Talk:MyOpenMath/Solutions/Maxwell's integral equations

https://en.wikiversity.org/w/index.php?title=Talk:MyOpenMath/Solutions/Maxwell%27s_integral_equations&oldid=2174778

Spherical
$$\text{If }r>R: \varepsilon_0\underbrace{\oint\vec E\cdot d\vec A}_{4\pi r^2E}= \underbrace{\int\rho d\tau}_{\frac{4\pi R^3}{3}\rho_0}$$

$$\text{If }rR:\varepsilon_0\underbrace{\oint\vec E\cdot d\vec A}_{2\pi rLE}= \underbrace{\rho_0\int d\tau}_{\pi R^2L\rho_0}$$

$$\text{If } r<R:\varepsilon_0\underbrace{\oint\vec E\cdot d\vec A}_{2\pi r LE}= \underbrace{\rho_0\int d\tau}_{\pi r^2 L\rho_0}$$

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

 * $$\text{If } r>R: \underbrace{\oint\vec B\cdot d\vec l}_{2\pi r B_\theta}=\mu_0I $$
 * $$\text{If } r<R: \underbrace{\oint\vec B\cdot d\vec l}_{2\pi r B_\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
$$\underbrace{\oint\vec B\cdot d\vec l}_{B_zL}=\mu_0 NI$$

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
$$\underbrace{\oint\vec B\cdot d\vec l}_{B_\theta 2\pi r}=\mu_0 N I$$