OpenStax University Physics/E&M

https://www.myopenmath.com/course/course.php?folder=0&cid=70881

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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

${\epsilon }_0=$ 8.85×10⁻¹² F/m = vacuum permittivity.

e = 1.602×10⁻¹⁹C: negative (positive) charge for electrons (protons)

$k_e=\frac{1}{4\pi {\epsilon }_0}=$ = 8.99×10⁹ m/F

$\overrightarrow{F}=Q\overrightarrow{E}$ where $\overrightarrow{E}=\frac{1}{4\pi {\epsilon }_0}\sum _{i=1}^N\frac{q_i}{r_{Pi}^2}{\hat{r}}_{Pi}$

$\overrightarrow{E}=\int \frac{dq}{r^2}\hat{r}$ where $dq=\lambda d\ell =\sigma da=\rho dV$

$E=\frac{\sigma }{2{\epsilon }_0}$ = field above an infinite plane of charge.

Chapter 6

$\mathrm{\Phi }=\overrightarrow{E}\cdot \overrightarrow{A}$ $\to \int \overrightarrow{E}\cdot d\overrightarrow{A}=\int \overrightarrow{E}\cdot \hat{n}dA$ = electric flux

$q_{enclosed}={\epsilon }_0\oint \overrightarrow{E}\cdot d\overrightarrow{A}$

$d\text{Vol}=dxdydz=r^{2}drdA$ where $dA=r^{2}d\varphi d\theta$

$A_{\text{sphere}}=r^2{\int }_0^{\pi }\sin \theta d\theta {\int }_0^{2\pi }d\varphi =4\pi r^2$

Chapter 7

$\mathrm{\Delta }{V}_{AB}=V_A-V_B=-{\int }_A^B\overrightarrow{E}\cdot d\overrightarrow{\ell }$ = electric potential

$\overrightarrow{E}=-\frac{\partial V}{\partial x}\hat{i}-\frac{\partial V}{\partial y}\hat{j}-\frac{\partial V}{\partial z}\hat{k}=-\overrightarrow{\mathrm{\nabla }}V$

$q\mathrm{\Delta }V$ = change in potential energy (or simply $U=qV$)

$Power=\frac{\mathrm{\Delta }U}{\mathrm{\Delta }t}=\frac{\mathrm{\Delta }q}{\mathrm{\Delta }t}V=IV=e\frac{\mathrm{\Delta }N}{\mathrm{\Delta }t}$

Electron (proton) mass = 9.11×10⁻³¹kg (1.67× 10⁻²⁷kg). Elementary charge = e = 1.602×10⁻¹⁹C.

$K=\frac{1}{2}m{v}^2$=kinetic energy. 1 eV = 1.602×10⁻¹⁹J

$V(r)=k\frac{q}{r}$ 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.

$C={\epsilon }_0\frac{A}{d}$ where A is area and d<<A^1/2 is gap length of parallel plate capacitor

$\text{Series}:\frac{1}{C_S}=\sum \frac{1}{C_i}.$   $\text{ Parallel:}{C}_P=\sum C_i.$

$u=\frac{1}{2}QV=\frac{1}{2}C{V}^2=\frac{1}{2C}{Q}^2$ = stored energy

$u_E=\frac{1}{2}{\epsilon }_0{E}^2$ = energy density

Chapter 9

Electric current: 1 Amp (A) = 1 Coulomb (C) per second (s)

Current=$I=dQ/dt=nq{v}_{d}A$, where

$(n,q,v_d,A)$ = (density, charge, speed, Area)

$I=\int \overrightarrow{J}\cdot d\overrightarrow{A}$ where $\overrightarrow{J}=nq{\overrightarrow{v}}_d$ =current density.

$\overrightarrow{E}=\rho \overrightarrow{J}$ = electric field where $\rho$ = resistivity

$\rho ={\rho }_0[1+\alpha (T-T_0)]$, and $R=R_0[1+\alpha \mathrm{\Delta }T]$,

where $R=\rho \frac{L}{A}$ is resistance

$V=IR$ and Power=$P=IV=I^{2}R=V^2/R$

Chapter 10

$V_{terminal}=\epsilon -I{r}_{eq}$ where $r_{eq}$=internal resistance and $\epsilon$=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: $q(t)=Q(1-e^{t/\tau })$ and $I=I_0{e}^{-t/\tau }$ where $\tau =RC$ is RC time, $Q=\epsilon C$ and $I_0=\epsilon /R$.

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

Chapter 11

cross product

$|\overrightarrow{a}\times \overrightarrow{b}|$$=ab\sin \theta \iff$ $(\overrightarrow{a}\times \overrightarrow{b}{)}_x=(a_y{b}_z-a_z{b}_y)$, $(\overrightarrow{a}\times \overrightarrow{b}{)}_y=(a_z{b}_x-a_x{b}_z)$, $(\overrightarrow{a}\times \overrightarrow{b}{)}_z=(a_x{b}_y-a_y{b}_x)$
Magnetic force: $\overrightarrow{F}=q\overrightarrow{v}\times \overrightarrow{B},$$d\overrightarrow{F}=I\overrightarrow{d\ell }\times \overrightarrow{B}$.
${\overrightarrow{v}}_d=\overrightarrow{E}\times \overrightarrow{B}/B^2$=EXB drift velocity
Circular motion (uniform B field): $r=\frac{mv}{qB}.$ Period=$T=\frac{2\pi m}{qB}.$

Hall effect

Dipole moment=$\overrightarrow{\mu }=NIA\hat{n}$. Torque=$\overrightarrow{\tau }=\overrightarrow{\mu }\times \overrightarrow{B}$. Stored energy=$U=\overrightarrow{\mu }\cdot \overrightarrow{B}$.
Hall field =$E=V/\ell =B{v}_d=\frac{IB}{neA}$
Lorentz force =$q(\overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B})$

