Gen Phys 2

Electrostatic energy and capacitors

Capacitors and capacitance

\begin{equation} C = \frac{Q}{V_{ab}} \end{equation} \begin{equation} C = \frac{Q}{V_{ab}} = \epsilon_0 \frac{A}{d} \end{equation}

Capacitors in series

\begin{equation} \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \, ... \end{equation}

Capacitors in parallel

\begin{equation} C_{eq} = C_1 + C_2 + C_3 \, + ... \end{equation}

Energy in a capacitor

\begin{equation} U = \frac{Q^2}{C} = \frac{1}{2} \, C \, V^2 = \frac{1}{2} \, Q \, V \end{equation} \begin{equation} u = \frac{1}{2} \epsilon_0 \, E^2 \end{equation}

Dielectrics

\begin{equation} C = K \, C_0 = K \epsilon_0 \frac{A}{d} = \epsilon \frac{A}{d} \end{equation} \begin{equation} u = \frac{1}{2} K \epsilon_0 \, E^2 = \frac{1}{2} \epsilon \, E^2 \end{equation} \begin{equation} \oint_S K \vec{E} \cdot d\vec{A} = \frac{Q_{enc \, free}}{\epsilon_0} \end{equation}