Two parallel narrow horizontal slits in an opaque vertical screen are separated. center$-$to$-$center by $2\mathrm{.}644$ mm$.$ These directly illuminated by yellow plane waves from a filtered discharge lamp. Horizontal fringes are formed on a vertical D viewing screen $4\mathrm{.}500$ m from the aperture plane. The center of the fifth bright band is $5\mathrm{.}000$ mm above the center of the zeroth or central bright band. $\backslash 3+2]$

$($a$)$ Determine the wavelength of the light in air.

$($b$)$ If the entire space is filed with clear soybean oil $($n $=1\mathrm{.}4729)$ where would the fifth fringe now appear?