State the meaning of critical angle

3.2.2 Refraction of Light

Key terminology

• Normal – an imaginary line drawn perpendicular to the surface at the point of incidence.
• 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.

Propagation inside a homogeneous medium

Within a uniform (homogeneous) transparent material a light ray travels in a straight line. It changes direction only when it reaches a boundary where the refractive index changes.

Refractive index

The refractive index of a medium, n, quantifies how much the speed of light is reduced in that medium:

\[

n = \frac{c}{v} = \frac{\sin i}{\sin r}

\]

where c is the speed of light in vacuum and v is the speed of light in the medium.

Example: In water \(\sin i / \sin r \approx 1.33\); therefore \(n_{\text{water}} \approx 1.33\).

Snell’s law – boundaries only

When a ray crosses a boundary between two media of different refractive indices, the angles of incidence and refraction are related by

\[

n1\sin i = n2\sin r

\]

where \(n1\) and \(n2\) are the refractive indices of the first and second medium respectively.

Classroom experiment – observing refraction and measuring the refractive index

1. Draw a straight line on a sheet of paper and place a rectangular glass slab (or a triangular prism) on the line.
2. Shine a narrow laser beam (or a ray of sunlight) onto the slab so that it enters one face and emerges from the opposite face.
3. At the entry and exit points:

   • Draw the normal.
   • Mark the incident ray and the emergent ray.
   • Measure the angle of incidence \(i\) and the angle of refraction \(r\) with a protractor.

4. Calculate the refractive index of the slab using \(n = \sin i / \sin r\). Compare your result with the accepted value (e.g. \(n_{\text{glass}}\approx1.50\)).
5. Repeat the procedure with the prism to see how the ray is bent twice (once on entry, once on exit).

Critical angle

The critical angle (\(c\)) is the smallest angle of incidence in the denser medium for which the angle of refraction in the rarer medium becomes exactly \(90^{\circ}\). At this angle the refracted ray travels along the interface.

Mathematical expression

For light passing from a denser medium (\(n1\)) to a rarer medium (\(n2\)) with \(n1 > n2\):

\[

\sin c = \frac{n2}{n1}

\]

If the angle of incidence exceeds \(c\), the light is totally internally reflected.

How to determine the critical angle experimentally

1. Fill a transparent tank with water (or any transparent liquid). Place a protractor on the side of the tank so that the normal at the water‑air interface is marked.
2. From below the water surface, shine a laser beam upward at the interface.
3. Gradually increase the angle of incidence and observe the emergent beam:

   • When the beam just grazes the surface (i.e. disappears from the air side), record that angle as the experimental critical angle \(c_{\text{exp}}\).
   • Compare \(c{\text{exp}}\) with the calculated value using \(\sin c = n{\text{air}}/n_{\text{water}}\).

Typical critical angles

Internal reflection and total internal reflection (TIR)

• Internal reflection – any reflection that occurs when a ray travelling in one medium meets the boundary with a second medium and is reflected back into the original medium.
• Total internal reflection (TIR) – internal reflection that occurs when the angle of incidence is greater than the critical angle; no refracted ray emerges from the interface.

Everyday examples

1. A fish looking up at the water surface sees the world above water only within a cone of about 48°; outside this cone the light is totally internally reflected, giving the fish a mirror‑like view of its own surroundings.
2. Optical fibres guide light by repeated total internal reflection inside a glass core (n≈1.50) surrounded by a cladding of lower refractive index (n≈1.45).

Simple classroom demonstration of TIR

Using the water tank described above, increase the laser’s angle of incidence until the beam no longer emerges from the water‑air interface but instead is reflected back into the water. This visualises total internal reflection at the critical angle.

Key points to remember

• The critical angle exists only when light travels from a denser to a rarer medium (\(n1 > n2\)).
• At the critical angle the refracted ray makes \(90^{\circ}\) with the normal, i.e. it runs along the interface.
• Incidence angles greater than the critical angle produce total internal reflection.
• A larger difference between the refractive indices → a smaller critical angle.
• Snell’s law (\(n1\sin i = n2\sin r\)) and the definition \(n = \sin i / \sin r\) are the tools used to calculate both the refractive index and the critical angle.