A silicon n$-$p$-$n BJT has doping concentrations of $5$x$1018$ cm$\text{'},2$x$10\text{'}7$ cm$?$ and $1016$ cm$*$ in the emitter, base and collector areas respectively. The width of its base is $10$ microns. Find emitter, collector, and base current densities of this transistor at emitter$-$base voltage $0\mathrm{.}5$ V and collector$-$emitter voltage $20$ V$.$ Calculate common$-$emitter and common$-$base current gains of this transistor.