So the material in this assignment hasn't actually been covered in lecture yet and I'm going into this a little blind. So I'd be very appreciative if someone could sanity check my answer for the first part here as the remaining questions depend on getting this correct.

I've got a capacitor circuit with a switch in it that goes from the state in the first image to the state in the second when the switch is flipped, and it would like me to derive the equations for the voltage across and current through the capacitor after the switch is flipped. in the interest of making my life easier, i convert that to the norton equivalent

where Rth=R+Rt and It=Vt/Rth

Summing the currents at the top node gets me \(It=C(\frac{dv}{dt})+\frac{v}{Rth}\) and I can solve that to get

\(v(t)=ItRth+(v(0)-ItRth)e^{(\frac{-t}{RthC})

for t>=0

similarly I can get the current

\(i(t)=(It-\frac{v(0)}{R})e^{(\frac{-t}{RthC}) \)

for t>0

The capacitor isn't connected to a source in the initial state so v(0)=0 and i(0)=0. Subbing that back in along with It=Vt/Rth gets me

\(v(t)=Vt-Vte^{(\frac{-t}{RthC}) t>=0

i(t)=(Vt/Rth)e^{(\frac{-t}{RthC}) t>0

i(t)=0 t<=0.\)

Have I done something horribly wrong here with my derivation or am I on the right track?\)

I've got a capacitor circuit with a switch in it that goes from the state in the first image to the state in the second when the switch is flipped, and it would like me to derive the equations for the voltage across and current through the capacitor after the switch is flipped. in the interest of making my life easier, i convert that to the norton equivalent

where Rth=R+Rt and It=Vt/Rth

Summing the currents at the top node gets me \(It=C(\frac{dv}{dt})+\frac{v}{Rth}\) and I can solve that to get

\(v(t)=ItRth+(v(0)-ItRth)e^{(\frac{-t}{RthC})

for t>=0

similarly I can get the current

\(i(t)=(It-\frac{v(0)}{R})e^{(\frac{-t}{RthC}) \)

for t>0

The capacitor isn't connected to a source in the initial state so v(0)=0 and i(0)=0. Subbing that back in along with It=Vt/Rth gets me

\(v(t)=Vt-Vte^{(\frac{-t}{RthC}) t>=0

i(t)=(Vt/Rth)e^{(\frac{-t}{RthC}) t>0

i(t)=0 t<=0.\)

Have I done something horribly wrong here with my derivation or am I on the right track?\)

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