Question

I need to derive the loop equation for an RC Circuit Q/C-IR=0 to find the equation V=V0e^(-t/RC) can someone walk me step by step through it.

Answer 1

For current leaving the capacitor you have
\[\huge V=IR=\frac{Q}{C} = -\frac{dq}{dt}R\] separate variables AND take reciprocals
\[\huge \frac{dq}{Q}=-\frac{dt}{RC}\] Now integrate
\[\huge \ln|Q|+C = -\frac{t}{RC}\]
\[\huge \ln|Q| = -\frac{t}{RC}+C\] raise both sides to e
\[\huge Q=Q_{0}e^{-\frac{t}{RC}}\]

Divide both sides by C to get;
\[\huge \frac{Q}{C}= \frac{Q_0}{C}e^{-\frac{t}{RC}}\]
Now remember that \[\huge V= \frac{Q}{C}, V_0 = \frac{Q_0}{C}\] so we have

\[\huge V = V_0e^{-\frac{t}{RC}}\]