# Physics Formulae/Electric Circuits Formulae

Lead Article: Tables of Physics Formulae

## DC Quantities

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Electrical Resistance ${\displaystyle R\,\!}$  ${\displaystyle R=V/I\,\!}$  Ω = V A-1 = J s C-2 [M][L]2 [T]-3 [I]-2
Resistivity, Scalar ${\displaystyle \rho \,\!}$  ${\displaystyle \rho ={\frac {RA}{l}}\,\!}$  Ω m [M]2 [L]2 [T]-3 [I]-2
Resistivity Temperature Coefficient,

Linear Temperature Dependance

${\displaystyle \alpha \,\!}$  ${\displaystyle \rho -\rho _{0}=\rho _{0}\alpha (T-T_{0})\,\!}$  K-1 [Θ]-1
Terminal Voltage for

Power Supply

${\displaystyle V_{\mathrm {ter} }\,\!}$  V = J C-1 [M] [L]2 [T]-3 [I]-1
Load Voltage for Circuit ${\displaystyle V_{\mathrm {load} }\,\!}$  V = J C-1 [M] [L]2 [T]-3 [I]-1
Internal Resistance of

Power Supply

${\displaystyle R_{\mathrm {int} }\,\!}$  ${\displaystyle R_{\mathrm {int} }={\frac {V_{\mathrm {ter} }}{I}}\,\!}$  Ω = V A-1 = J s C-2 [M][L]2 [T]-3 [I]-2

Circuit

${\displaystyle R_{\mathrm {ext} }\,\!}$  ${\displaystyle R_{\mathrm {ext} }={\frac {V_{\mathrm {load} }}{I}}\,\!}$  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

${\displaystyle {\mathcal {E}}\,\!}$  ${\displaystyle {\mathcal {E}}=V_{\mathrm {ter} }+V_{\mathrm {load} }\,\!}$  V = J C-1 [M] [L]2 [T]-3 [I]-1
Electrical Conductance ${\displaystyle G\,\!}$  ${\displaystyle G=1/R\,\!}$  S = Ω-1 [T]3 [I]2 [M]-1 [L]-2
Electrical Conductivity, Scalar ${\displaystyle \sigma \,\!}$  ${\displaystyle \sigma =1/\rho \,\!}$  Ω-1 m-1 [I]2 [T]3 [M]-2 [L]-2
Electrical Conductivity, Tensor ${\displaystyle {\boldsymbol {\sigma }},\sigma _{\mathrm {ij} }\,\!}$  ${\displaystyle \sigma _{\mathrm {ij} }{\begin{pmatrix}\ \sigma _{11}&\sigma _{12}&\sigma _{13}\\\ \sigma _{21}&\sigma _{22}&\sigma _{23}\\\ \sigma _{31}&\sigma _{32}&\sigma _{33}\end{pmatrix}}\,\!}$  Ω-1 m-1 [I]2 [T]3 [M]-2 [L]-2
Electrical Power ${\displaystyle P\,\!}$  ${\displaystyle P=VI\,\!}$  W = J s-1 [M] [L]2 [T]-3
emf Power ${\displaystyle P\,\!}$  ${\displaystyle P_{\mathrm {emf} }=I{\mathcal {E}}\,\!}$  W = J s-1 [M] [L]2 [T]-3
Resistor Power Dissipation ${\displaystyle P\,\!}$  ${\displaystyle P=I^{2}R=V^{2}/R\,\!}$  W = J s-1 [M] [L]2 [T]-3
 Resistors in Series ${\displaystyle R_{\mathrm {net} }=\sum _{i}R_{i}\,\!}$ Resistors in Parallel ${\displaystyle {\frac {1}{R_{\mathrm {net} }}}=\sum _{i}{\frac {1}{R_{i}}}\,\!}$
 Ohm's Law Scalar form ${\displaystyle V=IR\,\!}$  Vector Form ${\displaystyle \mathbf {J} =\sigma \mathbf {E} \,\!}$  Tensor Form, general applies to all points in a conductor ${\displaystyle \mathbf {J} _{i}=\sigma _{ij}\mathbf {E} _{j}\,\!}$ Kirchoff's Laws emf loop rule around any closed circuit ${\displaystyle \sum _{i}V_{i}=\sum _{i}I_{i}R_{i}=0\,\!}$  Current law at junctions ${\displaystyle I_{\mathrm {in} }=I_{\mathrm {out} }\,\!}$

## AC Quantitites

Quantity (Common Name/s) Common Name/s Quantity (Common Name/s) Quantity (Common Name/s) Quantity (Common Name/s)
Resistive Load Voltage ${\displaystyle V_{R}\,\!}$  ${\displaystyle V_{R}=I_{R}R\,\!}$  V = J C-1 [M] [L]2 [T]-3 [I]-1
Capacitive Load Voltage ${\displaystyle V_{C}\,\!}$  ${\displaystyle V_{C}=I_{C}X_{C}\,\!}$  V = J C-1 [M] [L]2 [T]-3 [I]-1
Inductive Load Voltage ${\displaystyle V_{L}\,\!}$  ${\displaystyle V_{L}=I_{L}X_{L}\,\!}$  V = J C-1 [M] [L]2 [T]-3 [I]-1
Capacitive Reactance ${\displaystyle X_{C}\,\!}$  ${\displaystyle X_{C}={\frac {1}{\omega _{\mathrm {d} }C}}\,\!}$  Ω-1 m-1 [I]2 [T]3 [M]-2 [L]-2
Inductive Reactance ${\displaystyle X_{L}\,\!}$  ${\displaystyle X_{L}=\omega _{d}L\,\!}$  Ω-1 m-1 [I]2 [T]3 [M]-2 [L]-2
AC Impedance ${\displaystyle Z\,\!}$  ${\displaystyle V=IZ\,\!}$

${\displaystyle Z={\sqrt {R^{2}-\left(X_{L}-X_{C}\right)^{2}}}\,\!}$

Ω-1 m-1 [I]2 [T]3 [M]-2 [L]-2
Phase Constant ${\displaystyle \phi \,\!}$  ${\displaystyle \tan \phi ={\frac {X_{L}-X_{C}}{R}}\,\!}$  dimensionless dimensionless
AC Circuit Resonant

Pulsatance

${\displaystyle \omega _{\mathrm {res} }\,\!}$  ${\displaystyle \omega _{\mathrm {d} }=\omega _{\mathrm {res} }=\omega ={\frac {1}{\sqrt {LC}}}\,\!}$  s-1 [T]-1
AC Peak Current ${\displaystyle I_{0}\,\!}$  ${\displaystyle I_{0}=I_{\mathrm {rms} }{\sqrt {2}}\,\!}$  A [I]
AC Root Mean

Square Current

${\displaystyle I_{\mathrm {rms} },{\sqrt {\langle I\rangle }}\,\!}$  ${\displaystyle I_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[I\left(t\right)\right]^{2}\mathrm {d} t}}\,\!}$  A [I]
AC Peak Voltage ${\displaystyle V_{0}\,\!}$  ${\displaystyle V_{0}=V_{\mathrm {rms} }{\sqrt {2}}\,\!}$  V = J C-1 [M] [L]2 [T]-3 [I]-1
AC Root Mean

Square Voltage

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