# Physics Formulae/Electric Circuits Formulae

Lead Article: Tables of Physics Formulae
This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Electric Circuits, Electronics.

## DC Quantities

| Quantity (Common Name/s) | (Common) Symbol/s | Defining Equation | SI Units | Dimension |
| Electrical Resistance | $R\,\!$ | $R=V/I\,\!$ | Ω = V A$^{-1}$ = J s C$^{-2}$ | [M][L]$^2$ [T]$^{-3}$ [I]$^{-2}$ |
| Resistivity, Scalar | $\rho \,\!$ | $\rho ={\frac {RA}{l}}\,\!$ | Ω m | [M]$^2$ [L]$^2$ [T]$^{-3}$ [I]$^{-2}$ |
| Resistivity Temperature Coefficient, Linear Temperature Dependance | $\alpha \,\!$ | $\rho -\rho _{0}=\rho _{0}\alpha (T-T_{0})\,\!$ | K$^{-1}$ | [Θ]$^{-1}$ |
| Terminal Voltage for Power Supply | $V_{\mathrm {ter} }\,\!$ | | V = J C$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ [I]$^{-1}$ |
| Load Voltage for Circuit | $V_{\mathrm {load} }\,\!$ | | V = J C$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ [I]$^{-1}$ |
| Internal Resistance of Power Supply | $R_{\mathrm {int} }\,\!$ | $R_{\mathrm {int} }={\frac {V_{\mathrm {ter} }}{I}}\,\!$ | Ω = V A$^{-1}$ = J s C$^{-2}$ | [M][L]$^2$ [T]$^{-3}$ [I]$^{-2}$ |
| Load Resistance of Circuit | $R_{\mathrm {ext} }\,\!$ | $R_{\mathrm {ext} }={\frac {V_{\mathrm {load} }}{I}}\,\!$ | Ω = V A$^{-1}$ = J s C$^{-2}$ | [M][L]$^2$ [T]$^{-3}$ [I]$^{-2}$ |
| Electromotive Force (emf), Voltage across entire circuit including power supply, external components and conductors | ${\mathcal {E}}\,\!$ | ${\mathcal {E}}=V_{\mathrm {ter} }+V_{\mathrm {load} }\,\!$ | V = J C$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ [I]$^{-1}$ |
| Electrical Conductance | $G\,\!$ | $G=1/R\,\!$ | S = Ω$^{-1}$ | [T]$^3$ [I]$^2$ [M]$^{-1}$ [L]$^{-2}$ |
| Electrical Conductivity, Scalar | $\sigma \,\!$ | $\sigma =1/\rho \,\!$ | Ω$^{-1}$ m$^{-1}$ | [I]$^2$ [T]$^3$ [M]$^{-2}$ [L]$^{-2}$ |
| Electrical Conductivity, Tensor | ${\boldsymbol {\sigma }},\sigma _{\mathrm {ij} }\,\!$ | $\sigma _{\mathrm {ij} }{\begin{pmatrix}\ \sigma _{11}&\sigma _{12}&\sigma _{13}\\\ \sigma _{21}&\sigma _{22}&\sigma _{23}\\\ \sigma _{31}&\sigma _{32}&\sigma _{33}\end{pmatrix}}\,\!$ | Ω$^{-1}$ m$^{-1}$ | [I]$^2$ [T]$^3$ [M]$^{-2}$ [L]$^{-2}$ |
| Electrical Power | $P\,\!$ | $P=VI\,\!$ | W = J s$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ |
| emf Power | $P\,\!$ | $P_{\mathrm {emf} }=I{\mathcal {E}}\,\!$ | W = J s$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ |
| Resistor Power Dissipation | $P\,\!$ | $P=I^{2}R=V^{2}/R\,\!$ | W = J s$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ |
| Resistors in Series | $R_{\mathrm {net} }=\sum _{i}R_{i}\,\!$ |
| Resistors in Parallel | ${\frac {1}{R_{\mathrm {net} }}}=\sum _{i}{\frac {1}{R_{i}}}\,\!$ |
| Ohm's Law | Scalar form $V=IR\,\!$ Vector Form $\mathbf {J} =\sigma \mathbf {E} \,\!$ Tensor Form, general applies to all points in a conductor $\mathbf {J} _{i}=\sigma _{ij}\mathbf {E} _{j}\,\!$ |
| Kirchoff's Laws | emf loop rule around any closed circuit $\sum _{i}V_{i}=\sum _{i}I_{i}R_{i}=0\,\!$ Current law at junctions $I_{\mathrm {in} }=I_{\mathrm {out} }\,\!$ |

