Question

A point source of white light is being viewed normally through a thin prism of refracting angle \[{{3}^{0}}\] having refractive index 1.49 and 1.51 for violet and red light respectively and source is placed \[6\pi \] cm away from the prism. Find the distance between the red and the violet images formed due to refraction by prism. (Take \[{{\pi }^{2}}=10\])
A) 0.01cm
B) 0.014cm
C) 0.016cm
D) 0.02cm

Answer 1

Hint: A prism is given along with its refractive indices for the red and violet lights. The different refractive indices for different colours result in different images separated by a distance which is proportional to the refractive indices in the prism and the angle of prism.

Complete answer:
The deviation is caused by the difference in refractive indices of different colours in two different media. The deviation is also dependent on the thickness of the prism or the angle of the prism.

                 

The deviation caused by the change in refractive index for a prism with A as the angle of prism is –
\[\delta =A(\mu -1)\]
The deviation caused by each colour is given by –
\[\begin{align}
  & {{\delta }_{R}}=A({{\mu }_{R}}-1) \\
 & {{\delta }_{V}}=A({{\mu }_{V}}-1) \\
\end{align}\]
Where, \[\delta \]is the deviation in the light ray,
A is the angle of prism,
\[\mu \]is the refractive index of the colour in the medium
Now, we need to deviation between the two colours red and violet, which is given by the difference between their respective deviations as –
\[{{\delta }_{R}}-{{\delta }_{V}}=A({{\mu }_{R}}-{{\mu }_{B}})\]
The deviation in image formed after the light rays travelling a distance is given by multiplying the distance with this deviation. We are given that the image is formed after a distance of \[6\pi \]. The deviation between the images of the colors red and violet is given by –
\[\begin{align}
  & {{({{\delta }_{R}}-{{\delta }_{V}})}_{6\pi }}=6\pi [A({{\mu }_{R}}-{{\mu }_{B}})] \\
 & \Rightarrow \text{ }{{({{\delta }_{R}}-{{\delta }_{V}})}_{6\pi }}=6\pi \times \dfrac{3\pi }{180}\times (1.51-1.49) \\
 & \Rightarrow \text{ }{{({{\delta }_{R}}-{{\delta }_{V}})}_{6\pi }}=0.02cm \\
\end{align}\]
The deviation between the images of two colours is 0.02cm.

So, the correct answer is “Option D”.

Note:
The deviation of the colours is due to the different wavelengths each colour possesses. The wavelength is a characteristic property of the light passing through the prism. The colours with higher wavelength deviate less and have a higher refractive index.