Question

A thin prism ${p_1}$ with angle 4 and made from glass with refractive index 1.54 is combined with another prism ${p_2}$ made from glass of refractive index 1.72 to produce dispersion without deviation. What is the angle of the prism ${p_2}$?

Answer 1

Here we will first know what does dispersion mean then we will also know about dispersion of light in prism and then by using the formula for finding the angle of deviation in a prism we will solve the given problem.

Formula used:
For dispersion without deviation:
$\left( {{\mu }_{1}}-1 \right)\times {{A}_{1}}=\left( {{\mu }_{2}}-1 \right){{A}_{2}}$
Where ${{\mu }_{1}}$ is the refractive index of prism ${p_1}$, ${{A}_{1}}$ is the angle of prism ${p_1}$, ${{\mu }_{2}}$ is the is the refractive index of prism ${p_2}$ and ${{A}_{2}}$ is the is the angle of prism ${p_2}$.

Complete step by step answer:
As white light passes through a glass prism, it breaks into its spectrum of colours (in order violet, indigo, blue, green, yellow, orange, and red), a phenomenon known as dispersion.As light travels from one medium to another, the speed at which it propagates varies, and it bends or refracts as a result.

Light is now refracted into the triangle's base as it passes through a prism. The above diagram clearly depicts the refraction of light through a prism. The wavelengths of various colours in the spectrum of light are different. As a result, the rate at which they bend varies depending on the wavelength, with violet bending the most due to its shortest wavelength and red bending the least due to its longest wavelength.

When white light is refracted through a prism, it disperses through its spectrum of colours as a result of this. Here the given data are
${{\mu }_{1}}$ is the refractive index of prism ${p_1}$ = 1.54
${{A}_{1}}$ is the angle of prism ${p_1}$ = 4
${{\mu }_{2}}$ is the is the refractive index of prism ${p_2}$ = 1.72
${{A}_{2}}$ is the angle of the prism ${p_2}$ = ?
Now, we know that
$\left( {{\mu }_{1}}-1 \right)\times {{A}_{1}}=\left( {{\mu }_{2}}-1 \right){{A}_{2}}$
Putting all the known quantities,
$\left( 1.54-1 \right)\times 4=\left( 1.72-1 \right)\times {{A}_{2}}$
${{A}_{2}}=2.16/0.72$
$\therefore {{A}_{2}}=3{}^\circ$

Therefore, the angle of prism ${p_2}$ is$3{}^\circ$.

Note: The following are the variables that influence the angle of deviation: The angle of incidence, the refracting angle of the prism, Refractive index of the material used in the prism and the wavelength of the light ray.