## AC Quantitites

| Quantity (Common Name/s) | Common Name/s | Quantity (Common Name/s) | Quantity (Common Name/s) | Quantity (Common Name/s) |
| Resistive Load Voltage | $V_{R}\,\!$ | $V_{R}=I_{R}R\,\!$ | V = J C$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ [I]$^{-1}$ |
| Capacitive Load Voltage | $V_{C}\,\!$ | $V_{C}=I_{C}X_{C}\,\!$ | V = J C$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ [I]$^{-1}$ |
| Inductive Load Voltage | $V_{L}\,\!$ | $V_{L}=I_{L}X_{L}\,\!$ | V = J C$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ [I]$^{-1}$ |
| Capacitive Reactance | $X_{C}\,\!$ | $X_{C}={\frac {1}{\omega _{\mathrm {d} }C}}\,\!$ | Ω$^{-1}$ m$^{-1}$ | [I]$^2$ [T]$^3$ [M]$^{-2}$ [L]$^{-2}$ |
| Inductive Reactance | $X_{L}\,\!$ | $X_{L}=\omega _{d}L\,\!$ | Ω$^{-1}$ m$^{-1}$ | [I]$^2$ [T]$^3$ [M]$^{-2}$ [L]$^{-2}$ |
| AC Impedance | $Z\,\!$ | $V=IZ\,\!$ $Z={\sqrt {R^{2}-\left(X_{L}-X_{C}\right)^{2}}}\,\!$ | Ω$^{-1}$ m$^{-1}$ | [I]$^2$ [T]$^3$ [M]$^{-2}$ [L]$^{-2}$ |
| Phase Constant | $\phi \,\!$ | $\tan \phi ={\frac {X_{L}-X_{C}}{R}}\,\!$ | dimensionless | dimensionless |
| AC Circuit Resonant Pulsatance | $\omega _{\mathrm {res} }\,\!$ | $\omega _{\mathrm {d} }=\omega _{\mathrm {res} }=\omega ={\frac {1}{\sqrt {LC}}}\,\!$ | s$^{-1}$ | [T]$^{-1}$ |
| AC Peak Current | $I_{0}\,\!$ | $I_{0}=I_{\mathrm {rms} }{\sqrt {2}}\,\!$ | A | [I] |
| AC Root Mean Square Current | $I_{\mathrm {rms} },{\sqrt {\langle I\rangle }}\,\!$ | $I_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[I\left(t\right)\right]^{2}\mathrm {d} t}}\,\!$ | A | [I] |
| AC Peak Voltage | $V_{0}\,\!$ | $V_{0}=V_{\mathrm {rms} }{\sqrt {2}}\,\!$ | V = J C$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ [I]$^{-1}$ |
| AC Root Mean Square Voltage | $V_{\mathrm {rms} },{\sqrt {\langle V\rangle }}\,\!$ | $V_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[V\left(t\right)\right]^{2}\mathrm {d} t}}\,\!$ | V = J C$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ [I]$^{-1}$ |
| AC emf, Root Mean Square | ${\mathcal {E}}_{\mathrm {rms} },{\sqrt {\langle {\mathcal {E}}\rangle }}\,\!$ | ${\mathcal {E}}_{\mathrm {rms} }={\mathcal {E}}_{\mathrm {m} }/{\sqrt {2}}\,\!$ | V = J C$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ [I]$^{-1}$ |
| AC Average Power | $\langle P\rangle \,\!$ | $\langle P\rangle ={\mathcal {E}}I_{\mathrm {rms} }\cos \phi \,\!$ | W = J s$^{-1}$ | [M] [L]$^2$ [T]$^{-3}$ |
| Capacitive Time Constant | $\tau _{C}\,\!$ | $\tau _{C}=RC\,\!$ | s | [T] |
| Inductive Time Constant | $\tau _{L}\,\!$ | $\tau _{L}=L/R\,\!$ | s | [T] |
| RC Circuits | RC Circuit Equation $Rq'+C^{-1}q={\mathcal {E}}\,\!$ RC Circuit Capacitor Charging $q=C{\mathcal {E}}(1-e^{-t/RC})\,\!$ |
| RL Circuits | RL Circuit Equation $Li''+Ri'={\mathcal {E}}\,\!$ RL Circuit Current Rise $I={\frac {\mathcal {E}}{R}}\left(1-e^{-t/\tau _{L}}\right)\,\!$ RL Circuit, Current Fall $I={\frac {\mathcal {E}}{R}}e^{-t/\tau _{L}}=I_{0}e^{-t/\tau _{L}}\,\!$ |
| LC Circuit | LC Circuit Equation $Lq''+q/C={\mathcal {E}}\,\!$ LC Circuit Resonance $\omega =1/{\sqrt {LC}}\,\!$ LC Circuit Charge $q=Q\cos(\omega t+\phi )\,\!$ LC Circuit Current $I=-\omega Q\sin(\omega t+\phi )\,\!$ LC Circuit electrical potential energy $U_{E}=q^{2}/2C=Q^{2}\cos ^{2}(\omega t+\phi )/2C\,\!$ LC circuit magnetic potential energy $U_{B}=Q^{2}\sin ^{2}(\omega t+\phi )/2C\,\!$ |
| RLC Circuits | RLC Circuit Equation $Lq''+Rq'+C^{-1}q={\mathcal {E}}\,\!$ RLC Circuit Charge $q=QeT^{-Rt/2L}\cos(\omega 't+\phi )\,\!$ |