A) Order of the first laser: 3, order of the second laser: 2
B) The overlap occurs at an angle of ![34.9^{\circ}](https://tex.z-dn.net/?f=34.9%5E%7B%5Ccirc%7D)
Explanation:
A)
The formula that gives the position of the maxima (bright fringes) for a diffraction grating is
![d sin \theta = m \lambda](https://tex.z-dn.net/?f=d%20sin%20%5Ctheta%20%3D%20m%20%5Clambda)
where
d is spacing between the lines in the grating
is the angle of the maximum
m is the order of diffraction
is the wavelength of the light
For laser 1,
![d sin \theta = m_1 \lambda_1](https://tex.z-dn.net/?f=d%20sin%20%5Ctheta%20%3D%20m_1%20%5Clambda_1)
For laser 2,
![d sin \theta = m_2 \lambda_2](https://tex.z-dn.net/?f=d%20sin%20%5Ctheta%20%3D%20m_2%20%5Clambda_2)
where
![\lambda_1 = 420 nm\\\lambda_2 = 630 nm](https://tex.z-dn.net/?f=%5Clambda_1%20%3D%20420%20nm%5C%5C%5Clambda_2%20%3D%20630%20nm)
Since the position of the maxima in the two cases overlaps, then the term
on the left is the same for the two cases, therefore we can write:
![m_1 \lambda_1 = m_2 \lambda_2\\\frac{m_1}{m_2}=\frac{\lambda_2}{\lambda_1}=\frac{630}{420}=\frac{3}{2}](https://tex.z-dn.net/?f=m_1%20%5Clambda_1%20%3D%20m_2%20%5Clambda_2%5C%5C%5Cfrac%7Bm_1%7D%7Bm_2%7D%3D%5Cfrac%7B%5Clambda_2%7D%7B%5Clambda_1%7D%3D%5Cfrac%7B630%7D%7B420%7D%3D%5Cfrac%7B3%7D%7B2%7D)
Therefore:
![m_1 = 3](https://tex.z-dn.net/?f=m_1%20%3D%203)
![m_2 = 2](https://tex.z-dn.net/?f=m_2%20%3D%202)
B)
In order to find the angle at which the overlap occurs, we use the 1st laser situation:
![d sin \theta = m_1 \lambda_1](https://tex.z-dn.net/?f=d%20sin%20%5Ctheta%20%3D%20m_1%20%5Clambda_1)
where:
N = 450 lines/mm = 450,000 lines/m is the number of lines per unit length, so the spacing between the lines is
![d=\frac{1}{N}=\frac{1}{450,000}=2.2\cdot 10^{-6} m](https://tex.z-dn.net/?f=d%3D%5Cfrac%7B1%7D%7BN%7D%3D%5Cfrac%7B1%7D%7B450%2C000%7D%3D2.2%5Ccdot%2010%5E%7B-6%7D%20m)
is the order of the maximum
is the wavelength of the laser light
Solving for
, we find the angle of the maximum:
![sin \theta = \frac{m_1 \lambda_1}{d}=\frac{(3)(420\cdot 10^{-9})}{2.2\cdot 10^{-6}}=0.572](https://tex.z-dn.net/?f=sin%20%5Ctheta%20%3D%20%5Cfrac%7Bm_1%20%5Clambda_1%7D%7Bd%7D%3D%5Cfrac%7B%283%29%28420%5Ccdot%2010%5E%7B-9%7D%29%7D%7B2.2%5Ccdot%2010%5E%7B-6%7D%7D%3D0.572)
So the angle is
![\theta=sin^{-1}(0.572)=34.9^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%3Dsin%5E%7B-1%7D%280.572%29%3D34.9%5E%7B%5Ccirc%7D)
Learn more about diffraction:
brainly.com/question/3183125
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