學院物理/電容與介電質

定義

$C=\frac{Q}{V}\begin{array}{c}\begin{array}{c}(\end{array}\frac{Col}{Volt}\begin{array}{c}=Faraday)\end{array}\end{array}=\frac{\sigma A}{V=\sigma d/ϵ}\because V=Ed=\frac{\sigma d}{ϵ}$

例題

一

圓柱電容

$\begin{array}{cc}& V=-\int Edr=-\int 2k\lambda rdr=-2k\lambda \int \frac{1}{r}dr=-2k\lambda \ln r\\ & V=\underset{a}{\overset{b}{\int }}\frac{2k\lambda }{r}dr=2k\lambda [\ln r{]}_a^b=2k\lambda \ln \frac{b}{a}\\ & C=\frac{Q}{V}=\frac{l}{2k\ln \frac{b}{a}}\end{array}$

示意圖

V

r

ln r

- ln r

$ab$

二

圓球電容

$\begin{array}{cc}& V=\underset{a}{\overset{b}{\int }}\frac{kQ}{r^2}dr=-kQ{\left[\frac{1}{r}\right]}_a^b=kQ(\frac{1}{a}-\frac{1}{b})\\ & C=\frac{Q}{V}=\frac{1}{k}\frac{ab}{b-a}\end{array}$