# OpenStax University Physics/E&M

Each link below leads two different variations of equation sheets used in the electromagnetism unit OpenStax University Physics (Volume 2 Unit 2).
* These equations can be given to students as they take chapter quizzes based on examples in the textbook OpenStax University Physics
* A study guide for these quizzes can be found at Quizbank/Electricity and Magnetism (calculus based)
* Most solutions for the questions can be found by reading the examples in Unit 2 of the textbook.

#### Chapter 5

$\varepsilon _{0}=$ 8.85×10$^{−12}$ F/m = vacuum permittivity.
e = 1.602×10$^{−19}$C: negative (positive) charge for electrons (protons)
$k_{e}={\tfrac {1}{4\pi \varepsilon _{0}}}=$ = 8.99×10$^9$ m/F
${\vec {F}}=Q{\vec {E}}$ where ${\vec {E}}={\tfrac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}{\tfrac {q_{i}}{r_{Pi}^{2}}}{\hat {r}}_{Pi}$
${\vec {E}}=\int {{\tfrac {dq}{{r}^{2}}}{\hat {r}}}$ where $dq=\lambda d\ell =\sigma da=\rho dV$
$E={\tfrac {\sigma }{2\varepsilon _{0}}}$ = field above an infinite plane of charge.

#### Chapter 6

$\Phi ={\vec {E}}\cdot {\vec {A}}$ $\to \int {\vec {E}}\cdot d{\vec {A}}=\int {\vec {E}}\cdot {\hat {n}}\,dA$ = electric flux
$q_{enclosed}=\varepsilon _{0}\oint {\vec {E}}\cdot d{\vec {A}}$
$d\,{\text{Vol}}=dxdydz=r^{2}drdA$ where $dA=r^{2}d\phi d\theta$
$A_{\text{sphere}}=r^{2}\int _{0}^{\pi }\sin \theta d\theta \int _{0}^{2\pi }d\phi =4\pi r^{2}$

#### Chapter 7

$\Delta V_{AB}=V_{A}-V_{B}=-\int _{A}^{B}{\vec {E}}\cdot d{\vec {\ell }}$ = electric potential
${\vec {E}}=-{\tfrac {\partial V}{\partial x}}{\hat {i}}-{\tfrac {\partial V}{\partial y}}{\hat {j}}-{\tfrac {\partial V}{\partial z}}{\hat {k}}=-{\vec {\nabla }}V$
$q\Delta V$ = change in potential energy (or simply $U=qV$)
$Power={\tfrac {\Delta U}{\Delta t}}={\tfrac {\Delta q}{\Delta t}}V=IV=e{\tfrac {\Delta N}{\Delta t}}$
Electron (proton) mass = 9.11×10$^{−31}$kg (1.67× 10$^{−27}$kg). Elementary charge = e = 1.602×10$^{−19}$C.
$K={\tfrac {1}{2}}mv^{2}$=kinetic energy. 1 eV = 1.602×10$^{−19}$J
$V(r)=k{\tfrac {q}{r}}$ near isolated point charge
Many charges: $V_{P}=k\sum _{1}^{N}{\frac {q_{i}}{r_{i}}}\to k\int {\frac {dq}{r}}$.

#### Chapter 8

$Q=CV$ defines capacitance.
$C=\varepsilon _{0}{\tfrac {A}{d}}$ where A is area and d<<A$^{1/2}$ is gap length of parallel plate capacitor
${\text{Series}}:\;{\tfrac {1}{C_{S}}}=\sum {\tfrac {1}{C_{i}}}.$ ${\text{ Parallel:}}\;C_{P}=\sum C_{i}.$
$u={\tfrac {1}{2}}QV={\tfrac {1}{2}}CV^{2}={\tfrac {1}{2C}}Q^{2}$ = stored energy
$u_{E}={\tfrac {1}{2}}\varepsilon _{0}E^{2}$ = energy density

