Talk:MyOpenMath/Solutions/Maxwell's integral equations

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Spherical

${\text{If }}r>R:\varepsilon _{0}\underbrace {\oint {\vec {E}}\cdot d{\vec {A}}} _{4\pi r^{2}E}=\underbrace {\int \rho d\tau } _{{\frac {4\pi R^{3}}{3}}\rho _{0}}$

${\text{If }}r<R:\varepsilon _{0}\underbrace {\oint {\vec {E}}\cdot d{\vec {A}}} _{4\pi r^{2}E}\quad =\underbrace {\int \rho d\tau } _{{\frac {4}{3}}\pi r^{3}\rho _{0}}$

Cylindrical

${\text{If }}r>R:\varepsilon _{0}\underbrace {\oint {\vec {E}}\cdot d{\vec {A}}} _{2\pi rLE}=\underbrace {\rho _{0}\int d\tau } _{\pi R^{2}L\rho _{0}}$

${\text{If }}r<R:\varepsilon _{0}\underbrace {\oint {\vec {E}}\cdot d{\vec {A}}} _{2\pi rLE}=\underbrace {\rho _{0}\int d\tau } _{\pi r^{2}L\rho _{0}}$

slab

$\varepsilon _{0}\underbrace {\oint {\vec {E}}\cdot d{\vec {A}}} _{A\left(E_{+}-E_{-}\right)}=\underbrace {\rho _{0}\int d\tau } _{Ab\rho _{0}=A\sigma },$

  where: $\left\{{\begin{array}{l l}{\vec {E}}_{+}=&E(x_{0}+L/2){\hat {x}}\\{\vec {E}}_{-}=&E(x_{0}-L/2){\hat {x}}\end{array}}\right.$

$\rightarrow \varepsilon _{0}{\frac {\partial E_{x}}{\partial x}}=\rho (x)$

cylindrical wire

${\text{If }}r>R:\underbrace {\oint {\vec {B}}\cdot d{\vec {l}}} _{2\pi rB_{\theta }}=\mu _{0}I$

${\text{If }}r<R:\underbrace {\oint {\vec {B}}\cdot d{\vec {l}}} _{2\pi rB_{\theta }}=\mu _{0}\underbrace {\int J_{0}dA} _{J_{0}\pi r^{2}}$

${\text{To find }}J_{0}:\;I=\underbrace {\int {\vec {J}}_{0}\cdot d{\vec {A}}} _{\pi R^{2}J_{0}}$

Solenoid

$\underbrace {\oint {\vec {B}}\cdot d{\vec {l}}} _{B_{z}L}=\mu _{0}NI$

Faraday

$\underbrace {\oint {\vec {E}}\cdot d{\vec {l}}} _{E_{\theta }2\pi r}=-{\frac {d}{dt}}\underbrace {\int {\vec {B}}\cdot d{\vec {A}}} _{BA}$

Maxwell

$\varepsilon _{0}\oint {\vec {E}}\cdot d{\vec {A}}=Q$

$I_{displacement}=\varepsilon _{0}{\frac {d}{dt}}\oint {\vec {E}}\cdot d{\vec {A}}$

$I_{conventional}=\int _{S}{\vec {J}}_{0}\cdot d{\vec {A}}$

$\oint _{\partial S}{\vec {B}}\cdot d{\vec {l}}=$

$\mu _{0}\left(\int _{S}{\vec {J}}\cdot d{\vec {A}}+\varepsilon _{0}{\frac {d}{dt}}\int _{S}{\vec {E}}\cdot d{\vec {A}}\right)$

toroid=

$\underbrace {\oint {\vec {B}}\cdot d{\vec {l}}} _{B_{\theta }2\pi r}=\mu _{0}NI$