Question

In a spectrometer experiment, monochromatic light is incident normally on a diffraction grating having \[4.5 \times {10^5}\] lines per meter. The second order line is seen at an angle \[30^\circ \] to the normal. What is the wavelength of the light?
A. 200 nm
B. 556 nm
C. 430 nm
D. 589 nm

Answer 1

Hint: Calculate the spacing in the slits of grating from the number of lines given. Use the relation between angle of diffraction for \[{n^{th}}\] order diffraction pattern and wavelength of light used.

Formula Used: The angle of diffraction for \[{n^{th}}\] order diffraction pattern is given as,
\[d\sin \theta = n\lambda \]
Here, d is the spacing in slits, \[\theta \] is the angle of diffraction, n is the order of diffraction and \[\lambda \] is the wavelength of light used.

Complete step by step answer:
We know that the angle of diffraction for \[{n^{th}}\] order diffraction pattern is given as,
\[d\sin \theta = n\lambda \]
Here, d is the spacing in slits, \[\theta \] is the angle of diffraction, n is the order of diffraction and \[\lambda \] is the wavelength of light used.

Rearrange the above equation for \[\lambda \] as follows,
\[\lambda = \dfrac{{d\sin \theta }}{n}\]
Now, the spacing in the slits is equal to the reciprocal of the number of lines per meter. Therefore, the spacing in the slits is,
\[d = \dfrac{1}{{4.5 \times {{10}^5}\,{m^{ - 1}}}}\]
\[ \Rightarrow d = 2.22 \times {10^{ - 6}}\,m\]

We have been given second order as seen at an angle \[30^\circ \], therefore, \[n = 2\] and \[\theta = 30^\circ \]. On substituting the above values in the equation for \[\lambda \], we get,
\[\lambda = \dfrac{{\left( {2.22 \times {{10}^{ - 6}}\,m} \right)\sin \left( {30^\circ } \right)}}{2}\]
\[ \Rightarrow \lambda = \dfrac{{1.11 \times {{10}^{ - 6}}\,m}}{2}\]
\[ \Rightarrow \lambda = 5.55 \times {10^{ - 7}}\,m\]
\[ \Rightarrow \lambda = 555 \times {10^{ - 9}}\,m\]
We know that, \[1\,nm = {10^{ - 9}}\,m\], therefore,
\[\lambda = 555\,nm\]

Therefore, the wavelength of light used is close to 556 nm.

Hence, the correct option is (B).

Note:The number of lines through a slit is generally given in the number of lines per mm as the width of the slit is 1 mm. Now, if the number of lines is 300 lines per mm, then the spacing in the slit is, \[\dfrac{{{{10}^{ - 3}}\,m}}{{300}} = 3.33 \times {10^{ - 6}}\,m\].