$($b$)$ A step$-$index optical fibre with a core radius of $a=15m$ supports $50$ modes at $850$ nm $.$ The cladding refractive index of the fibre is $1\mathrm{.}5.$

$($i$)$ Find the core refractive index and the acceptance angle of the fiber if light enters from air into the fibre.

$($ii$)$ Determine the maximum data rate the fibre can support if the light at $1\mathrm{.}55m$ wavelength is launched into the fibre over a transmission distance of $50$ km $.$

$(10$ Marks$)$