With the use of the formula SinФ = nλ / d, there are 16 spectral orders which can be seen when it is illuminated by white light.
Given that a grating has 2000 slits/cm. That is,
d = 0.01 / 2000
d = 5 x [tex]10^{-6}[/tex] m
The wavelength λ = (700 - 400) nm
λ = 300 x [tex]10^{-9}[/tex] m
To calculate how many full spectral orders that can be seen (400 to 700 nm) when it is illuminated by white light, we will use the below formula
SinФ = nλ / d
Φ = [tex]Sin^{-1}[/tex] (nλ / d)
When n = 1
Φ = [tex]Sin^{-1}[/tex](300 x [tex]10^{-9}[/tex]/ 5 x [tex]10^{-6}[/tex] )
Φ = 3.4 degrees
when n = 2
Φ = [tex]Sin^{-1}[/tex](2 x 300 x [tex]10^{-9}[/tex]/ 5 x [tex]10^{-6}[/tex] )
Ф = 6.9 degrees
When n = 3
Ф = [tex]Sin^{-1}[/tex](3 x 300 x [tex]10^{-9}[/tex]/ 5 x [tex]10^{-6}[/tex] )
When n = 16
Ф = [tex]Sin^{-1}[/tex](16 x 300 x [tex]10^{-9}[/tex]/ 5 x [tex]10^{-6}[/tex] )
Ф = [tex]Sin^{-1}[/tex](0.96)
Ф = 73.7 degrees
when n = 17
Ф = [tex]Sin^{-1}[/tex](17 x 300 x [tex]10^{-9}[/tex]/ 5 x [tex]10^{-6}[/tex] )
Ф = [tex]Sin^{-1}[/tex](1.05)
Ф = Error ( that is, it does not exist)
Therefore, there are 16 spectral orders which can be seen when it is illuminated by white light.
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