Coulombs law

The force

F\bf{\it{F}}

on a point charge

q1q_1

at the point

r1\bf{r_1}

caused by a point charge

q2q_2

at the point

r2\bf{r_2}
F=q1q24πε0∣r1−r2∣2(r1−r2)∣r1−r2∣F = \frac{q_1 q_2}{4\pi \varepsilon_0 |r_1 - r_2|^2}\frac{(r_1-r_2)}{|r_1-r_2|}

Electric Field Strength

From a point charge

qq

in

r′\bm{r}'
E(r)=q4πϵ0R2eR\bm{E}(\bm{r}) = \frac{q}{4\pi \epsilon_0 R^2}\bm{e}_R

From charge distribution

E(r)=∫14πϵ0R2eRdq(r′),\bm{E}(\bm{r}) = \int \frac{1}{4\pi\epsilon_0 R^2}\bm{e}_R dq(\bm{r}'),
dq(r′)={ρtot(r′)dv′=ρ(r′)+ρp(r′))dv′ρtot,s(r′)dS′=ρs(r′)+ρp,s(r′))dS′ρl(r′)dl′dq(\bm{r}') = \begin{cases} \rho_{tot}(\bm{r}')dv' = \rho (\bm{r}') + \rho_p (\bm{r}'))dv'\\ \rho_{tot,s}(\bm{r}')dS' = \rho_s (\bm{r}') + \rho_{p,s} (\bm{r}'))dS'\\ \rho_l(\bm{r}')dl' \end{cases}

From point dipole

p=pez\bm{p} = p\bm{e}_z
E(r)=p4πϵ0r3(2cos⁡(θ)er+sin⁡(θ)eθ)\bm{E}(\bm{r}) = \frac{p}{4\pi \epsilon_0 r^3}(2\cos(\theta)\bm{e}_r + \sin(\theta)\bm{e}_\theta)

From line charge

ρl\rho_l
E(r)=ρl2πϵ0rcerc\bm{E}(\bm{r}) = \frac{\rho_l}{2\pi \epsilon_0 r_c}\bm{e}_{r_c}

From dipole line

pl=plex\bm{p}_l = p_l\bm{e}_x
E(r)=pl2πϵ0rc2(cos⁡(φ)erc+sin⁡(φ)eφ)\bm{E}(\bm{r}) = \frac{p_l}{2\pi \epsilon_0 r_c^2}(\cos(\varphi)\bm{e}_{r_c} + \sin(\varphi)\bm{e}_\varphi)

Electrical Potential

E=−∇V\bm{E} = -\bm{\nabla}V

From pointsource

qq

in

r′\bm{r}'
V(r)=q4πϵ0RV(\bm{r}) = \frac{q}{4\pi\epsilon_0 R}

From charge distribution

V(r)=∫14πϵ0Rdq(r′)V(\bm{r}) = \int\frac{1}{4\pi\epsilon_0 R}dq(\bm{r}')

From point dipole

p=pez\bm{p} = p\bm{e}_z
V(r)=p⋅r4πϵ0r3=pcos⁡(θ)4πϵ0r2V(\bm{r}) = \frac{\bm{p} \cdot \bm{r} }{4\pi\epsilon_0 r^3} = \frac{p\cos(\theta)}{4\pi\epsilon_0 r^2}

From line charge

ρl\rho_l
V(r)=ρl2πϵ0ln⁡(1rc)V(\bm{r}) = \frac{\rho_l}{2\pi\epsilon_0}\ln\left(\frac{1}{r_c}\right)

From dipole line

pl=plex\bm{p}_l = p_l \bm{e}_x
V(r)=pl2πϵ0cos⁡(φ)rcV(\bm{r}) = \frac{p_l}{2\pi\epsilon_0}\frac{\cos(\varphi)}{r_c}

Electrical flow density

Where

D\bm D

is defined by

∇D=ρ\bm\nabla \bm D = \rho

Gauss law, where

en\bm e_n

is the unit normal to the volume surface pointing outwards

P,E\bm P, \bm E

and

D\bm D

:

{D=ϵ0E+P(valid generally)D=ϵrϵ0E\begin{cases} \bm D = \epsilon_0\bm E + \bm P \qquad(\textup{valid generally}) \\ \bm D = \epsilon_r\epsilon_0\bm E \end{cases}

Polarization Charge

ρp=−∇⋅Pspace charge density\rho_p = -\bm\nabla\cdot\bm P \qquad \textup{space charge density}
ρp,s=en1⋅(P1−P2)surface charge density\rho_{p,s} = \bm e_{n1}\cdot(\bm P_1 - \bm P_2) \qquad \textup{surface charge density}

where the unit normal

en1\bm e_{n1}

is directed from 1 to 2.

Boundary Conditions

{Et  continuousρs=en2⋅(D1−D2)\begin{cases} E_t \;\textup{continuous} \\ \rho_s = \bm e_{n2}\cdot(\bm D_1 - \bm D_2) \end{cases}

where

ρs\rho_s

is free surface charge density and

en2\bm e_{n2}

is directed from volume 2 to volume 1.

Electrostatic Energy

We=12∑iQiViW_e = \frac 12 \sum_iQ_iV_i
We=12∫ρV dvW_e = \frac 12 \int\rho V\: dv
We=12∫E⋅D dvW_e = \frac 12 \int\bm E\cdot\bm D \: dv

Maxwell's voltage

∣T∣=12E⋅DE is a bisector to en and T|\bm T| = \frac 12\bm E\cdot\bm D \qquad \bm E \textup{ is a bisector to } \bm e_n \textup{ and } \bm T

Torque on Electrical Dipole

Te=p×E\bm T_e = \bm p\times\bm E