$12\mathrm{.}15$ For an $Al-Si{O}_2-$Si capacitor, the doping of the substrate is $N_A={10}^{14}c{m}^{-3}$ and the oxide thickness is $80$ Angstroms. Given that the flatband voltage, $V_{FB},$ is $-0\mathrm{.}4$ V $,$ calculate at strong inversion

$($a$)$ the surface potential.

$($b$)$ the maximum depletion layer width.

$($c$)$ the gate voltage.