, $n_2,\theta_2$ on the refracted side.
2. Write Snell's law and rearrange for the unknown (often $\sin\theta_2 = \dfrac{n_1\sin\theta_1}{n_2}$).
3. Substitute, evaluate $\sin\theta_2$, then take the inverse sine. Keep full precision until the final answer and quote it to 3 s.f.

A quick self-check: if the ray enters a denser medium the refraction angle should be smaller than the incidence angle; if it enters a less dense medium it should be larger. If your numbers disagree, you have probably swapped $n_1$ and $n_2$.

For air, $n \approx 1.00$, so when a ray goes from air into glass, $\sin\theta_1 = n_{\text{glass}}\sin\theta_2$, i.e. $n_{\text{glass}} = \dfrac{\sin\theta_1}{\sin\theta_2}$. This is the basis of the glass-block experiment.

You can also combine $n=c/v$ with the wave equation: since frequency is unchanged on refraction, $n = \dfrac{c}{v} = \dfrac{\lambda_{\text{vacuum}}}{\lambda_{\text{medium}}}$ — the wavelength shortens by the same factor that the speed falls.

When light passes from a denser medium towards a less dense one, it bends away from the normal. As the angle of incidence increases, the refracted ray bends more and more until, at one special angle — the critical angle $C$ — the refracted ray travels right along the boundary, i.e. the angle of refraction is exactly 90°.

Putting $\theta_2 = 90°$ into Snell's law (denser medium 1, less dense medium 2): $n_1\sin C = n_2\sin 90° = n_2 \;\Rightarrow\; \sin C = \frac{n_2}{n_1}$ For the common case where the second medium is air ($n_2 = 1$) and $n_1 = n$: $\boxed{\sin C = \frac{1}{n}}$

Total internal reflection (TIR) happens when the angle of incidence is greater than the critical angle: now there is no refracted ray and all the light is reflected back into the denser medium, obeying the law of reflection. Two conditions are needed — examiners want both:

1. The light must be travelling in the optically denser medium (heading towards the less dense one).
2. The angle of incidence must exceed the critical angle, $\theta > C$.

Applications of TIR

• Optical fibres: a thin glass core is wrapped in lower-index cladding; light is guided along the fibre by repeated total internal reflection at the core–cladding boundary (use $\sin C = n_{\text{cladding}}/n_{\text{core}}$, not $1/n$). The cladding protects the surface and prevents cross-talk between fibres.
• Reflecting prisms: a 45° glass prism (critical angle $\approx 42°$) turns light through 90° or 180° by TIR — used in binoculars and periscopes, giving brighter, sharper images than a silvered mirror.

A standard Unit 2 practical determines the refractive index of glass by ray tracing through a rectangular block.

Method

1. Place the block on paper and draw round its outline.
2. Shine a single ray from a ray box into one long face. Mark the incident ray with two well-spaced dots before the block and the emergent ray with two dots after it.
3. Remove the block and join the dots to reconstruct the incident, refracted and emergent rays.
4. At the entry point draw the normal (perpendicular to the face) and measure the angle of incidence $\theta_1$ and angle of refraction $\theta_2$ — both from the normal — with a protractor.
5. Repeat for several different angles of incidence.

Analysis. From air into glass, Snell's law gives $\sin\theta_1 = n\sin\theta_2$, so $n = \dfrac{\sin\theta_1}{\sin\theta_2}$. The best approach is to plot $\sin\theta_1$ (y-axis) against $\sin\theta_2$ (x-axis): the points lie on a straight line through the origin whose gradient is the refractive index $n$. Using a best-fit line over many angles averages out random error.

Reducing uncertainty

• Use large angles of incidence so the protractor reading has a smaller percentage error.
• Mark each ray with two widely spaced points and draw it with a sharp pencil and a ruler.
• Take readings at several angles and use the gradient of a graph rather than a single calculation, so one bad reading does not dominate.

Polarisation restricts the oscillations of a wave to a single plane perpendicular to the direction of travel. Unpolarised light from a lamp or the Sun oscillates in all planes perpendicular to its direction of travel; after a polarising filter it oscillates in one plane and is said to be plane-polarised.

The crucial point: only transverse waves can be polarised, because only they have oscillations perpendicular to the direction of travel, giving a choice of plane to select. A longitudinal wave (like sound) oscillates along the direction of travel, so there is nothing perpendicular to restrict — it cannot be polarised. The fact that light can be polarised is therefore direct evidence that light is a transverse wave (the oscillating electric and magnetic fields are perpendicular to the direction of travel).

Two filters: the demonstration

• The first filter (polariser) transmits only the oscillation component along its transmission axis, producing plane-polarised light.
• A second filter (analyser) transmits only the component along its axis. Rotating it changes the transmitted intensity: a maximum when the two axes are parallel, falling to (nearly) zero when the axes are crossed at 90°.

This bright→dark cycle on rotation is impossible for a longitudinal wave, which is why the experiment proves light is transverse.

Applications

• Polaroid sunglasses: glare reflected from water or roads is partly horizontally polarised, so lenses with a vertical transmission axis absorb the glare while passing other light.
• Stress analysis: transparent plastic models viewed between crossed polarisers show coloured fringes that reveal where stress concentrates.
• LCD screens: liquid crystals rotate the plane of polarised light between two crossed polarisers to switch pixels light or dark.