A silicon $pn$ junction diode is characterized by a junction capacitance $C_j$ defined as

$C_j=\frac{K_{}{}_{0}A}{W}$

where $K_s=11\mathrm{.}8$ for silicon, ${}_0$ is the vacuum permittivity, $A=$ the cross$-$sectional area of the junction, and $W$ is known as the depletion width of the junction. Width $W$ depends not only on how the diode is fabricated, but also on the voltage applied to its two terminals. It can be computed using

$W=\sqrt[\phantom{2}]{\frac{2K_s{}_0}{qN}(V_{bb}-V_A)}$

Thus, diodes are often used in electronic circuits since they can be viewed in this context as voltage$-$controlled capacitors. Assuming parameter values of $N=5\mathrm{.}0{10}^{18}c{m}^{-3},V_{bi}=0\mathrm{.}62V,$ and using $q=1\mathrm{.}6{10}^{-19}C,$ calculate the capacitance of a diode with cross$-$sectional area $A=2m^2$ at applied voltages of $V_A=-1,-3,$ and $-10$ V $.$