Question

Light of wavelength 650 nm is normally incident on the rear of a grating. The first bright fringe (other than the central one) is at an angle of 5 ° with respect to the normal.
1)Find the number of slits per centimeter in the grating.


_________slits/cm


2)Two rays of light of wavelength 650 nm and 420 nm are normally incident on a different grating. If the grating has 5000 slits/cm, what is the angular seperation of of the two light rays' second order maximum.


|θ650 -θ420| =_________°

Answer 1

Answer

given,

wavelength (λ) = 650 nm

angle = 5°

using bragg's law

$sin \theta = \dfrac{n \lambda}{d}$

$d= \dfrac{n \lambda}{sin \theta}$

$d= \dfrac{1 \times 650 \times 10^{-9}\ m}{sin5^0}$

d = 7.46 x 10⁻⁴ cm

number of slits per centimeter

  = $\dfrac{1}{d}\\\Rightarrow \dfrac{1}{7.46\times 10^{-4}}\\\Rightarrow 1340 split per centimeter.$

b) wavelength of two rays  650 nm and 420 nm

 $d = \dfrac{1}{5000}$

     d =  2 x 10⁻6 m

    we now,

$sin \theta = \dfrac{n \lambda}{d}$

for 650 nm

$sin \theta = \dfrac{2\times 650\times 10^{-9}}{2\times 10^{-6}}$

$\theta =sin^{-1}(0.65)$

θ = 40.54°

for 450 nm

$sin \theta = \dfrac{2\times 450\times 10^{-9}}{2\times 10^{-6}}$

$\theta =sin^{-1}(0.45)$

θ = 24.83°

now, difference

|θ_{650} -θ_{420}| =40.54°-24.83°

|θ_{650} -θ_{420}| =19.71°