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.

Explanation

`lambda=0.5 mm, d=1 mm, D=1.0 m`

For minima, `Deltax=d sin theta=(2n+1)lambda/2`
`sin theta=((2n+1)lambda)/(2d)=((2n+1)(0.5)xx10^(-3))/(2xx10^(-3))=(2n+1)/4`
`n=0, sin theta=1//4`
`tan theta=1/sqrt(15)=y/D=y/1implies y=1/sqrt(15)m`
`n=1, sin theta=3//4`
`tan theta=3/sqrt(7)=y/1impliesy=3/sqrt(7)`
y-coordinates of all minima on screen: `+-1/sqrt(15)m, +-3/sqrt(7)m`
(b)
For central maximum
`d sin 30^(@)=d sin thetaimpliestheta=30^(@)`
`y_(0)/D=tantheta=tan 30^(@) implies y_(0)=1/sqrt(3)m`
For first minimum
`d sin theta-d sin 30^(@) =+-lambda/2`
`sintheta=sin30^(@)+-lambda/(2d)`
`=1/2+-0.5/(2xx1)=1/2+-1/4`
`sintheta=3/4,1/4`
`tantheta=3/sqrt(7),1/sqrt(15)`
Position of first minima on either side of maximum
`y/D=tantheta implies y=tantheta`
`y=3sqrt(7)m` and `1/sqrt(15)m`