コンデンサの直列接続と静電容量

コンデンサの定義式

\begin{align}
v_c=\frac{1}{C} \int i dt
\end{align}

より

\begin{align}
E & =v_{c1}+v_{c2} \cdots+v_{cn} =\frac{1}{C_{1}} \int i dt + \frac{1}{C_{2}} \int i dt+\cdots +\frac{1}{C_{n}} \int i dt \\
&= \left ( \frac{1}{C_{1}} + \frac{1}{C_{2}}+ \cdots +\frac{1}{C_{n}}\right ) \int i dt \\
\end{align}

\(Q= \displaystyle \int i dt\)より

\begin{align}
Q= \frac{1}{\dfrac{1}{C_{1}} + \dfrac{1}{C_{2}}+ \cdots +\dfrac{1}{C_{n}}} E
\end{align}

特に２つの場合は

\begin{align}
Q & = \frac{C_{1} C_{2}}{C_{1} + C_{2}} E \\
\end{align}