William Burnett

2022-01-11

Two slits are 0.800 m from a screen, separated by 0.0720 mm. The two slits allow coherent light of wavelength to pass through. The separation between the first minimum and the center of the central maximum in their interference pattern on the screen is 3.00 mm. What is the intensity at spots on the screen that are (a) 2.00 mm and (b) 1.50 mm from the center of the central maximum if the intensity at the peak of the central maximum is 0.0600 W/m2?

Deufemiak7

Expert

The intensity at any point can be determined using the relation:
$I={I}_{max}{\mathrm{cos}}^{2}\left(\frac{\varphi }{2}\right)$
where $\varphi$ is the phase difference corresponding to the given point.
The path difference corresponding to the position y can be given by the following relation:
$\mathrm{△}x=\frac{yd}{D}$
This path difference for the first minimum is equal to half of the wavelength. Hence, calculate the wavelength using the relation.
$\frac{\lambda }{2}=\frac{\left(3×{10}^{-3}\right)\left(0.072×{10}^{-3}\right)}{\left(0.800\right)}$

Now, calculate the path difference corresponding to y=2 mm from the central maximum.
$\mathrm{△}x=\frac{yd}{D}$
$=\frac{\left(2×{10}^{-3}\right)\left(0.072×{10}^{-3}\right)}{0.800}$

Now, calculate the phase difference corresponding to the path difference.
$\varphi =\frac{2\pi }{\lambda }×\mathrm{△}x$
$=\frac{2\pi }{54×{10}^{-8}}×\left(18×{10}^{-8}\right)$
$=\frac{2\pi }{3}$
The intensity at the point can be determined as follows:
$I={I}_{max}{\mathrm{cos}}^{2}\left(\frac{2\frac{\pi }{3}}{2}\right)$

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