A coherent parallel beam of microwaves of wavelength `lambda = 0.5 mm` falls on aYoung'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 figure

If the incient beam falls normally on the double-slit apparatus, find the order of the interference minima on the screen

Options

A. Only the first order minima are possible
B. Only the first order and second minima are possible
C. Total six minima appear on the screen
D. Total eight minima appear on the screen

Answer

a

Explanation

Position of a point on a screen is
`y = D tan theta = 1 tan theta`
For first minima,
`n = 1, sin theta_(1) = (1)/(4), tan theta_(1) = (1)/(sqrt 15)`
For second minima,
`n = 2, sin theta_(2) = (3)/(4), tan theta_(2) = (3)/(sqrt 7)`
So, the positions of minima are
`y_(1) = tan theta_(1) = (1)/(sqrt 15) = 0.258 m`
`y_(2) = tan theta_(2) = (3)/(sqrt 7) = 1.13 m`
The minima are symmetrically placed on either of central mixima, therefore will be 4 minima at positions `+- 0.258 m` and `+- 1.13 m` on the screen.
a. When incident rays are incident normally, the waves arriving at slits are in phase, zero path difference before slit. Path difference after slits, at point P, is `d sin theta`.
Condition for minima at y-axis is
`d sin theta = (2 n - 1) (lambda)/(2)`
`sin theta = ((2 n -1)lambda)/(2d) = ((2 n - 1)(0.5))/(2 xx 1) = ((2 n - 1))/(4)`
As `sin theta le 1, ((2n - 1)/(4)) le` or `n le 2.5`
Hence, only first-order and second-order minima are possible.

b. 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 difference for first minima is
`d sin theta - d sin phi = +- (lambda)/(2)`
`sin theta = sin phi +- (lambda)/(2)`
`= sin 30^(@) +- (0.5)/(2 xx 1) = (3)/(4)` or `(1)/(4)`
`tan theta = (3)/(sqrt 7)` and `(1)/(sqrt 15)`
So, the position of first minima on either side of central maxima is
`y = D tan theta = (3)/(sqrt 7)` and `(1)/(15) m`