A spectrometer gives the following reading when used to measure the angle of a prism main scale reading: \[58\cdot 5\text{ degree}\], vernier scale reading: 9 divisions. Given that 1 division of the main scale corresponds to \[0\cdot 5\] degree. Total Divisions on the vernier scale is 30 and match with 29 divisions of the main scale. The angle of the prism from above data?
A. \[58\cdot 59\text{ degree}\]
B. \[58\cdot 77\text{ degree}\]
C. \[58\cdot 65\text{ degree}\]
D. 59 degree

Hint: Spectrometer is an optical instrument used to study the spectra of different sources of light and to measure the refractive indices of materials. It consists of basically three parts: collimator, prism table and telescope. Here it is used to measure the angle of a prism.

Complete step by step answer:
GIVEN
Main scale Reading (MSR) \[=58\cdot 5\text{ degree}\]
Vernier scale Reading (VSR) = 9 Divisions
1 Division of main scale (1MSD) \[=0\cdot 5{}^\circ \]
Also 30 vernier scale divisions are equal to 29 main scale divisions.
So, \[30\text{ VSD}=29\text{ MSD}\]
\[\Rightarrow \text{1 VSD}=\dfrac{29}{30}\text{ MSD}\]
Least count of vernier: It is the smallest value that can be measured by any instrument.
L.C. of vernier \[=1\text{MSD}-1\text{VSD}\]
\[=\text{0}\cdot \text{5}{}^\circ -\dfrac{29}{30}\times 0\cdot 5{}^\circ \]
\[=\dfrac{0\cdot 5{}^\circ }{30}\]
\[\text{Reading of vernier = main scale reading + (vernier scale reading } \times \text{ Least count)}\]
Reading of vernier \[\text{=58}\cdot \text{5}{}^\circ \text{+}\left( 9\times \dfrac{0\cdot 5{}^\circ }{30} \right)\]
\[=58\cdot 65{}^\circ \]

So, the correct answer is “Option C”.

Note:
At the time of calculating the least count of vernier it is needed to multiply the value of 1 main scale division to vernier scale division.