Gen Phys 2

DC circuits

Resistors

\begin{equation} R_{eq} = R_1 + R_2 + R_3 \, + \, ... \qquad \textnormal{(Resistors in series)} \end{equation} \begin{equation} \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \, + \, ... \qquad \textnormal{(Resistors in parallel)} \end{equation}

Kirchoff's rule

\begin{equation} \sum I = 0 \qquad \textnormal{(Junction rule)} \end{equation} \begin{equation} \sum V = 0 \qquad \textnormal{(Loop rule)} \end{equation}

Circuits and EMF

The emf is given by \( \xi \). \begin{equation} V_{ab} = \xi - Ir \qquad \textnormal{(Source with internal resistance)} \end{equation}

Energy and power in circuits

\begin{equation} P = V_{ab} I \qquad \textnormal{(General circuit element)} \end{equation} \begin{equation} P = V_{ab} I = I^2 R = \frac{V_{ab}^2}{R} \qquad \textnormal{(Power into a resistor)} \end{equation}

R-C circuits

Capacitor charging, \( \xi \) = emf. \begin{equation} q = C \, \xi (1 - e^{-t/RC}) = Q_f (1 - e^{-t/RC}) \qquad \textnormal{(Charge as function of time)} \end{equation} \begin{equation} i = \frac{dq}{dt} = \frac{\xi}{R} e^{-t/RC} = I_{0} e^{-t/RC} \qquad \textnormal{(Current as function of time)} \end{equation} Capacitor discharging \begin{equation} q = Q_0 \, e^{-t/RC} \qquad \textnormal{(Charge as function of time)} \end{equation} \begin{equation} i = \frac{dq}{dt} = - \frac{Q_0}{RC} e^{-t/RC} = I_{0} e^{-t/RC} \qquad \textnormal{(Current as function of time)} \end{equation}