In a double slit experiment, instead of taking slits of equal widths, one slit is made twice as wide as the other. Then in the interference pattern

$A$ parallel beam of light of wavelength $500 \ nm$ is incident at an angle $30^o$ with the normal to the slit plane in a Young's double-slit experiment. The intensity due to each slit is $I_o$. Point $O$ is equidistant from $S_1$ and $S_2$. The distance between the slits is $1 \ mm$.

As shown in the figure,in Young's double slit experiment,a thin plate of thickness $t = 10\,\mu m$ and refractive index $\mu_1 = 1.2$ is inserted in front of slit $S_1$. The experiment is conducted in air $(\mu = 1)$ and uses a monochromatic light of wavelength $\lambda = 500\,nm$. Due to the insertion of the plate,the central maxima is shifted by a distance of $x\beta_0$,where $\beta_0$ is the fringe-width before the insertion of the plate. The value of $x$ is $.............$

If a transparent medium of refractive index $\mu = 1.5$ and thickness $t = 2.5 \times 10^{-5} \, m$ is inserted in front of one of the slits of Young's Double Slit experiment,how much will be the shift in the interference pattern? The distance between the slits is $0.5 \, mm$ and that between slits and screen is $100 \, cm$. (Answer in $cm$)

In Young's double-slit experiment,a glass plate of refractive index $1.5$ and thickness $5 \times 10^{-4} \,cm$ is kept in the path of one of the light rays. Then

In a Young's double slit experiment,each of the two slits $A$ and $B$,as shown in the figure,are oscillating about their fixed center with a mean separation of $0.8 \ mm$. The distance between the slits at time $t$ is given by $d = (0.8 + 0.04 \sin \omega t) \ mm$,where $\omega = 0.08 \ rad \ s^{-1}$. The distance of the screen from the slits is $1 \ m$ and the wavelength of the light used to illuminate the slits is $6000 \ \mathring A$. The interference pattern on the screen changes with time,while the central bright fringe (zeroth fringe) remains fixed at point $O$.
$(1)$ The $8^{\text{th}}$ bright fringe above the point $O$ oscillates with time between two extreme positions. The separation between these two extreme positions,in micrometer $(\mu m)$,is. . . . .
$(2)$ The maximum speed in $\mu m/s$ at which the $8^{\text{th}}$ bright fringe will move is. . . . .