If a parallel RLC circuit has a $3$k Ohm resistor and a $2$ H inductor, what are the possible values of

the capacitor for $(1)$ overdamped; $(2)$ critically damped, and $(3)$ underdamped circuit

behaviors?

$($b$)$ What if that was a series circuit instead?

$($c$)$ Consider the critically damped circuit in series, and add a voltage supply of $10$V aligned with the

capacitor so that the plus of both are on the same side $($i$.$e$.$ short those two terminals$).$ If that

voltage supply turns on at t$=0+,$ what is the ic$($t$)$ for t$>0$ if the dvc$(0)/$dt$=100$V$/$s and vc$(0)=0?$

$2.$ In the circuit below, if the capacitor value is $250$ mF and its polarity follows the same as V$1$ and V$2$

$($that is$,$ the positive sign is up$,$ the negative is down$),$ what is vc$($t$)$ for t$>0$ if V$1=20$V and V$2=10$V$?$

Assume iL$(0)=10$mA and diL$(0)/$dt $=-50$ mA$/$s$.$