Relation Between Critical Angle And Refractive Index

In Optics, The angle of incidence to which the angle of refraction is 90° is called the critical angle. The ratio of velocities of a light ray in the air to the given medium is a refractive index. Thus, the relation between the critical angle and refractive index can be established as the Critical angle is inversely proportional to the refractive index.

Critical Angle And Refractive Index

The relationship between critical angle and refractive index can be mathematically written as –

\(\begin{array}{l}SinC=\frac{1}{\mu _{b}^{a}}\end{array} \)

Where,

C is the critical angle.

μ is the refractive index of the medium.

a and b represent two mediums in which light rays travel.

Critical Angle And Refractive Index Formula

Formula SI Unit
Critical angle to Refractive index
\(\begin{array}{l}SinC=\frac{1}{\mu _{b}^{a}}\end{array} \)
degree
Refractive index to Critical angle
\(\begin{array}{l}\mu _{b}^{a}=\frac{1}{sinC}\end{array} \)
No SI unit

Critical angle and refractive index relation derivation

The relationship between critical angle and refractive index can be derived as –

Consider a ray of light,

  • Let the angle of incidence i be critical angle C
  • Let the angle of refraction r=90º
  • Refractive index of the rarer medium be μa
  • Refractive index of the denser medium be μb

Applying Snell’s Law

  • \(\begin{array}{l}\frac{sin\;i}{sin\;r}=\frac{\mu _{a}}{\mu _{b}}\end{array} \)
  • \(\begin{array}{l}\Rightarrow \mu _{b}sinC=\mu _{a}sin90^{0}\end{array} \)
  • \(\begin{array}{l}\Rightarrow \frac{\mu _{b}}{\mu _{a}}=\frac{1}{sinC}\end{array} \)

Thus, we arrive at a formula expressing the critical angle and refractive index relation –

\(\begin{array}{l}\mu _{b}^{a}=\frac{1}{sinC}\end{array} \)

Hope you understood the relation and conversion between the Critical Angle and Refractive Index in Optics.

Physics Related Topics:

Optical density
Phase angle
Refraction and Dispersion of Light through a Prism
Polarisation by scattering

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