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

## Chapter 10

#### Direct-Current Circuits

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

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

#### For quiz at QB/d_cp2.10

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

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

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

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

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