In a modified YDSE, sources `S` is kept in front of slit `S_(1)`. Find the phase difference at point O that is equidistant from slits `S_(1)` and `S_(2)` and point P that is in front of slit `S_(1)` in the following situations.

A liquid of refractive index `mu` is filled between the screen and slits.

Options

A. `(2pi)/(lambda) = [[ sqrt(d^(2) + x_(0)^(2)) + x_(0)] + (mu d^(2))/(2D)]`
B. `(2pi)/(lambda) = [[ sqrt(d^(2) + x_(0)^(2)) - x_(0)] + (mu d^(2))/(2D)]`
C. `(2pi)/(lambda) = [[ sqrt(d^(2) - x_(0)^(2)) + x_(0)] + (mu d^(2))/(2D)]`
D. `(2pi)/(lambda) = [[ sqrt(d^(2) - x_(0)^(2)) - x_(0)] + (mu d^(2))/(2D)]`

Answer

b

Explanation

While calculating path difference, it must be remembered that is is equal to difference of optical path lengths from sources S to point on the screen.
`:. Delta x_("total") = [Delta x]_("before slits") + [Delta x]_("after slits")`
` = (SS_(2) - SS_(1)) + (S_(2) P - S_(1) P)` ltbgt a. When liquid is filled between the slits and screen, then
`[S_(2) P]_("liquid") = [mu S_(2) P]_("air")`
`[S_(1) P]_("liquid") = [mu S_(1) P]_("air")`
At point O: `mu S_(2) O = mu S_(1) O`
Noth path difference is introduced after slits. So,
`Delta x_("total") = SS_(2) - SS_(1) = sqrt(d^(2) + X_(0)^(2)) - x_(0)`
Thus, phase diffenence,
`Delta phi = (2pi)/(lambda) (sqrt(d^(2) + X_(0)^(2)) - x_(0))`
At point P: `[Delta x]_("before slits") = sqrt(d^(2) + x_(0)^(2)) - x_(0)`
`[Delta x]_("after slits") = mu S_(2) P - mu S_(1) ap = mu (S_(2)P - S_(1)P)`
`= (mu yd)/(D) = (mu d^(2))/(2 D) ("as" y = (d)/(2))`
Thus phase difference,
`Delta phi = (2pi)/(lambda) [(sqrt(d^(2) + x_(0)^(2)) - x_(0)) + (mu d^(2))/(2 D)]`
b. When liquid is filled between the source and slits:
At point O: `(Delta x)_("before slits") = (SS_(2) + SS_(1))_("liquid")`
`(mu SS_(2) + mu SS_(1))_("air")`
`= mu (sqrt(d^(2) + x_(0)^(2)) - x_(0))`
`(Delta x)_("after slits") = S_(2) O + S_(1) O = 0`
`(Delta x)_("total") = mu (sqrt(d^(2) + x_(0)^(2)) - x_(0))`
Thus, phase differnce at P,
` Delta phi = (2pi)/(lambda) (mu sqrt(d^(2) + x_(0)^(2)) - x_(0))`
At point P: `(Delta x)_("before slits") = (SS_(2) + SS_(1))_("liquid") = (mu SS_(2) + mu SS_(1))_("air")`
`(Delta x)_("after slits") = (S_(2)P - S_(1)P)_("air") = (y d)/(D) = (d^(2))/(2D)`
`(Delta x)_("total") = [(mu sqrt(d^(2) + x_(0)^(2)) - x_(0)) + (d^(2))/(2D)]`
Thus, phase difference at P,
`Delta phi = (2 pi)/(lambda) [mu sqrt(D^(2) + x_(0)^(2)) - x_(0) + (d^(2))/(2D)]`