Given a plan $p^{+}-n$ Si step junction diode with an $n-$side doping of $N_D=\frac{{10}^{15}}{c}{m}^3$ and $T=300K,$ determine

$($a$)$ The approximate $V_{BR}$ of the diode.

$($b$)$ The depletion width at the breakdown voltage.

$($c$)$ The maximum magnitude of the electric field in the depletion region at the breakdown voltage.