$8\mathrm{.}2$ A silicon pn junction has impurity doping concentrations of Nd $21015$ cm$3$ and Na $81015$ cm$3.$ Determine the minority carrier concentrations at the edges of the space charge region for $($a$)$ Va $0\mathrm{.}45$ V$,($b$)$ Va $0\mathrm{.}55$ V$,$ and $($c$)$ Va $0\mathrm{.}55$ V$.$