A monochromatic light of `lambda=500 nm` is incident on two identical slits separated by a distance of `5xx10^(-4)m`. The interference pattern is seen on a screen placed at a distance of `1 m` from the plane of slits. A thin glass plate of thickness `1.5xx10^(-6)m` and refractive index `mu=1.5` is placed between one of the slits and the screen. Find the intensity at the center of the screen if the intensity is `I_(0)` in the absence of the plate. Also find the lateral shift of the central maxima and number of fringes crossed through center.

Explanation

In case of interference
`I_(R) = I_(1) + I_(2) + 2 (sqrt (I_(1) I_(2))) cos phi`
Now, as for identical slits `I_(1) = I_(2) = I,` so
`I_(R) = 2 (1 + cos phi) = 4 I cos^(2) (phi // 2)`
But for central maxima,
`phi = 0^(@),` and here `I_(R) = I_(0)` (given)
`I_(0) = 4 I cos (0^(@)) = 4 I`
Hence, `I_(R) = I_(0) cos^(2) ((phi)/(2))`
Now, when the glass plate is introduced, path difference between the waves at the position of central maxima will become
`Delta x = ( mu - 1) t = (1.5 - 1) 1.5 xx 10^(-6) = 7.5 xx 10^(-7) m`
`:. phi = (2 pi)/(lambda) (Delta x) = (2 pi)/(5 xx 10^(-7)) xx 7.5 xx 10^(-7) = 3 pi`
So, intensity at central maxima will now be
`I_(R) = I _(0) cos^(2) ((3 pi)/(2)) = I_(0) xx 0 = 0`
Also, from theory of intergerence, fringe shift
`y_(0) = (D)/(d) (mu - 1) t`
Which in the light of Eq. (ii) and given data becomes
`y_(0) = 7.5 xx 10^(-7) = 1.5 xx 10^(-3) m = 1.5 mm`.