To solve this problem it is necessary to apply the Bragg's law which allows to study the directions in which the diffraction of X-rays on the surface of a crystal produces constructive interference.
Extrapolating the equation and obtaining its mathematical meaning we have to
[tex]dsin\theta = n\lambda[/tex]
Where
d = Separation between slits
[tex]\lambda =[/tex] wavelength
n = Order number representing the number of repetition of the spectrum
[tex]\theta =[/tex] Angle between the source and the screen at this case are perpendicular
At the same time we have that the grating for this case is given as
[tex]d = \frac{1}{N} = \frac{1}{410*10^{-3}}m = 2.439*10^{-6}m[/tex]
Using the previous equation to find the order number we have that
[tex]dsin\theta = n\lambda[/tex]
For the first wavelength
[tex]n_{red} = \frac{dsin\theta}{\lambda_1}[/tex]
[tex]n_{red} = \frac{(2.439*10^{-6})(sin90)}{700*10^{-9}}[/tex]
[tex]n_{red} = 3.4842[/tex]
For the second wavelength
[tex]n_{violet} = \frac{dsin\theta}{\lambda_2}[/tex]
[tex]n_{violet} = \frac{(2.439*10^{-6})(sin90)}{400*10^{-9}}[/tex]
[tex]n_{violet} = 6.0975[/tex]
Therefore the number of orders in which the visible spectrum exists is
[tex]\Delta n = n_{violet}-n_{red}[/tex]
[tex]\Delta n = 6.0975-3.4842[/tex]
[tex]\Delta n = 2.6133 \approx 3[/tex]
Therefore the number of order can one see the entire visible spectrum 3.
