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, B, C

Explanation

(a) The path difference `(trianglex)` from the ray starting from `S_(1)` and `S_(2)` and reaching a point P will be
`trianglex=dsintheta`
We know that the path difference for minimum intensity is
`(2m-`)(lamda)/(2)` where `m=1,2,3`…
`:'` `dsintheta=(2m-1)(lamda)/(2)`
`impliessintheta=((2m-1)lamda)/(2xx1.0)=((2m-1)0.5)/(2xx1.0)=(2m-1)/(4)`
Also `-1lesinthetale1`. Therefore, possible values of m are `+-1`,`+-2`,0
From `trianglePOQ`
+-`y=Dtantheta=(Dsintheta)/(sqrt(1-sin^(2)theta))` ...(i)
Positions of minima
For `m=+1`,`sintheta(1)/(4)` and `y=0.26`
`m=-1`, `sintheta=-(3)/(4)` and `y=-1.13`m
`m=+2`, `sintheta=(3)/(4)` `:'` `y=+1.13`m
`m=0`, `sintheta=-(1)/(4)` `:'` `y=-0.26m` (b) WHEN THE INCIDENT BEAM MAKES AN ANGLE OF `30^(CIRC)` WITH X-AXIS
Two cases arise as shown by the following two fringes.
Path difference between ray 1 and 2 reaching `P=S_(2)M-NS_(1)`
`:'``trianglex_(1)=dsintheta-dsinalpha`... (Case 1)
Path difference between ray 1 and 2 reaching `P=NS_(1)+S_(1)M`
`trianglex_(2)=disinalpha+dsintheta`....(Case2)
Position of Central maxima: Path difference should be zero. Therefore `trianglex_(1)=0` or `trianglex_(2)=0`
`impliesdsinalpha=dsintheta`
`impliessintheta=(1)/(2)` [`:'`a=30^circ]
From equation (i), `y=0.58m`
for first minima, `dsintheta+dsinalpha=(lamda)/(2)`
`impliesdsintheta=(lamda)/(2)+dsinalpha`
`:'``sintheta=(lamda)/(2d)+sinalpha=(0.5)/(2xx1)+sin30^circ=(1)/(4)+(1)/(2)=(3)/(4)`
From equation (i), `y=1.15m`
For first minima on the other side
N119`dsinalpha+dsintheta=(lamda)/(2)impliessintheta=(-1)/(4)`
`:'` From (i), `y=-0.26`m