Question

Two coherent point sources `S_(1)` and `S_(2)` vibrating in phase emit light of wavelength `lambda`. The separation between the sources is `2lambda`. Consider a line passing through `S_(2)` and perpendicular to line `S_(1) S_(2)`. Find the position of farthest and nearest minima.
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Answer 1

At `P`: path difference
`Deltax=S_(1)P-S_(2)P=sqrt(x^(2)+(2lambda)^(2))-x=sqrt(x^(2)+4lambda^(2))-x`
For minima
`Deltax=(2n+1)lambda/2`
`sqrt(x^(2)+4lambda^(2))-x=(2n-1)lambda/2`
`n=0, sqrt(x^(2)+4lambda^(2))-x=lambda/2`
`sqrt(x^(2)+4lambda^(2))=(x+lambda/2)^(2)`
`x^(2)+4lambda^(2)=x^(2)+lambdax+lambda^(2)/4implies x=(15 lambda)/4`
`n=1, sqrt(x^(2)+4lambda^(2))-x=(3lambda)/2`
`sqrt(x^(2)+4lambda^(2))=(x+(3lambda)/2)`
`x^(2)+4lambda^(2)=x^(2)+3lambdax+(9lambda^(2))/4 impliesx=(7lambda)/12`
`n=2, sqrt(x^(2)+4lambda^(2))-x=(5lambda)/2`
`sqrt(x^(2)+4lambda^(2))=(x+(5lambda)/2)^(2)`
`x^(2)+4 lambda^(2)=x^(2)+5lambdax+(25lambda^(2))/4 implies x=-ve`
`x_(min)=(7lambda)/12`