#### Chapter 9

Electric current: 1 Amp (A) = 1 Coulomb (C) per second (s)
Current=$I=dQ/dt=nqv_{d}A$, where
$(n,q,v_{d},A)$ = (density, charge, speed, Area)
$I=\int {\vec {J}}\cdot d{\vec {A}}$ where ${\vec {J}}=nq{\vec {v}}_{d}$ =current density.
${\vec {E}}=\rho {\vec {J}}$ = electric field where $\rho$ = resistivity
$\rho =\rho _{0}\left[1+\alpha (T-T_{0})\right]$, and $R=R_{0}\left[1+\alpha \Delta T\right]$,
where $R=\rho {\tfrac {L}{A}}$ is resistance
$V=IR$ and Power=$P=IV=I^{2}R=V^{2}/R$

#### Chapter 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 $I(t)=-{\tfrac {Q}{RC}}e^{-t/\tau }$

#### Chapter 11

$|{\vec {a}}\times {\vec {b}}|$$=ab\sin \theta \Leftrightarrow$ $({\vec {a}}\times {\vec {b}})_{x}=(a_{y}b_{z}-a_{z}b_{y})$, $({\vec {a}}\times {\vec {b}})_{y}=(a_{z}b_{x}-a_{x}b_{z})$, $({\vec {a}}\times {\vec {b}})_{z}=(a_{x}b_{y}-a_{y}b_{x})$ Magnetic force: ${\vec {F}}=q{\vec {v}}\times {\vec {B}},\;$$d{\vec {F}}=I{\overrightarrow {d\ell }}\times {\vec {B}}$. ${\vec {v}}_{d}={\vec {E}}\times {\vec {B}}/B^{2}$=EXB drift velocity Circular motion (uniform B field): $r={\tfrac {mv}{qB}}.\;$ Period=$T={\tfrac {2\pi m}{qB}}.\;$
Dipole moment=${\vec {\mu }}=NIA{\hat {n}}$. Torque=${\vec {\tau }}={\vec {\mu }}\times {\vec {B}}$. Stored energy=$U={\vec {\mu }}\cdot {\vec {B}}$. Hall field =$E=V/\ell =Bv_{d}={\tfrac {IB}{neA}}$ Lorentz force =$q\left({\vec {E}}+{\vec {v}}\times {\vec {B}}\right)$

#### Chapter 12

Free space permeability $\mu _{0}=4\pi \times 10^{-7}$ T·m/A Force between parallel wires ${\tfrac {F}{\ell }}={\tfrac {\mu _{0}I_{1}I_{2}}{2\pi r}}$ Biot–Savart law ${\vec {B}}={\tfrac {\mu _{0}}{4\pi }}\int \limits _{wire}{\frac {Id{\vec {\ell }}\times {\hat {r}}}{r^{2}}}$ Ampère's Law:$\oint {\vec {B}}\cdot d{\vec {\ell }}=4\pi \mu _{0}I_{enc}$ Magnetic field inside solenoid with paramagnetic material =$B=\mu nI$ where $\mu =(1+\chi )\mu _{0}$= permeability

#### Chapter 13

Magnetic flux $\Phi _{m}=\int _{S}{\vec {B}}\cdot {\hat {n}}dA$ Motional $\varepsilon =B\ell v$ if ${\vec {v}}\perp {\vec {B}}$
Electromotive "force" (volts) $\varepsilon =-N{\tfrac {d\Phi _{m}}{dt}}=\oint {\vec {E}}\cdot d{\vec {\ell }}$ rotating coil $\varepsilon =NBA\omega \sin \omega t$

#### Chapter 14

Unit of inductance = Henry (H)=1V·s/A
Mutual inductance: $M{\tfrac {dI_{2}}{dt}}=N_{1}{\tfrac {d\Phi _{12}}{dt}}=-\varepsilon _{1}$ where $\Phi _{12}$=flux through 1 due to current in 2. Reciprocity: $M{\tfrac {dI_{1}}{dt}}=-\varepsilon _{2}$
Self-inductance: $N\Phi _{m}=LI\rightarrow \varepsilon =-L{\tfrac {dI}{dt}}$
$L_{\text{solenoid}}\approx \mu _{0}N^{2}A\ell$, $L_{\text{toroid}}\approx {\tfrac {\mu _{0}N^{2}h}{2\pi }}\ln {\tfrac {R_{2}}{R_{1}}}$, Stored energy=${\tfrac {1}{2}}LI^{2}$
$I(t)={\tfrac {\varepsilon }{R}}\left(1-e^{-t/\tau }\right)$ in LR circuit where $\tau =L/R$.
$q(t)=q_{0}\cos(\omega t+\phi )$ in LC circuit where $\omega ={\sqrt {\tfrac {1}{LC}}}$

