Question

In the measurement of the angle of a prism using a spectrometer, the readings of first reflected image are $Vernier\ I:{{320}^{{}^\circ }}{{40}^{\prime}};VernierII:{{140}^{\circ }}30^{\prime}$ and those of the second reflected image are $VernierI:{{80}^{{}^\circ }}{{38}^{\prime}};VernierII:{{260}^{{}^\circ }}{{24}^{\prime}}$. Then the angle of prism is:
$\text{A.}\quad 59\circ 58\prime$
$\text{B.}\quad 59\circ 56\prime$
$\text{C.}\quad 60\circ 2\prime$
$\text{D.}\quad 60\circ 4\prime$

Answer 1

Hint: In the spectrometer, we have a telescope attached to see the spectrum. We have to fix the angle by rotation to the point having a maximum intensity of colored light. After fixing the angle, we note the readings of the two Vernier scales attached to it. After that, we rotate the telescope to the other side to observe the intensity again. Hence the difference in Vernier reading gives us the angular distance between the two points which corresponds to maximum intensities.

Formula used:
$\phi = \dfrac{|difference\ of\ Vernier\ readings|}2$

Complete step by step answer:
For Vernier 1:
$1^{st}$ Reading = $320^{\circ}40\prime$
$2^{nd}$ Reading =$80^{\circ}38\prime$
Hence the difference in reading i.e. Spectral distance = $320^{\circ}40\prime-80^{\circ}38\prime=240^{\circ}2\prime$

For Vernier 2:
$1^{st}$ Reading = $260^{\circ}24\prime$
$2^{nd}$ Reading = $140^{\circ}30\prime$
Hence the difference in reading i.e. Spectral distance = $260^{\circ}24\prime-140^{\circ}30\prime=119^{\circ}24\prime$

Now, using
Angle of prism ($\phi$)=$\dfrac{|difference\ of\ Vernier \ readings|}{2}$
$\phi = \dfrac{240^{\circ}2\prime-119^{\circ}24\prime}2 = \dfrac{120^{\circ}8\prime}2 = 60^{\circ}4\prime$

So, the correct answer is “Option D”.

Additional Information:
Generally we think that a prism is a part of glass or a device made up of glass, having a triangular shape. But this is not so true. Any object made up of refracting medium, having faces through which light can pass can be termed as a prism. Prism works on the principle that when light passes through it, it undergoes refraction in such a way that light gets separated into a spectrum of different colors i.e. different wavelengths.

Note:
A spectrometer is a scientific instrument used to separate and measure spectral components of a physical phenomenon like the light. Spectrometers are generally present in engineering labs and are used to analyze the spectrum of light produced by sodium lamps and measure it precisely.