Chapter 12

Free space permeability ${\mu }_0=4\pi \times 10^{-7}$ T·m/A
Force between parallel wires $\frac{F}{\ell }=\frac{{\mu }_0{I}_1{I}_2}{2\pi r}$
Biot–Savart law $\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\underset{wire}{\int }\frac{Id\overrightarrow{\ell }\times \hat{r}}{r^2}$
Ampère's Law:$\oint \overrightarrow{B}\cdot d\overrightarrow{\ell }=4\pi {\mu }_0{I}_{enc}$
Magnetic field inside solenoid with paramagnetic material =$B=\mu nI$ where $\mu =(1+\chi ){\mu }_0$= permeability

Chapter 13

Magnetic flux ${\mathrm{\Phi }}_m={\int }_S\overrightarrow{B}\cdot \hat{n}dA$
Motional $\epsilon =B\ell v$ if $\overrightarrow{v}\perp \overrightarrow{B}$
Electromotive "force" (volts) $\epsilon =-N\frac{d{\mathrm{\Phi }}_m}{dt}=\oint \overrightarrow{E}\cdot d\overrightarrow{\ell }$
rotating coil $\epsilon =NBA\omega \sin \omega t$

Chapter 14

Unit of inductance = Henry (H)=1V·s/A

Mutual inductance: $M\frac{d{I}_2}{dt}=N_1\frac{d{\mathrm{\Phi }}_{12}}{dt}=-{\epsilon }_1$ where ${\mathrm{\Phi }}_{12}$=flux through 1 due to current in 2. Reciprocity: $M\frac{d{I}_1}{dt}=-{\epsilon }_2$

Self-inductance: $N{\mathrm{\Phi }}_m=LI\to \epsilon =-L\frac{dI}{dt}$

$L_{\text{solenoid}}\approx {\mu }_0{N}^{2}A\ell$, $L_{\text{toroid}}\approx \frac{{\mu }_0{N}^{2}h}{2\pi }\ln \frac{R_2}{R_1}$, Stored energy=$\frac{1}{2}L{I}^2$

$I(t)=\frac{\epsilon }{R}(1-e^{-t/\tau })$ in LR circuit where $\tau =L/R$.

$q(t)=q_0\cos (\omega t+\varphi )$ in LC circuit where $\omega =\sqrt{\frac{1}{LC}}$

Chapter 15

AC voltage and current $v=V_0\sin (\omega t-\varphi )$ if $i=I_0\sin \omega t.$
RMS values $I_{rms}=\frac{I_0}{\sqrt{2}}$ and $V_{rms}=\frac{V_0}{\sqrt{2}}$
Impedance $V_0=I_{0}X$
Resistor $V_0=I_0{X}_R,\varphi =0,$ where $X_R=R$
Capacitor $V_0=I_0{X}_C,\varphi =-\frac{\pi }{2},$ where $X_C=\frac{1}{\omega C}$
Inductor $V_0=I_0{X}_L,\varphi =+\frac{\pi }{2},$ where $X_L=\omega L$
RLC series circuit $V_0=I_{0}Z$ where $Z=\sqrt{R^2+{(X_L-X_C)}^2}$ and $\varphi ={\tan }^{-1}\frac{X_L-X_C}{R}$
Resonant angular frequency ${\omega }_0=\sqrt{\frac{1}{LC}}$
Quality factor $Q=\frac{{\omega }_0}{\mathrm{\Delta }\omega }=\frac{{\omega }_{0}L}{R}$
Average power $P_{ave}=\frac{1}{2}{I}_0{V}_0\cos \varphi =I_{rms}{V}_{rms}\cos \varphi$
Transformer voltages and currents $\frac{V_S}{V_P}=\frac{N_S}{N_P}=\frac{I_P}{I_S}$

Chapter 16

Displacement current $I_d={\epsilon }_0\frac{d{\mathrm{\Phi }}_E}{dt}$ where ${\mathrm{\Phi }}_E=\int \overrightarrow{E}\cdot d\overrightarrow{A}$ is the electric flux.

Maxwell's equations: ${ϵ}_0{\mu }_0=1/c^2$
${\oint }_S\overrightarrow{E}\cdot \mathrm{d}\overrightarrow{A}=\frac{1}{{ϵ}_0}{Q}_{in}$
${\oint }_S\overrightarrow{B}\cdot \mathrm{d}\overrightarrow{A}=0$
${\oint }_C\overrightarrow{E}\cdot \mathrm{d}\overrightarrow{\ell }=-{\int }_S\frac{\partial \overrightarrow{B}}{\partial t}\cdot \mathrm{d}\overrightarrow{A}$
${\oint }_C\overrightarrow{B}\cdot \mathrm{d}\overrightarrow{\ell }={\mu }_{0}I+{ϵ}_0{\mu }_0\frac{\mathrm{d}{\mathrm{\Phi }}_E}{\mathrm{d}t}$

$\frac{{\partial }^2{E}_y}{\partial x^2}={\epsilon }_0{\mu }_0\frac{{\partial }^2{E}_y}{\partial t^2}$ and $\frac{E_0}{B_0}=c$

Poynting vector $\overrightarrow{S}=\frac{1}{{\mu }_0}\overrightarrow{E}\times \overrightarrow{B}$=energy flux

Average intensity $I=S_{ave}=\frac{c{\epsilon }_0}{2}{E}_0^2=\frac{c}{2{\mu }_0}{B}_0^2=\frac{1}{2{\mu }_0}{E}_0{B}_0$

Radiation pressure $p=I/c$ (perfect absorber) and $p=2I/c$ (perfect reflector).

Index

$ℰ\mathrm{\&}ℳ$ (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

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