Question

According to Cauchy, the Refractive index is
A. Directly proportional to the wavelength of light
B. Directly proportional to the square of the wavelength of light
C. Inversely proportional to the wavelength of light
D. Inversely proportional to the square of the wavelength of light

Answer 1

Hint: An empirical/dispersion relationship between the wavelength of the light and the refractive index for a particular transparent material is the Cauchy’s transmission equation. The empirical coefficients of Cauchy’s equation represent the nature of the material.
Formula used:
\[n(\lambda )=A+\dfrac{B}{{{\lambda }^{2}}}+\dfrac{C}{{{\lambda }^{4}}}+.....\]

Complete step-by-step solution:
The Cauchy’s equation is given by,
\[n(\lambda )=A+\dfrac{B}{{{\lambda }^{2}}}+\dfrac{C}{{{\lambda }^{4}}}+.....\]
Where n is the refractive index, \[\lambda \] is the wavelength and A, B, C, ….. are the coefficients coated for \[\lambda \] as the vacuum wavelengths (empirical coefficients).
As the wavelength is in terms of nanometers, the wavelengths with the higher powers can be omitted.
So, the equation reduces as follows.
\[n(\lambda )=A+\dfrac{B}{{{\lambda }^{2}}}\]
The refractive index is inversely proportional to the square of the wavelength of light.
As, according to Cauchy’s equation, the refractive index is inversely proportional to the square of the wavelength of light, thus, option (D) is correct.

Additional information:
This Cauchy’s equation is only valid for the regions of the normal dispersion in the visible wavelength.
The parallel beams of light incident on a lens will bend more towards the axis in the case blue light when compared to the red light, as \[{{\lambda }_{r}}> {{\lambda }_{b}}\] hence \[{{n}_{r}}< {{n}_{b}}\].
The red light gets less dispersed than the violet light because the refractive index is inversely proportional to the wavelength of the light, as, the wavelength of violet light is less than that of the red light.

Note: The things to be on your finger-tips for further information on solving these types of problems are: There are different Cauchy’s equations: They are: Cauchy’s functional equation, Cauchy momentum equation, and Cauchy’s equation in optics. All these belong to different concepts.