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essential electromagnetism (SI) - Coggle Diagram
essential electromagnetism (SI)
stationary charges 3D\(\Rightarrow\) electric field \(\boldsymbol{E}\) is
not rotating
\(\boldsymbol{\nabla}\times\boldsymbol{E}=\boldsymbol{0}\)
\(\oint\boldsymbol{E}\cdot d\boldsymbol{l}=0\)
is a sink/source
\(\boldsymbol{\nabla}\cdot\boldsymbol{E}=\frac{1}{\varepsilon_0}\rho\)
\(\oint\boldsymbol{E}\cdot d\boldsymbol{a}=\frac{1}{\varepsilon_0}Q_{\text{enclosed}}\)
derived from
\(\boldsymbol{E}(\boldsymbol{r})=\frac{1}{4\pi\varepsilon_0}\int\frac{\rho(\boldsymbol{r}')}{\eta^2}\,\hat{\boldsymbol{\eta}}\,d\tau'\)
steady current 3D\(\Rightarrow\) magnetic field \(\boldsymbol{B}\) is
not a sink/source
\(\boldsymbol{\nabla}\cdot\boldsymbol{B}=0\)
\(\oint\boldsymbol{B}\cdot d\boldsymbol{a}=0\)
derived from
\(B(r)=\frac{\mu_0}{4\pi}\int\frac{J(r')\times\hat{\eta}}{\eta^2}\,d\tau'\)
rotating
\(\boldsymbol{\nabla}\times\boldsymbol{B}=\mu_0\boldsymbol{J}\)
\(\oint\boldsymbol{B}\cdot d\boldsymbol{l}=\mu_0I_{\text{enclosed}}\)
what is steady current?
\(\frac{\partial\boldsymbol{J}}{\partial t}=\boldsymbol{0}\)
\(\frac{\partial p}{\partial t}=0\)
what is magnetic field?
A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field.
magnetic extras
current density
surface
\(\boldsymbol{F}=\int\,(\boldsymbol{K}\times\boldsymbol{B})\,da\)
volume
\(\boldsymbol{F}=\int\,(\boldsymbol{J}\times\boldsymbol{B})\,d\tau\)
line
\(\boldsymbol{F}=\int\,(\boldsymbol{I}\times\boldsymbol{B})\,dl\)
amount of charge per second that flows through unit area of a chosen cross section.
continuity equation
\(\boldsymbol{\nabla}\cdot\boldsymbol{J}=-\frac{\partial p}{\partial t}\)
torque
\(\boldsymbol{N}=\boldsymbol{m}\times\boldsymbol{B}\)
force
\(\boldsymbol{F}=\boldsymbol{\nabla}(\boldsymbol{m}\cdot\boldsymbol{B})\)
auxillery field
\(\boldsymbol{H}=\frac{1}{\mu_0}\boldsymbol{B}-\boldsymbol{M}\)
free current \(\boldsymbol{J}_f\)
bound current \(\boldsymbol{J}_b\)
magnetization \(\boldsymbol{M}\)
material is either paramagnet, diamagnets or ferromagnets.
diamagnet
paramagnet
ferromagnet
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.
electric extras
electric suspectibility \(\chi_e\)
the degree of polarization of a dielectric material in response to an applied electric field.
\(\boldsymbol{P}=\varepsilon_0\chi_e\boldsymbol{E}\)
polarization \(\boldsymbol{P}\)
\(\boldsymbol{P}=\frac{\boldsymbol{p}}{V}\)
relative permitivity \(\varepsilon_r\)
\(\varepsilon_r=\frac{\varepsilon}{\varepsilon_0}=1+\chi_e\)
the factor by which an electric field is reduced inside a material relative to vacuum
displacement \(\boldsymbol{D}\)
\(\boldsymbol{D}=\varepsilon_0\boldsymbol{E}+\boldsymbol{P}\)
\(\boldsymbol{\nabla}\times\boldsymbol{D}=\boldsymbol{\nabla}\times\boldsymbol{P}\)
dipole moment \(\boldsymbol{p}\)
\(\boldsymbol{p}=\alpha\boldsymbol{E}\)
\(\boldsymbol{F}=(\boldsymbol{p}\cdot\boldsymbol{\nabla})\boldsymbol{E}\)
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
electrons that are in atoms and molecules, they cannot move.
\(\rho_b=-\boldsymbol{\nabla}\cdot\boldsymbol{P}\)
\(\sigma_b=\boldsymbol{P}\cdot\hat{\boldsymbol{n}}\)
free charges
electrons that can move freely, not bound
\(\rho_f=\boldsymbol{\nabla}\cdot\boldsymbol{D}\)
\(\sigma_f=\boldsymbol{D}\cdot\hat{\boldsymbol{n}}\)
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.
math details
divergence theorem
\(\oint\boldsymbol{A}\cdot d\boldsymbol{a}=\int (\boldsymbol{\nabla}\cdot\boldsymbol{A})\,d\tau\)
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
\(\boldsymbol{A}\cdot(\boldsymbol{B}+\boldsymbol{C})=\boldsymbol{A}\cdot\boldsymbol{B}+\boldsymbol{A}\cdot\boldsymbol{C}\)
\(\boldsymbol{A}\times(\boldsymbol{B}+\boldsymbol{C})=\boldsymbol{A}\times\boldsymbol{B}+\boldsymbol{A}\times\boldsymbol{C}\)
notation
source point \(\boldsymbol{r}'\), where electric charge is located
field point \(\boldsymbol{r}'\), where electric charge is located
\(d\boldsymbol{l}=dx\,\hat{\boldsymbol{x}}+dy\,\hat{\boldsymbol{y}}+dz\,\hat{\boldsymbol{z}}\)
\(\boldsymbol{\eta}=\boldsymbol{r}-\boldsymbol{r}'\)
\(\boldsymbol{\nabla}=\frac{\partial}{\partial x}\hat{\boldsymbol{x}}+\frac{\partial}{\partial y}\hat{\boldsymbol{y}}+\frac{\partial}{\partial z}\hat{\boldsymbol{z}}\)
\(d\tau=dx\,dy\,dz\)
\(d\boldsymbol{a}=\boldsymbol{n}da\)