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 and with a mean separation of $0.8 \mathrm{~mm}$. The distance between the slits at time $t$ is given by $d=(0.8+0.04 \sin \omega t) \mathrm{mm}$, where $\omega=0.08 \mathrm{rad} \mathrm{s}^{-1}$. The distance of the screen from the slits is $1 \mathrm{~m}$ and the wavelength of the light used to illuminate the slits is $6000$ Å . The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point $O$.

The maximum speed in $\mu \mathrm{m} / \mathrm{s}$ at which the $8^{\text {th }}$ bright fringe will move is ______.

A point source $\mathrm{S}$ emits unpolarized light uniformly in all directions. At two points $\mathrm{A}$ and $\mathrm{B}$, the ratio $r=I_A / I_B$ of the intensities of light is 2 . If a set of two polaroids having $45^{\circ}$ angle between their pass-axes is placed just before point $\mathrm{B}$, then the new value of $r$ will be __________.

A Young's double slit interference arrangement with slits S₁ and S₂ is immersed in water (refractive index = 4/3) as shown in the figure. The positions of maxima on the surface of water are given by x² = p²m²$$\lambda$$² $$-$$ d², where $$\lambda$$ is the wavelength of light in air (refractive index = 1). 2d is the separation between the slits and m is an integer. The value of p is