Problem $1(35$ points$)-$ For the circuit shown, the switch has been open a long time and is closed at time $\backslash (\backslash$mathrm$\{$t$\}-0\backslash ).$

a$)$ Find the voltage across the capacitor just before the switch is closed.

b$)$ Find the final voltage across the capacitor.

c$)$ Find the time constant, $\backslash (\backslash$tau $\backslash )$

d$)$ Find the voltage across the capacitor, $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$c$\}\}(\backslash$mathrm$\{$t$\})\backslash )$ for all $\backslash (\backslash$mathrm$\{$t$\}>0\backslash ).$

$($e$)$ What is the energy stored in the energy stored in the capacitor as $\backslash (\backslash$mathrm$\{$t$\}\backslash$rightarrow $\backslash$infty $\backslash )?$

Problem $2(35$ points$)-$ For the circuit shown, the switch has been open a long time and is closed at $\backslash (\backslash$mathrm$\{$t$\}=0\backslash ).$

a$)$ Find the initial current in the inductor, $\backslash (\backslash$mathrm$\{$i$\}\_\{\backslash$mathrm$\{$L$\}\}(0+\backslash )).$

b$)$ Find the final current in the inductor, $\backslash (\backslash$mathrm$\{$i$\}\_\{\backslash$mathrm$\{$L$\}\}(\backslash$infty$)\backslash ).$

c$)$ Find the time constant, $\backslash (\backslash$tau $\backslash ).$

d$)$ Find $\backslash ($ i$\_\{$L$\}($t$)\backslash )$ for all $\backslash ($ t$>0\backslash ).$

e$)$ Find the voltage across R$2$ just after the switch is closed.

Problem $3(30$ points$)-$ For the circuit shown, the capacitor is charged to $10$ V before the switch is closed at $\backslash (\backslash$mathrm$\{$t$\}=0\backslash ).$

a$)$ Find the natural frequency, $\backslash (\backslash$omega$\_\{$n$\}\backslash ).$

b$)$ Find the damping ratio, $\backslash ($ s $\backslash ).$

c$)$ Is this circuit overdamped, underdamped or critically damped?

d$)$ Sketch the voltage across the capacitor as a function of time.

e$)$ If R$1$ is increased $($with no changes to L or C $),$ what is the effect on $\backslash (\backslash$varsigma $\backslash )?$