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 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.

Answer

(a)`+-(1)/(sqrt(15))m +-(3)/(sqrt(7))m` (b)`(1)/(sqrt(15)),+(3)/(sqrt(17))m`

Explanation

`d=sintheta=(2n-1)(lambda)/(2)`
`sintheta=((2n-1)lambda)/(2d) rArr sintheta=(2n-1)/(2)xx(0.5xx10^(-3))/(10^(-3)) rArr sintheta=((2n-1))/(4)`
For first minima `n=1`
`sintheta=+-(1)/(4) rArr tantheta=+-(1)/(sqrt(15)) rArr tantheta=(y_(1))/(D)=+-(1)/(sqrt(15)) rArr y_(1)=+-(1)/(sqrt(15))m`
for second minima `n=2`
`sintheta=+-(3)/(4) rArr tan theta=+-(3)/(sqrt(7)) rArr tantheta=(y_(2))/(D)=pm(3)/(sqrt(7)) rArr y_(2)=+-(3)/(sqrt(7))m` (b) For point`P`to be centeral maximum
`dsin30^(2)+S_(1)P-S_(2)P=0`
`dsin30^(@)=(S_(2)P-S_(1)P)`
`d sin30^(@)=d sintheta`
`theta=30^(@)`
`tan30^(@)=(y_(0))/(D) rArr y_(0)=(1)/(sqrt(3))m`
`P_(1)` and `P_(2)` to be minima
For `P_(1)` to be minima
`dsin30^(@)+S_(1)P_(1)-S_(2)P_(1)=(lambda)/(2) rArr dsin30^(@)+dsin theta=(lambda)/(2)`
`sintheta=sin30^(@)-(lambda)/(2d) rArr sintheta=(1)/(2)-(1)/(4)`
`sintheta=(1)/(4)`or `tantheta=(1)/(sqrt(15)) rArr (y_(1))/(D)=(1)/(sqrt(15)) rArr y_(1)=(1)/(sqrt(15))m`
For `P_(2)` to be minima
`S_(2)P-(dsin30^(@)+S_(1)P)=(lambda)/(2) rArr (S_(2)P-S_(1)P)-dsin 30^(@)=(lambda)/(2)`
`-dsin30^(@)+dsintheta=(lambda)/(2) rArr sin theta=(lambda)/(2d)+sin30^(@)`
`rArr sin theta=(3)/(4) rArr tantheta=(3)/(2sqrt(7)) rArr y_(2)=(3)/(sqrt(7))m`