OpenStax University Physics/E&M/Direct-Current Circuits

Chapter 10

Direct-Current Circuits

Terminal voltage ${\displaystyle V_{terminal}=\varepsilon -Ir_{eq}}$ where ${\displaystyle r_{eq}}$ is the internal resistance and ${\displaystyle \varepsilon }$ is the electromotive force.
▭ Resistors in series and parallel: ${\displaystyle R_{series}=\sum _{i=1}^{N}R_{i}}$ ▭ ${\displaystyle R_{parallel}^{-1}=\sum _{i=1}^{N}R_{i}^{-1}}$
▭ Kirchoff's rules. Loop:${\displaystyle \sum I_{in}=\sum I_{out}}$ Junction: ${\displaystyle \sum V=0}$

▭ ${\displaystyle V_{terminal}^{series}=\sum _{i=1}^{N}\varepsilon _{i}-I\sum _{i=1}^{N}r_{i}}$ ▭ ${\displaystyle V_{terminal}^{parallel}=\varepsilon -I\sum _{i=1}^{N}\left({\frac {1}{r_{i}}}\right)^{-1}}$ where ${\displaystyle r_{i}}$ is internal resistance of each voltage source.
▭ Charging an RC (resistor-capacitor) circuit: ${\displaystyle q(t)=Q\left(1-e^{-t/\tau }\right)}$ and ${\displaystyle I=I_{0}e^{-t/\tau }}$ where ${\displaystyle \tau =RC}$ is RC time, ${\displaystyle Q=\varepsilon C}$ and ${\displaystyle I_{0}=\varepsilon /R}$.
▭ Discharging an RC circuit: ${\displaystyle q(t)=Qe^{-t/\tau }}$ and ${\displaystyle I(t)=-{\tfrac {Q}{RC}}e^{-t/\tau }}$

For quiz at QB/d_cp2.10

${\displaystyle V_{terminal}=\varepsilon -Ir_{eq}}$ where ${\displaystyle r_{eq}}$=internal resistance and ${\displaystyle \varepsilon }$=emf.

${\displaystyle R_{series}=\sum _{i=1}^{N}R_{i}}$ and ${\displaystyle R_{parallel}^{-1}=\sum _{i=1}^{N}R_{i}^{-1}}$

Kirchhoff Junction:${\displaystyle \sum I_{in}=\sum I_{out}}$ and Loop: ${\displaystyle \sum V=0}$

Charging an RC (resistor-capacitor) circuit: ${\displaystyle q(t)=Q\left(1-e^{t/\tau }\right)}$ and ${\displaystyle I=I_{0}e^{-t/\tau }}$ where ${\displaystyle \tau =RC}$ is RC time, ${\displaystyle Q=\varepsilon C}$ and ${\displaystyle I_{0}=\varepsilon /R}$.

Discharging an RC circuit: ${\displaystyle q(t)=Qe^{-t/\tau }}$ and ${\displaystyle I(t)=-{\tfrac {Q}{RC}}e^{-t/\tau }}$