Assume R$1\backslash (=50\backslash )$ kohms, $\backslash (\backslash$mathrm$\{$R$\}2=22\backslash )$ kohms, and V$1\backslash (=10\backslash )$ Volts. Initial voltage on C $1$ is $2$ volts.

$1.$ Determine the Charge equations for the circuit above assuming $\backslash (\backslash$mathrm$\{$Cl$\}=10\backslash$mathrm$\{$uF$\}\backslash ).$ Create a table to show the voltage and current of the capacitor for $\backslash (0,1,3\backslash ),$ and $5$ time constants. Create a graph of voltage and current vs$.$ time in Excel.

$2.$ For the network below:

a$.$ Determine the mathematical expressions for the current $\backslash ($ i$\_\{\backslash$mathrm$\{$L$\}\}\backslash )$ and the voltage $\backslash ($ v$\_\{\backslash$mathrm$\{$L$\}\}\backslash )$ when the switch is closed.

b$.$ Repeat part $($a$)$ if the switch is opened after a period of five time constants has passed.

c$.$ Sketch the waveforms of parts $($a$)$ and $($b$)$ on the same set of axes with continuous time.