Part C$-$Application $-$ Full Solutions are Required $(15$ marks$)$

$1.$ In a classroom demonstration, a beam of coherent light of wavelength $550$ nm is incident perpendicularly onto a pair of slits. Each slit has a width $\backslash ($ w $\backslash )$ of $\backslash (1\mathrm{.}2\backslash$times $10^\{-6\}\backslash$mathrm$\{$~m$\}\backslash ),$ and the distance $\backslash ($ d $\backslash )$ between the centers of the slits is $\backslash (1\mathrm{.}8\backslash$times $10^\{-5\}\backslash$mathrm$\{$~m$\}\backslash ).$ The class observes light and dark fringes on a screen that is a distance $\backslash ($ L $\backslash )$ of $2\mathrm{.}2$ m from the slits. Your notebook shows the following setup for the demonstration.

a$.$ Calculate the frequency of the light. $(2$ marks$)$

b$.$ Calculate the distance between two adjacent dark fringes on the screen. $(2$ marks$)$

The entire apparatus is now immersed in a transparent fluid having index of refraction $1\mathrm{.}4.$

c$.$ What is the frequency of the light in the transparent fluid? $(1$ mark$)$

d$.$ Does the distance between the dark fringes increase, decrease, or remain the same? Explain your reasoning. $(2$ marks$)$