A coherent parallel beam of microwaves of wavelength `lambda = 0.5 mm` falls on a Young's double- slit apparatus. The separation between the slits is 1.0 mm. The intensity of microwaves is measured on a screen placed parallel to the plane of the slits at a distance of 1.0 m from it as shown in Fig. 2.42.
If the incident beam makes an angle or `30^(@)` with the x-axis (as in the dotted arrow shown in the figure), find the y-coordinates of the first minima on either side of the central maximum.

Explanation

Path difference before slits `= d sin phi`
Path difference after slits `= d sin theta`
As path of rays before slits is longer at `S_(1)` and `S_(2) P gt S_(1) P` after slits, so net path defference for first minima is
`Delta x = d sin theta - d sin phi = +- (lambda)/(2)`
`implies sin theta = sin phi +- (lambda)/(2d)`
`(y)/(D) = sin 30^(@) +- (0.5)/(2 xx 1) = (3)/(4)` or `(1)/(4)`
`implies tan theta = (3)/(sqrt 7)` and `(1)/(sqrt 15)`
So, the position of first minima on either side of the central maxima is
`y = D tan theta = (3)/(sqrt 7)` and `(1)/(sqrt 15)` m
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