Derivation of Prism Formula

Prism in Physics is defined as a transparent, polished flat optical element that reflects light. These can be made from any transparent material with wavelengths that they are designed for. The most commonly used material are glass, fluorite, and plastic.

Prisms called dispersive prism are used to break the light into its spectral colors. Other uses of prisms are to split light into its components with the polarisation of light or to reflect light. Following are the types of prisms:

  • Dispersive prisms: These are used to break the light into their constituent spectral colors. A triangular prism, amici prism grism are few examples of the dispersive prism.
  • Reflective prisms: These are used to reflect light in order to invert, rotate, deviate or displace the light beam. Pentaprism, dove prism, retroreflector prism are some examples of reflective prisms.
  • Polarising prisms: These are used to split the beam of light by varying the polarization. Nicol prism, Glan-Taylor prism are some the examples of polarising prisms.
  • Beam-splitting prisms: These are used to split beams into two or more beams. Beam splitter cube and dichroic prism are the examples of the beam-splitting prism.
  • Deflecting prisms: These are used to deflect the beam of light at a fixed angle. Wedge prism is the example of deflecting prism.

Derivation of prism formula

\(\mu =\frac{sini}{sinr}\) (by Snell’s law)

\(\delta =i_{1}-r_{1}+i_{2}-r_{2}…\) (eq.1)

\(\delta =i_{1}+i_{2}-(r_{1}+r_{2})\) \(\angle ALO+\angle AMO=2rt\angle s\) (from quadrilateral and ∠ALO=∠AMO=90°)

\(\angle LAM+\angle LOM=2rt\angle s\) (sum of four ∠s of a quadrilateral = 4 rt∠s) (eq.2)

\(\angle r_{1}+\angle r_{2}+\angle LOM=2rt\angle s\) (eq.3)

\(\angle LAM =\angle r_{1}+\angle r_{2}\) (comparing eq.2 and eq.3)

\(A =\angle r_{1}+\angle r_{2}\) \(\delta =i_{1}+i_{2}-A\) (substituting A in eq.1)

\(i_{1}+i_{2}=A+\delta\) \(\angle i_{1}=\angle i_{2}\) \(\angle r_{1}=\angle r_{2}=\angle r\) \(\angle ALM=\angle LMA=90^{\circ}-\angle r\)

Thus, AL = LM and LM ∥ BC

\(\angle A=\angle r_{1}+\angle r_{2}\) \(A=2r\) (since, \(\angle r_{1}=\angle r_{2}=\angle r\))

\(r=\frac{A}{2}\) \(i_{1}+i_{2}=A+\delta\) \(i_{1}+i_{1}=A+\delta_{m}\) \(2i_{1}=A+\delta_{m}\) \(i_{1}= \frac{A+\delta_{m}}{2}\) \(∴\mu =\frac{sin\frac{A+\delta _{m}}{2}}{sin\frac{A}{2}}\)

Thus, above is the prism formula.

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