A) Order of the first laser: 3, order of the second laser: 2
B) The overlap occurs at an angle of [tex]34.9^{\circ}[/tex]
Explanation:
A)
The formula that gives the position of the maxima (bright fringes) for a diffraction grating is
[tex]d sin \theta = m \lambda[/tex]
where
d is spacing between the lines in the grating
[tex]\theta[/tex] is the angle of the maximum
m is the order of diffraction
[tex]\lambda[/tex] is the wavelength of the light
For laser 1,
[tex]d sin \theta = m_1 \lambda_1[/tex]
For laser 2,
[tex]d sin \theta = m_2 \lambda_2[/tex]
where
[tex]\lambda_1 = 420 nm\\\lambda_2 = 630 nm[/tex]
Since the position of the maxima in the two cases overlaps, then the term [tex]d sin \theta[/tex] on the left is the same for the two cases, therefore we can write:
[tex]m_1 \lambda_1 = m_2 \lambda_2\\\frac{m_1}{m_2}=\frac{\lambda_2}{\lambda_1}=\frac{630}{420}=\frac{3}{2}[/tex]
Therefore:
[tex]m_1 = 3[/tex]
[tex]m_2 = 2[/tex]
B)
In order to find the angle at which the overlap occurs, we use the 1st laser situation:
[tex]d sin \theta = m_1 \lambda_1[/tex]
where:
N = 450 lines/mm = 450,000 lines/m is the number of lines per unit length, so the spacing between the lines is
[tex]d=\frac{1}{N}=\frac{1}{450,000}=2.2\cdot 10^{-6} m[/tex]
[tex]m_1 = 3[/tex] is the order of the maximum
[tex]\lambda_1 = 420 nm = 420\cdot 10^{-9} m[/tex] is the wavelength of the laser light
Solving for [tex]\theta[/tex], we find the angle of the maximum:
[tex]sin \theta = \frac{m_1 \lambda_1}{d}=\frac{(3)(420\cdot 10^{-9})}{2.2\cdot 10^{-6}}=0.572[/tex]
So the angle is
[tex]\theta=sin^{-1}(0.572)=34.9^{\circ}[/tex]
Learn more about diffraction:
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