figure shows three equidistant slits illuminated by a monochromatic parallel beam of light. Let `BP_0 -AP_0 = lambda/3` and `Dgtgtlambda`.

(a) Show that d = `sqrt ((2lambdaD)//3)`
(b) Show that the intensity at `P_0` is three times the intensity due to any of the three slits
individually.

Answer

A

Explanation

a. `BP-AB=1/3`
`rarr sqrt(D^2+d^2)-D=lamda/3`
`rarr D^2+d^2=D^2+(lamda^2/9)+(2lamdaD)/3`
`rarr d=theta(2lamdD)/3`
(neglecting the term `lamda^2/9` as it is very small)
b. To fing the inensity at P we have to consider the inteference of light waves coming from all the three slits.
`Here CP-AP-sqrt(D^2+4d^2)-D`
`=sqrt(D^2+8lamdaD)-D`
`=D{1+(8lamda)/(3lamda)}^(1/2)-D`
`=D{1+(8lamda)/(3D)xx2+..}-D`
`=41/3`
`[using binomial expansion]
`So, the corresponding phase difference between waves from C and A is
`phi_c=(2pix)/lamda=(2pix4lamda)/(3lamda)`
`=(8pi)/3=(2pi+(2pi)/3)`
`=(2pi)/3` ..............i
`Again theta_B=(2pilamda)/(3lamda)=(2pi)/3`.......i
so it can be said that light from B and C are i same phase as they have same phase difference with respect to A.
So, `R=sqrt((2)^2)+r^2+2xx2rxxrcos((2pi)/3)`
[using vector method]
`=sqrt(4r^2)+r^2+2r^2.sqrt3r`
`:. l_(p0)=K (sqrt3r^2)=3Kr^2=3l`
As the resulting amplitude is m3 times the intensity due to individual slits.