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Mathematics for Physics

82 Refraction

A phenomenon in which an electromagnetic wave changes direction and speed as it crosses into a medium with a different refractive index that is used to aim a transmitted ray, where the refractive index is the factor by which the wave speed is reduced from \(c\).

Snell’s law. The transmitted ray bends according to Snell’s law. This principle is used to compute the angle of refraction from the two indices.

Snell’s law is

\[ n_{1}\sin\theta_{i} = n_{2}\sin\theta_{t} \]

where

  • \(n_{1}\) is the index of refraction of the incident medium.
  • \(n_{2}\) is the index of refraction of the transmitted medium.
  • \(\theta_{i}\) is the angle of incidence.
  • \(\theta_{t}\) is the angle of transmission.

The index of refraction. The wave speed in a linear medium is \(c/n\). This principle is used to relate the index to the permittivity and permeability of the medium.

The index of refraction is

\[ n = \sqrt{\dfrac{\epsilon\mu}{\epsilon_{0}\mu_{0}}} \]

where

  • \(n\) is the index of refraction.
  • \(\epsilon\) is the permittivity of the medium.
  • \(\mu\) is the permeability of the medium.
  • \(\epsilon_{0}\) is the permittivity of free space.
  • \(\mu_{0}\) is the permeability of free space.

Total internal reflection. When \(n_{1} > n_{2}\) and the incidence angle exceeds the critical angle, there is no transmitted ray. This principle is used to describe total internal reflection.

The critical angle is

\[ \theta_{c} = \arcsin\Bigl(\dfrac{n_{2}}{n_{1}}\Bigr) \]

where

  • \(\theta_{c}\) is the critical angle.
  • \(n_{1}\) is the index of the denser medium.
  • \(n_{2}\) is the index of the rarer medium.