Answer :
Answer:
The maximun distance is [tex]z_1 = z_2 = 0.0138m[/tex]
Explanation:
From the question we are told that
The wavelength are [tex]\lambda _ 1 = 540nm (green) = 540 *10^{-9}m[/tex]
[tex]\lambda_2 = 450nm(blue) = 450 *10^{-9}m[/tex]
The distance of seperation of the two slit is [tex]d = 0.180mm = 0.180 *10^{-3}m[/tex]
The distance from the screen is [tex]D = 1.53m[/tex]
Generally the distance of the bright fringe to the center of the screen is mathematically represented as
[tex]z = \frac{m \lambda D}{d}[/tex]
Where m is the order of the fringe
For the first wavelength we have
[tex]z_1 = \frac{m_1 (549 *10^{-9} * (1.53))}{0.180*10^{-3}}[/tex]
[tex]z_1=0.00459m_1 m[/tex]
[tex]z_1= 4.6*10^{-3}m_1 m ----(1)[/tex]
For the second wavelength we have
[tex]z_2 = m_2 \frac{450*10^{-9} * 1.53 }{0.180*10^{-3}}[/tex]
[tex]z_2 = 0.003825m_2[/tex]
[tex]z_2 = 3.825 *10^{-3} m_2 m[/tex] ----(2)
From the question we are told that the two sides coincides with one another so
[tex]zy_1 =z_2[/tex]
[tex]4.6*10^{-3}m_1 m = 3.825 *10^{-3} m_2 m[/tex]
[tex]\frac{m_1}{m_2} = \frac{3.825 *10^{-3}}{4.6*10^{-3}}[/tex]
Hence for this equation to be solved
[tex]m_1 = 3[/tex]
and [tex]m_2 = 4[/tex]
Substituting this into the equation
[tex]z_1 = z_2 = 3 * 4.6*10^{-3} = 4* 3.825*10^{-3}[/tex]
Hence [tex]z_1 = z_2 = 0.0138m[/tex]