Answer :
Answer:
The value is [tex]\theta = 1.63^o[/tex]
Explanation:
From the question we are told that
The wavelength of the first spectral line is [tex]\lambda _1 = 417 nm = 417*10^{-9} \ m[/tex]
The wavelength of the second spectral line is [tex]\lambda _2 = 388 nm = 388 *10^{-9} \ m[/tex]
The diffraction grating is [tex]k =456 lines/mm = 456000 \ lines / m[/tex]
Generally the condition for constructive interference is
[tex]dsin (\theta ) = n * \lambda[/tex]
Here n is the order of maxima which is n = 2 given that order of the spectral lines is second order
d is this the distance of slit separation which is evaluated as
[tex]d = \frac{1}{k}[/tex]
=> [tex]d = \frac{1}{456000}[/tex]
=> [tex]d = 2.19298 *10^{-6} \ m[/tex]
Considering the first spectral lines with [tex]\lambda _1 = 417 nm = 417*10^{-9} \ m[/tex]
[tex]\theta_1 = sin ^{-1} [\frac{ 2 * 417 * 10^{-9}}{2.19298 *10^{-6}} ][/tex]
=> [tex]\theta_1 = 22.35^o[/tex]
Considering the first spectral lines with [tex]\lambda _2 = 388 nm = 388 *10^{-9} \ m[/tex]
[tex]\theta_2 = sin ^{-1} [\frac{ 2 * 388 * 10^{-9}}{2.19298 *10^{-6}} ][/tex]
=> [tex]\theta_2 = 20.72^o[/tex]
Generally the angular separation is
[tex]\theta = \theta_1 - \theta_2[/tex]
=> [tex]\theta = 22.35 - 20.72[/tex]
=> [tex]\theta = 1.63^o[/tex]