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Question: In a spectrometer experiment, monochromatic light is incident normally on a diffraction grating havi...

In a spectrometer experiment, monochromatic light is incident normally on a diffraction grating having 4.5×1054.5 \times {10^5} lines per meter. The second order line is seen at an angle 30∘30^\circ to the normal. What is the wavelength of the light?
A. 200 nm
B. 556 nm
C. 430 nm
D. 589 nm

Explanation

Solution

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

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

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

Rearrange the above equation for λ\lambda as follows,
λ=dsin⁡θn\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=14.5×105 m−1d = \dfrac{1}{{4.5 \times {{10}^5}\,{m^{ - 1}}}}
⇒d=2.22×10−6 m\Rightarrow d = 2.22 \times {10^{ - 6}}\,m

We have been given second order as seen at an angle 30∘30^\circ , therefore, n=2n = 2 and θ=30∘\theta = 30^\circ . On substituting the above values in the equation for λ\lambda , we get,
λ=(2.22×10−6 m)sin⁡(30∘)2\lambda = \dfrac{{\left( {2.22 \times {{10}^{ - 6}}\,m} \right)\sin \left( {30^\circ } \right)}}{2}
⇒λ=1.11×10−6 m2\Rightarrow \lambda = \dfrac{{1.11 \times {{10}^{ - 6}}\,m}}{2}
⇒λ=5.55×10−7 m\Rightarrow \lambda = 5.55 \times {10^{ - 7}}\,m
⇒λ=555×10−9 m\Rightarrow \lambda = 555 \times {10^{ - 9}}\,m
We know that, 1 nm=10−9 m1\,nm = {10^{ - 9}}\,m, therefore,
λ=555 nm\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, 10−3 m300=3.33×10−6 m\dfrac{{{{10}^{ - 3}}\,m}}{{300}} = 3.33 \times {10^{ - 6}}\,m.