Electrostatics — Gauss, Capacitors & Dielectrics (Medium)

Quick Recap — Capacitors and Dielectrics

• Parallel plate: $C=\dfrac{\varepsilon_0 A}{d}$; with a full dielectric $C\to KC$. Isolated sphere $C=4\pi\varepsilon_0 R$.
• Slab of thickness $t$ ($t<d$): $C=\dfrac{\varepsilon_0 A}{d-t+t/K}$ (position-independent). A conducting slab uses $K\to\infty$: $C=\dfrac{\varepsilon_0 A}{d-t}$.
• Two dielectrics side by side (parallel): $C=\dfrac{\varepsilon_0 A}{2d}(K_1+K_2)$; stacked (series): $C=\dfrac{2\varepsilon_0 A}{d}\cdot\dfrac{K_1K_2}{K_1+K_2}$.
• Isolated ($Q$ fixed): raising $d$ raises $U$ ($U=\dfrac{Q^2 d}{2\varepsilon_0 A}$); inserting $K$ lowers $U$ to $U/K$. Battery-connected ($V$ fixed): raising $d$ lowers $U$; inserting $K$ raises $U$ to $KU$.
• Energy density $u=\tfrac12\varepsilon_0 E^2$; electrostatic pressure on a plate $=\dfrac{\sigma^2}{2\varepsilon_0}$.

Worked mini-example. $A=100$ cm$^2=10^{-2}$ m$^2$, $d=1$ mm: $C=\dfrac{(8.85\times10^{-12})(10^{-2})}{10^{-3}}=88.5$ pF; adding a $K=4$ dielectric makes it $354$ pF.