Problem $2-$ Michelson interferometer $(5$ points$)$

A Michelson interferometer is used to measure the refractive index of a gas. The gas is allowed

to flow into an evacuated glass cell of length L placed in one arm of the interferometer. The

wavelength is $.$

$($a$)$ If N fringes are counted $($moving past the middle of the interference pattern$)$ as the

pressure in the cell changes from vacuum to atmospheric pressure, what is the index

of refraction n in terms of N$,$ $,$ and L$?(3$ points$)$

$($b$)$ How many fringes would be counted if the gas were carbon dioxide $($n $=1\mathrm{.}00045)$ for

a $10\mathrm{.}0$ cm cell length, using sodium light at $589$ nm$?(2$ points$)$