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.

Explanation

(a) `BP_(0)-AP_(0)=lambda/3`
`(D^(2)+d^(2))^(1/2)-D=D(1+d^(2)/D^(2))^(1//2)-D`
`=D(1+d^(2)/D^(2))-D=d^(2)/(2D)=lambda/3`
`d=sqrt((2Dlambda)/3)`
(b) `CP_(0)-AP_(0)=[D^(2)+(2d)^(2)]^(1//2)-D=D(1+(4d^(2))/D^(2))^(1/2-D`
`=D(1+(4d^(2))/(2D^(2)))-D`
`=(2d^(2))/D=2/D.(2Dlambda)/3=(4lambda)/3`
Phase difference between waves from `B` and `A` reaching at `P`
`=(2pi)/lambda(BP_(0)-AP_(0))=(2pi)/lambda.lambda/3=(2pi)/3`
Phase difference between waves from `C` and `A` reaching at `P`
`=(2pi)/lambda(CP_(0)-AP_(0))=(2pi)/lambda.(4lambda)/3=(8pi)/3=2pi+(2pi)/3`
`A: y_(1)=a sin omegat`
`B: y_(2)=a sin (omegat+2pi//3)`
`C: y_(3)=a sin (omegat+8pi//3)=a sin (2pi+omegat+2pi//3)`
`=a sin (omegat+2pi//3)`
Resultant wave:
`y=y_(1)+y_(2)+y_(3)`
`=a sin omegat+2a sin (omegat+(2pi)/3)`
`=a sin omegat+2a[sin omegai cos((2pi)/3)+ cos omegat sin ((2pi)/3)]`
`=2a sin (2pi//3) cos omegat`
`=2a.sqrt(3)/2 cos omegat`
`y=sqrt(3) a cos omegat`
`I prop a^(2), I_(R) prop (sqrt(3)a)^(2)`
`I_(R)/I=3`
`I_(R)=3I`