Define and use the terms normal, angle of incidence and angle of refraction

3.2.2 Refraction of Light

Learning Objective

Define and use the terms normal, angle of incidence, angle of refraction and refractive index when analysing the refraction of light at a plane surface, and apply these ideas to critical‑angle and total‑internal‑reflection situations.

Key Definitions

• Normal: An imaginary line drawn perpendicular to the surface at the point where the incident ray meets the surface.
• Angle of incidence (i): The angle between the incident ray and the normal.
• Angle of refraction (r): The angle between the refracted ray and the normal.
• Refractive index (n): The ratio of the speed of light in vacuum to its speed in a given medium,

  \[

  n=\frac{c}{v}

  \]

  (dimensionless). It quantifies how much the medium slows the light.

• Critical angle (ic): The smallest angle of incidence (in the denser medium) for which the refracted ray travels along the boundary. For \(n1>n_2\)

  \[

  \sin ic=\frac{n2}{n1}\qquad\text{or}\qquad ic=\sin^{-1}\!\left(\frac{n2}{n1}\right).

  \]

• Total internal reflection (TIR): When the angle of incidence exceeds the critical angle, the light is reflected completely back into the original (denser) medium; no refracted ray emerges.

Geometrical Picture

When a ray passes from medium 1 (\(n1\)) to medium 2 (\(n2\)) at a plane boundary, its speed changes, causing the ray to bend. The bend is described by the angles measured from the normal, not from the surface.

Snell’s Law

The relationship between the two media is given by

\[

n1\sin i = n2\sin r

\]

Worked Example – Refraction

Light travels from air (\(n1=1.00\)) into water (\(n2=1.33\)). If the angle of incidence is \(30^{\circ}\), find the angle of refraction.

1. Apply Snell’s law: \(1.00\sin30^{\circ}=1.33\sin r\).
2. \(\sin30^{\circ}=0.500\) → \(0.500 = 1.33\sin r\).
3. \(\sin r = 0.500/1.33 = 0.376\) (keep three significant figures).
4. \(r = \sin^{-1}(0.376) = 22.0^{\circ}\) (round to the same number of significant figures as the given angle).

Link to Other Syllabus Topics

Refraction connects directly to:

• Wave speed: \(v = f\lambda\). Since the frequency \(f\) does not change at a boundary, a change in speed \(v\) implies a change in wavelength \(\lambda\), which is why the ray bends.
• Electromagnetic spectrum: Different wavelengths (e.g., red vs. blue light) have slightly different refractive indices – a phenomenon called dispersion, which underlies the formation of rainbows.

Experiments

1. Refraction at a Plane Surface – Determining the Refractive Index

1. Set up a ray‑box (or laser pointer) so that a narrow beam strikes a rectangular glass slab placed on a flat white sheet.
2. Mark the point where the beam meets the first surface of the slab.
3. Draw the normal at this point (use a ruler to make a line perpendicular to the surface).
4. Measure the angle between the incident ray and the normal with a protractor – this is the angle of incidence \(i\) (record to the nearest 0.5°).
5. Observe the refracted ray inside the slab, draw its path and measure the angle between this ray and the normal – this is the angle of refraction \(r\).
6. Repeat for at least five different incident angles (e.g., 10°, 20°, 30°, 40°, 50°) and record the values of \(i\) and \(r\) in a table.
7. For each pair, calculate the refractive index of the slab using

   \[

   n_{\text{slab}} = \frac{\sin i}{\sin r}

   \]

   (air is taken as \(n=1.00\)).

8. Average the calculated values and compare with the accepted value (e.g., \(n_{\text{glass}}\approx1.50\)).
9. Uncertainty check: Estimate the uncertainty in each angle (± 0.5°) and propagate it to obtain an uncertainty for each \(n\). Discuss possible sources of error (parallax, imperfect normal, surface scratches).

2. Observing Total Internal Reflection

1. Fill a transparent rectangular tank with water and place a laser pointer so the beam strikes the water‑air interface from below.
2. Adjust the angle of the beam (using a rotating mount with a protractor) until the refracted ray disappears and the beam is reflected back into the water.
3. Record the incident angle at which this occurs – this is the experimental critical angle \(i_c\).
4. Measure the reflected angle (the angle between the reflected ray and the normal). It should be equal to the incident angle, confirming the law of reflection.
5. Calculate the theoretical critical angle using

   \[

   ic = \sin^{-1}\!\left(\frac{n{\text{air}}}{n_{\text{water}}}\right)

   \]

   and compare with the measured value, including the ± 0.5° uncertainty.

Critical Angle and Total Internal Reflection

Critical Angle

Only possible when light travels from a denser medium (\(n1>n2\)) to a rarer medium.

Example – Water to Air

• \(n{\text{water}}=1.33,\; n{\text{air}}=1.00\)
• \(i_c = \sin^{-1}\!\left(\frac{1.00}{1.33}\right)=\sin^{-1}(0.752)=49.0^{\circ}\) (to three significant figures).
• Any incident angle in water greater than \(49^{\circ}\) produces total internal reflection.

Everyday Applications of TIR

• Fibre‑optic cables – light is confined within the glass core by repeated total internal reflection.
• Diamond sparkle – the high refractive index (\(n\approx2.42\)) gives a very small critical angle, causing many internal reflections.
• Mirage over a hot road – a steep temperature gradient creates a rapid change in refractive index, leading to TIR and the appearance of “water”.

Summary Table

Exam Checklist (AO1–AO3)

1. Identify the point of incidence on the surface.
2. Draw the normal – a short line perpendicular to the surface at that point.
3. Label the incident ray, refracted ray, angle of incidence \(i\) and angle of refraction \(r\).
4. State the refractive indices of the two media (or give the critical angle if required).
5. Apply Snell’s law (or the critical‑angle formula) to find the unknown quantity.
6. When calculating, keep appropriate significant figures and include an uncertainty estimate.
7. Check the direction of bending:

   • Towards the normal if entering a denser medium (\(n2>n1\)).
   • Away from the normal if entering a rarer medium (\(n21\)).

8. If \(i>i_c\), state that total internal reflection occurs and no refracted ray exists.
9. For experimental questions, comment on sources of error (e.g., protractor reading, imperfect normal, surface imperfections) and how they affect the result.

Common Mistakes & How to Avoid Them

• Measuring from the surface – always measure angles from the normal, not from the surface itself.
• Swapping \(n1\) and \(n2\) – \(n1\) belongs to the medium containing the incident ray; \(n2\) to the medium of the refracted ray.
• Ignoring the normal on curved surfaces – even on a curved surface, the normal is the line perpendicular to the surface at the point of incidence.
• Assuming a critical angle exists for any pair of media – a critical angle only exists when \(n1>n2\). If \(n1\le n2\) the light always refracts out of the surface.
• Forgetting total internal reflection – when \(i>i_c\) the refracted ray does not exist; the ray is completely reflected.
• Neglecting significant figures and uncertainties – record angles to the precision of your instrument and propagate the uncertainties through calculations.