essential electromagnetism (SI)

stationary charges 3D $\Rightarrow$ electric field $\boldsymbol{E}$ is

steady current 3D$\Rightarrow$ magnetic field $\boldsymbol{B}$ is

not a sink/source

not rotating

is a sink/source

derived from

rotating

$\boldsymbol{\nabla}\times\boldsymbol{B}=\mu_0\boldsymbol{J}$

$\boldsymbol{\nabla}\cdot\boldsymbol{B}=0$

$\oint\boldsymbol{B}\cdot d\boldsymbol{l}=\mu_0I_{\text{enclosed}}$

$\oint\boldsymbol{B}\cdot d\boldsymbol{a}=0$

$\boldsymbol{\nabla}\times\boldsymbol{E}=\boldsymbol{0}$

$\oint\boldsymbol{E}\cdot d\boldsymbol{l}=0$

$\boldsymbol{\nabla}\cdot\boldsymbol{E}=\frac{1}{\varepsilon_0}\rho$

$\oint\boldsymbol{E}\cdot d\boldsymbol{a}=\frac{1}{\varepsilon_0}Q_{\text{enclosed}}$

magnetic extras

electric extras

current density

surface

volume

line

amount of charge per second that flows through unit area of a chosen cross section.

$B(r)=\frac{\mu_0}{4\pi}\int\frac{J(r')\times\hat{\eta}}{\eta^2}\,d\tau'$

math details

$\boldsymbol{E}(\boldsymbol{r})=\frac{1}{4\pi\varepsilon_0}\int\frac{\rho(\boldsymbol{r}')}{\eta^2}\,\hat{\boldsymbol{\eta}}\,d\tau'$

notation

source point $\boldsymbol{r}'$, where electric charge is located

divergence theorem

$\oint\boldsymbol{A}\cdot d\boldsymbol{a}=\int (\boldsymbol{\nabla}\cdot\boldsymbol{A})\,d\tau$

field point $\boldsymbol{r}'$, where electric charge is located

$d\boldsymbol{l}=dx\,\hat{\boldsymbol{x}}+dy\,\hat{\boldsymbol{y}}+dz\,\hat{\boldsymbol{z}}$

$\boldsymbol{\eta}=\boldsymbol{r}-\boldsymbol{r}'$

$\boldsymbol{\nabla}=\frac{\partial}{\partial x}\hat{\boldsymbol{x}}+\frac{\partial}{\partial y}\hat{\boldsymbol{y}}+\frac{\partial}{\partial z}\hat{\boldsymbol{z}}$

$d\tau=dx\,dy\,dz$

$d\boldsymbol{a}=\boldsymbol{n}da$

integration by parts

$uv=\int\,u\,du+\int\,v\,dv$

curl theorem

$\int (\boldsymbol{\nabla}\times\boldsymbol{A})\cdot d\boldsymbol{a}=\oint\boldsymbol{A}\cdot d\boldsymbol{l}$

gradient theorem

$\int_\boldsymbol{a}^\boldsymbol{b}\boldsymbol{\nabla A}\cdot d\boldsymbol{l}=\oint\boldsymbol{A}\cdot d\boldsymbol{l}$

distribution

$\boldsymbol{A}\cdot(\boldsymbol{B}+\boldsymbol{C})=\boldsymbol{A}\cdot\boldsymbol{B}+\boldsymbol{A}\cdot\boldsymbol{C}$

$\boldsymbol{A}\times(\boldsymbol{B}+\boldsymbol{C})=\boldsymbol{A}\times\boldsymbol{B}+\boldsymbol{A}\times\boldsymbol{C}$

electric suspectibility $\chi_e$

polarization $\boldsymbol{P}$

relative permitivity $\varepsilon_r$

$\varepsilon_r=\frac{\varepsilon}{\varepsilon_0}=1+\chi_e$

the degree of polarization of a dielectric material in response to an applied electric field.

$\boldsymbol{P}=\frac{\boldsymbol{p}}{V}$

the factor by which an electric field is reduced inside a material relative to vacuum

$\boldsymbol{P}=\varepsilon_0\chi_e\boldsymbol{E}$

displacement $\boldsymbol{D}$

$\boldsymbol{D}=\varepsilon_0\boldsymbol{E}+\boldsymbol{P}$

dipole moment $\boldsymbol{p}$

$\boldsymbol{p}=\alpha\boldsymbol{E}$

materials are either conductors, containing unlimited of charges that are free to move, or dielectrics (insulators), where all charges are attached to specific atoms/molecules on a tight leash making them only capable to move a a bit inside the atoms/molecules. A large enough electric field can ionize a dielectric, tearing apart molecules/atoms making it a conductor.

bound charges

free charges

electrons that can move freely, not bound

electrons that are in atoms and molecules, they cannot move.

$\rho_b=-\boldsymbol{\nabla}\cdot\boldsymbol{P}$

$\rho_f=\boldsymbol{\nabla}\cdot\boldsymbol{D}$

$\sigma_b=\boldsymbol{P}\cdot\hat{\boldsymbol{n}}$

$\sigma_f=\boldsymbol{D}\cdot\hat{\boldsymbol{n}}$

$\boldsymbol{\nabla}\times\boldsymbol{D}=\boldsymbol{\nabla}\times\boldsymbol{P}$

$\boldsymbol{F}=(\boldsymbol{p}\cdot\boldsymbol{\nabla})\boldsymbol{E}$

continuity equation

$\boldsymbol{F}=\int\,(\boldsymbol{J}\times\boldsymbol{B})\,d\tau$

$\boldsymbol{F}=\int\,(\boldsymbol{K}\times\boldsymbol{B})\,da$

$\boldsymbol{F}=\int\,(\boldsymbol{I}\times\boldsymbol{B})\,dl$

$\boldsymbol{\nabla}\cdot\boldsymbol{J}=-\frac{\partial p}{\partial t}$

torque

what is steady current?

$\frac{\partial\boldsymbol{J}}{\partial t}=\boldsymbol{0}$

$\frac{\partial p}{\partial t}=0$

$\boldsymbol{N}=\boldsymbol{m}\times\boldsymbol{B}$

force

$\boldsymbol{F}=\boldsymbol{\nabla}(\boldsymbol{m}\cdot\boldsymbol{B})$

auxillery field

$\boldsymbol{H}=\frac{1}{\mu_0}\boldsymbol{B}-\boldsymbol{M}$

click to edit

what is magnetic field?

A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field.

free current $\boldsymbol{J}_f$

bound current $\boldsymbol{J}_b$

click to edit

magnetization $\boldsymbol{M}$

click to edit

material is either paramagnet, diamagnets or ferromagnets.

diamagnet

paramagnet

ferromagnet

In an dielectric/insulator, you have two types of charges: free charges and bound charges. D is due to free charges, P the polarization is due to bound charges, and E is for the total charges, both free and bound.

In an dielectric/insulator, you have two types of currents: free currents and bound currents. H is due to free currents, M the magnetization is due to bound currents, and B is for the total currents, both free and bound.