The figure shows the interference pattern obtained in a double$-$slit experiment using light with a wavelength of $450$ nm$.$ The slits are $2\mathrm{.}0$ um apart with a $12\mathrm{.}5$ m wide viewing screen that is $3\mathrm{.}0$ m away from the slits to produce $9$ bands on the viewing screen as shown below.

a$.$ Which fringe results from a path length difference of $2$$?$ Is it fringe A$,$ B$,$ C$,$ D or E$?$

b$.$ From center$-$to$-$center of the bands, what is the distance from band E to band B$?$

c$.$ How many bands would there be if we shined light with a wavelength of $800$ nm instead?

d$.$How would the picture above change if we replaced the double slit with a diffraction grating with $500$ lines per mm$?$