#### Chapter 15

AC voltage and current $v=V_{0}\sin(\omega t-\phi )$ if $i=I_{0}\sin \omega t.$ RMS values $I_{rms}={\tfrac {I_{0}}{\sqrt {2}}}$ and $V_{rms}={\tfrac {V_{0}}{\sqrt {2}}}$
Impedance $V_{0}=I_{0}X$ Resistor $V_{0}=I_{0}X_{R},\;\phi =0,$ where $X_{R}=R$ Capacitor $V_{0}=I_{0}X_{C},\;\phi =-{\tfrac {\pi }{2}},$ where $X_{C}={\tfrac {1}{\omega C}}$ Inductor $V_{0}=I_{0}X_{L},\;\phi =+{\tfrac {\pi }{2}},$ where $X_{L}=\omega L$ RLC series circuit $V_{0}=I_{0}Z$ where $Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2}}}$ and $\phi =\tan ^{-1}{\frac {X_{L}-X_{C}}{R}}$ Resonant angular frequency $\omega _{0}={\sqrt {\tfrac {1}{LC}}}$ Quality factor $Q={\tfrac {\omega _{0}}{\Delta \omega }}={\tfrac {\omega _{0}L}{R}}$ Average power $P_{ave}={\frac {1}{2}}I_{0}V_{0}\cos \phi =I_{rms}V_{rms}\cos \phi$ Transformer voltages and currents ${\tfrac {V_{S}}{V_{P}}}={\tfrac {N_{S}}{N_{P}}}={\tfrac {I_{P}}{I_{S}}}$

#### Chapter 16

Displacement current $I_{d}=\varepsilon _{0}{\tfrac {d\Phi _{E}}{dt}}$ where $\Phi _{E}=\int {\vec {E}}\cdot d{\vec {A}}$ is the electric flux.
Maxwell's equations: $\epsilon _{0}\mu _{0}=1/c^{2}$ $\oint _{S}{\vec {E}}\cdot \mathrm {d} {\vec {A}}={\frac {1}{\epsilon _{0}}}Q_{in}\qquad$ $\oint _{S}{\vec {B}}\cdot \mathrm {d} {\vec {A}}=0$ $\oint _{C}{\vec {E}}\cdot \mathrm {d} {\vec {\ell }}=-\int _{S}{\frac {\partial {\vec {B}}}{\partial t}}\cdot \mathrm {d} {\vec {A}}$ $\oint _{C}{\vec {B}}\cdot \mathrm {d} {\vec {\ell }}=\mu _{0}I+\epsilon _{0}\mu _{0}{\frac {\mathrm {d} \Phi _{E}}{\mathrm {d} t}}$
${\frac {\partial ^{2}E_{y}}{\partial x^{2}}}=\varepsilon _{0}\mu _{0}{\frac {\partial ^{2}E_{y}}{\partial t^{2}}}$ and ${\tfrac {E_{0}}{B_{0}}}=c$
Poynting vector ${\vec {S}}={\tfrac {1}{\mu _{0}}}{\vec {E}}\times {\vec {B}}$=energy flux
Average intensity $I=S_{ave}={\tfrac {c\varepsilon _{0}}{2}}E_{0}^{2}={\tfrac {c}{2\mu _{0}}}B_{0}^{2}={\tfrac {1}{2\mu _{0}}}E_{0}B_{0}$
Radiation pressure $p=I/c$ (perfect absorber) and $p=2I/c$ (perfect reflector).

## Index

${\mathcal {E}}\&{\mathcal {M}}$ (Vol. 2):  5:Electric Charges and Fields  6:Gauss's Law  7:Electric Potential  8:Capacitance  9:Current and Resistance  10:Direct-Current Circuits  11:Magnetic Forces and Fields  12:Sources of Magnetic Fields  13:Electromagnetic Induction  14:Inductance  15:Alternating-Current Circuits  16:Electromagnetic Waves
Subpages
* ↑ transclusions between OpenStax_University_Physics/E&M#Index and Quizbank/Electricity and Magnetism (calculus based)/Equations