Overview: Refraction of Light at a Plane Interface

The refractive index of a medium is the parameter that tells how much slower light travels in that medium compared to vacuum.

Mathematically,
n = \[\frac{\text{velocity of light in vacuum}}{\text{velocity of light in medium}}=\frac{c}{v}\]

where c = 3 × 10⁸ ms^-1.

When the velocity of light in a medium is compared with that in another medium, the parameter is called the relative refractive index.

“The perpendicular distance between the emergent ray and the direction of the incident ray is called the lateral shift.”

The critical angle for two given media is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90°.

When a ray of light, travelling from a denser medium to a rarer medium, is incident at the interface of the two media at an angle greater than the critical angle for the two media, the ray is 'totally' reflected back into the denser medium.

\[^1n_2=\frac{v_1}{v_2}=\frac{n_2}{n_1}\]

Lateral shift (d) = t sin (i − r) sec r

\[^2n_3=\frac{n_3}{n_2}\]

\[_1n_2=\frac{1}{\sin C}\cdot\]

Statement

When a light ray, after undergoing any number of reflections and refractions, has its direction reversed, it retraces its entire original path. This is called the principle of reversibility of light.

Explanation / Proof

Consider a light ray passing from medium 1 to medium 2 and suffering refraction at the boundary.
Let the angle of incidence be i and the angle of refraction be r.

By Snell’s law, the refractive index of medium 2 with respect to medium 1 is:

¹n₂ = \[\frac {sin ⁡i}{sin⁡ r}\]​

Now, suppose the refracted ray is reflected back and retraces the path in the reverse direction. In this case, the angle of incidence becomes r, and the angle of refraction becomes i.

Again, by Snell’s law, the refractive index of medium 1 with respect to medium 2 is:

²n₁ = \[\frac {sin ⁡r}{sin ⁡i}\]​

Multiplying the two equations:

¹n₂ × ²n₁ = 1

This shows that the ray follows the same path in the reverse direction, proving the reversibility of the light path.

Conclusion

Hence, a light ray always retraces its original path when its direction is reversed, even after multiple reflections and refractions. This establishes the principle of reversibility of light.

• Refraction occurs due to a change in the speed of light when it passes from one medium to another.
• The greater the change in speed, the greater is the bending of light at the boundary of the two media.
• According to Snell’s law:
  If v₁ > v₂​, the ray bends towards the normal (rarer to denser medium).
  If v₁ < v₂​, the ray bends away from the normal (denser to rarer medium).

• Refractive index indicates the direction of bending of light at a boundary (towards or away from the normal).
• It gives the ratio of the speeds of light in vacuum and in the medium:
  n = \[\frac {c}{v}\]So, a higher refractive index means a lower speed of light in the medium.
• The frequency of light remains unchanged during refraction, but the wavelength changes; hence, the refractive index also gives information about the wavelength of light in a medium.

• An object in a denser medium appears raised when viewed from a rarer medium due to refraction.
• Real depth is the actual depth of the object; apparent depth is the depth at which it appears.
• Refractive index is given by:
  n = \[\frac{\text{Real depth}}{\text{Apparent depth}}\]​
• Normal displacement is the difference between real and apparent depths:
  d = Real depth − Apparent depth
• For a medium of thickness t:
  d = t (1 − \[\frac {1}{n}\])

• Mirage is caused by total internal reflection in hot air layers, making objects appear inverted, as in water reflections.
• Diamonds sparkle because light undergoes repeated total internal reflections due to their small critical angle.
• Totally reflecting prisms use total internal reflection to reflect light efficiently.
• Right-angled prisms can turn light by 90° or 180° using total internal reflection.
• Prisms are better than mirrors because they reflect almost all light and produce clear images.
• Optical fibres guide light by total internal reflection and are used in communication and medical imaging.