Capacitor 充电放电公式

充电

设电容的电荷量为 $Q$，电流为 $I$，电容为 $C$，电阻为 $R$，初始电荷为 $Q_0$
根据基尔霍夫电压定律:

$$\begin{array}{rl}\frac{Q}{C}+IR& =0\\ I& =-\frac{dQ}{dt}\\ \frac{Q}{C}-R\frac{dQ}{dt}& =0\\ R\frac{dQ}{dt}& =\frac{Q}{C}\\ \frac{dQ}{dt}& =\frac{Q}{RC}\\ \frac{dQ}{Q}& =-\frac{dt}{RC}\\ \int \frac{dQ}{Q}& =-\int \frac{dt}{RC}\\ \ln |Q|& =-\frac{t}{RC}+C_1\\ Q& =e^{C_1}{e}^{-t/RC}\\ Q& =Q_0{e}^{-t/RC}\\ I& =-\frac{d}{dt}\left(Q_0{e}^{-t/RC}\right)\\ I& =Q_0\frac{1}{RC}{e}^{-t/RC}\\ I& =\frac{Q_0}{R}{e}^{-t/RC}\\ V& =\frac{Q_0}{C}{e}^{-t/RC}\end{array}$$ $$\boxed{Q=Q_0{e}^{-t/RC}}\boxed{I=\frac{Q_0}{R}{e}^{-t/RC}}\boxed{V=\frac{Q_0}{C}{e}^{-t/RC}}$$