Respuesta :
Answer:
The answer is "[tex]1.96^{\circ} \ and \ 3.94^{\circ}[/tex]"
Explanation:
[tex]N= 1000 \frac{slit}{cm} \\\\\lambda=550 nm\ \ =550 \times 10^{-9} \ m\\\\width=1.6 \ cm[/tex]
We can find the distance between any two adjacent slits by using
[tex]\to d= \frac{1.6}{N}\\\\ \therefore\\ \to d= \frac{1.6}{1000} \ cm= 1.6 \times 10^{-5} m\\[/tex]
We know that the angle of the mth fringe is given by
[tex]\sin \theta_m = \frac{m\lambda}{d}\\\\ \sin \theta_1 = \frac{1\lambda}{d}\\\\\theta_1 = \sin^{-1} \frac{1\lambda}{d}\\\\\theta_1 = \sin^{-1}(\frac{1 \times 550 \times 10^{-9}}{1.6 \times 10^{-5}})\\\\ \theta_1=1.96^{\circ}\\\\[/tex]
Using the same way:
[tex]\sin \theta_2 = \frac{2\times \lambda}{d}\\\\\theta_2 = \sin^{-1} \frac{2\times \lambda}{d}\\\\\theta_2 = \sin^{-1}(\frac{2 \times 550 \times 10^{-9}}{1.6 \times 10^{-5}})\\\\\theta_2=3.94^{\circ}\\\\[/tex]
By using the Huygens formula, the angles of the first two orders are 4 degrees and 8 degrees approximately.
Given that a 1.6cm wide diffraction grating has 1000 slits.
The slit spacing = (1.6 x [tex]10^{-2}[/tex]) / 1000
The slit spacing = 1.6 x [tex]1.6^{-5}[/tex]
Also, it is given that It is illuminated by light of wavelength 530 nm
Sin∅ = nλ/d
When n = 1
Sin∅ = 530 x [tex]10^{-9}[/tex]/1.6 x [tex]10^{-5}[/tex]
Sin∅ = 0.033125
∅ = [tex]Sin^{-1}[/tex] ( 0.033125)
∅ = 1.898 x 2
The angle of the first order = 2∅
2∅ = 4 degrees approximately
When n = 2
Sin∅ = nλ/d
Sin∅ = 2 x 530 x [tex]10^{-9}[/tex]/1.6 x [tex]10^{-5}[/tex]
Sin∅ = 0.06625
∅ = [tex]Sin^{-1}[/tex] ( 0.06625)
The angle of the second order = 2∅
2∅ = 3.7986 x 2
2∅ = 8 degrees approximately
Therefore, the angles of the first two orders are 4 degrees and 8 degrees approximately.
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