Properties of a perfect conductor:
- No resistance $R=0$
- All points on or in a conductor have the same $\phi$
- $\Delta\phi=IR=I\times0=0$
- $\vec E=0$
- $\rho=0$
- All excess charges are on the surface
$$ I=\int_S \vec j \cdot \text d \vec A \; ; \; \vec j= \text{current dens.} $$
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/f9f676e5-dbb1-4bf8-baec-8c84a9be37e1/capacitor.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/f9f676e5-dbb1-4bf8-baec-8c84a9be37e1/capacitor.png" width="40px" /> A capacitor is a passive device that stores charge when connected to a potential
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$$ Q=C\Delta\phi $$
where the Capacitance $C$ is in Farads $\left ( F=CV^{-1}=C^2J^{-1}\right )$
$$ \begin{aligned} E&=\frac{\sigma}{\epsilon_0}=\frac{Q}{\epsilon_0 A} \\ \Delta\phi&=-\int^0_d \vec E \cdot \text d \vec l = \frac{Qd}{\epsilon_0 A} \\ C&=\frac{Q}{\Delta\phi}=\frac{\epsilon_0 A}{d} \end{aligned} $$
$$ \begin{aligned} \vec E&=\frac{\lambda}{2\pi r \epsilon_0}\,\hat r \\ \Delta\phi&=-\int^{r_a}_{r_b}\vec E\cdot\text d\vec l=\frac{\lambda}{2\pi\epsilon_0}\ln\left( \frac{r_b}{r_a} \right) \\ \frac{C}{l}&=\frac{Q}{\Delta\phi l}=\frac{\lambda l}{\frac{\lambda l}{2\pi\epsilon_0}\ln\left( \frac{r_b}{r_a} \right)}=\frac{2\pi\epsilon_0}{\ln\left(\frac{r_b}{r_a}\right)} \end{aligned} $$
$$ \begin{aligned} \vec E&=\frac{q}{4\pi\epsilon_0 r^2}\,\hat r \\ \Delta\phi&=-\int^{r_a}_{r_b}\vec E\cdot\text d\vec l = \frac{q}{4\pi\epsilon_0}\left( \frac{1}{r_a}-\frac{1}{r_b} \right)\\ &=\frac{q(r_b-r_a)}{4\pi\epsilon_0r_ar_b}\\ C&=\frac{q}{\Delta\phi}=\frac{4\pi\epsilon_0r_ar_b}{(r_b-r_a)} \end{aligned} $$
