Constants

Elementary Charge: $e=1.602\times 10^{-19}$C

Mass of Electron: $m=9.109\times 10^{-31}\text{kg}$

Mass of Proton: $m=1.672\times 10^{-27}\text{kg}$

Vacuum Permittivity: ${ϵ}_0=8.854\times 10^{-12}$

Coulomb Constant: $k=9.0\times 10^9\frac{N{m}^2}{C^2}$

Gauss's Law

Flux Equations

Flux through a surface

${\Phi }_e=E_{\perp }A=EA\cos (\theta )$ where $\theta$ is the angle between the normal line and the flux lines or the angle between $\hat{n}$ and $\vec{E}$.

Flux Integral

${\Phi }_e=\oint \vec{E}\cdot d\vec{A}$

When using a Gaussian sphere:

${\Phi }_e=\frac{q}{4\pi {ϵ}_0{r}^2}\cdot 4\pi r^2=\frac{q}{{ϵ}_0}$

Gauss Law Combined

${\Phi }_e=\oint \vec{E}\cdot d\vec{A}=\frac{Q_{in}}{{ϵ}_0}$

Most General Use

Flux Around a Charged Wire

${\Phi }_e=2\pi rLE=\frac{Q_{in}}{{ϵ}_0}=\frac{\lambda L}{{ϵ}_0}$

Notice this the same as the sphere, except the surface area is just the perimeter of a circle $\times$ length.

Flux of Infinite Plane of Charge

$Q_{in}=\eta A$

${\Phi }_e=2EA=\frac{Q_{in}}{{ϵ}_0}=\frac{\eta A}{{ϵ}_0}$

Area divides out revealing:

$E_{plane}=\frac{\eta }{2{ϵ}_0}$

Electric Potential

Uniform Electric Field

$U_{elec}=U_0+qEs$

$U_0$ is the potential at the negative plate (often $U_0=0$)

$s$ is the distance from the negative plate

Difference for Negative and Positive Charges

In general, movement against the electric field is negative energy.

Positive

• Field direction is "downhill." Potential energy decreases as the charge speeds up.
• Potential goes down and kinetic goes up as it moves to the negative.

Negative

• Field direction is "uphill." Potential energy increases as the charge slows.
• Potential goes up and kinetic goes down as it moves toward the negative.
• The movement of an electron towards the positive (natural movement) is negative mechanical energy: $E_{mech}=K_i+(-e)Ed$. $K_i$ is zero.

Potential Equations

Potential Energy of Point Charges

$U_{elec}=K\frac{q_q{q}_2}{r}$ or $\frac{1}{4\pi {ϵ}_0}\frac{q_1{q}_2}{r}$

Escape speed

$K_f+U_f=K_i+U_i$

$v_i=\sqrt{\frac{K{q}_1{q}_2}{m{r}_i}}\leftarrow$ Unclear when the usage of this is.

Potential of a Dipole

$U_{dipole}=-pE\cos (\varphi )=-\vec{p}\cdot \vec{E}$

Where $\vec{p}$ is the dipole vector

Electric Potential Energy from Voltage

$U_q=qV$

Where 1 volt = $1J/C$

Example: A proton would loose $-100e$ moving through $100V$. Potential is negative even though the charge is positive:

Electric Potential inside a Parallel-Plate Capacitor

$V=Es$, where s is the distance form the negative electrode

Field can also be defined by $E=\frac{\Delta V_C}{d}$. This is more common.

Electric Potential from a Point Charge

$V=k\frac{q}{r}$

Electric Potential from a Ring

$V_{ringonaxis}=k\frac{Q}{\sqrt{R^2+z^2}}$, R is radius, z is distance.

Electric Potential from Charged Disk

$V_{diskonaxis}=\frac{Q}{2\pi {ϵ}_0{R}^2}(\sqrt{R^2+z^2}-z)$

Potential and Field

$\Delta V=V_f-V_i=-{\int }_{s_i}^{s_f}{E}_{s}ds$

Electric Field from Potential

$E_s=-\frac{dV}{ds}\leftarrow$ Ensuring that a $\Delta$ is always $final-initial$ is important. The this equation is bad for varying electric fields.

Energy from Electric Field or Potential

Because $U_{elec}=U_0+qEs$, the integral above ($-{\int }_{s_i}^{s_f}{E}_{s}ds$) can be multiplied by charge to get field because $\text{Voltage}=J/C$.

Field Equations

Field of radial rod and shell:

$E(r)=\frac{\lambda }{2\pi {ϵ}_{0}r}$, $\lambda$ is equal to the charge density of all enclosed charges.

Capacitors

Charge on Capacitor

$Q=C\Delta V$

Parallel Plate Capacitor

$E=\frac{Q}{{ϵ}_{0}A}$ or $C=\frac{Q}{\Delta V_c}=\frac{{ϵ}_{0}A}{d}$

Equivalent Capacitance

Parallel

$C_{eq}=C_1+C_2+C_3+\cdots$

Series

$C_{eq}=(\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots {)}^{-1}$

Energy of Capacitor

$U_C=\frac{1}{2}C(\Delta V{)}^2$

Dielectric Constant

$\kappa =\frac{E_0}{E}$

Capacitance is then increased proportionally:

$C=\kappa C_0$

This is designed to let air be $\kappa =1$

Change in Voltage

$\Delta V=V_f-V_i=-{\int }_{s_i}^{s_f}{E}_{s}ds$

Concentric Cylinders

$C=\frac{2\pi {ϵ}_{0}l}{\ln (\frac{R_2}{R_1})}$

Concentric Spheres

$C=\frac{4\pi {ϵ}_0}{\frac{1}{R_1}-\frac{1}{R_2}}$