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Obtain the formula of path difference at a point on the screen in Young's double slit experiment in term of x, d and D. |
Answer» Solution : One wavefront out of wavefronts emerging from slit S incident on slits S, and S, at the same TIME hence `S_(1)` and `S_(2)` act as coherent sources. Suppose distance between `S_(1)` and `S_(2)` is d which in the order to mm and screen GG. is placed at distance D from PERPENDICULAR bisector of `S_(1)S_(2)`. D is in the order of meter. Interference obtain at a point P on a screen. The maximum or minimum intensity of LIGHT at P depend on the path difference of `S_(1)` and `S_(2)`. Let OP=X. `S_(1)MbotGG.` and draw `S_(2)NbotGG.`. So `S_(1)M=D` and `PM=x-(d)/(2)andS_(2)N=DandPN=x+(d)/(2)` From `DeltaPMS_(1)` `S_(1)P^(2)=S_(1)M^(2)+PM^(2)` `=D^(2)+(x-(d)/(2))^(2)""......(1)` and from `DeltaPNS_(2)` `S_(2)P^(2)=S_(2)M^(2)+PN^(2)` `=D^(2)+(x+(d)/(2))^(2)""......(2)` `S_(2)P^(2)-S_(1)P^(2)=D^(2)+(x+(d)/(2))^(2)-D^(2)+(x-(d)/(2))^(2)` `:.S_(2)P^(2)-S_(1)P^(2)=(x+(d)/(2))^(2)-(x-(d)/(2))^(2)` `=x^(2)+xd+(d^(2))/(4)-x^(2)+xd-(d^(2))/(4)` `:.(S_(2)P-S_(1)P)(S_(2)P+S_(1)P)=2xd` But x and d are very small compare to D so `S_(1)P=S_(2)P=D` can be taken `:.(S_(2)P-S_(1)P)(D+D)=2xd` `:.S_(2)P-S_(1)P=(2xd)/(2D)` `:.S_(2)P-S_(1)P=(xd)/(D)` Hence, path difference of `S_(1)` and `S_(2)` from `P=(xd)/(D)` If `S_(2)P-S_(1)P` (path difference) `=nlamda` where n=0, 1, 2, ......, the intensity of light at point P is maximum (constructive interence). Hence, `nlamda=(x_(n)d)/(D)` `:.xorx_(n)=(nlamdaD)/(d)` where `n=0,pm1,pm2,......,` constructive interference obtained of `n^(th)` order. If `S_(2)P-S_(1)P` (path difference ) `=(n+(1)/(2))lamda` `:.(x_(n)d)/(D)=(n+(1)/(2))lamda` `:.x_(n)=(n+(1)/(2))(lamdaD)/(d)` where `n=0,pm,1pm2,......,` at point P then intensity of light will be minimum (zero) and destructive interference obtained. The distance of bright fringe from the mid point of central bright fringe is x but minimum of zeroth order cannot be obtained hence the formula of destructive interference is `(n-(1)/(2))lamda` where `n=pm1,pm2,......,` |
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