# 欧姆定律

${\displaystyle V\propto I}$

${\displaystyle R\ {\stackrel {def}{=}}\ {\frac {V}{I}}}$

${\displaystyle V=IR}$

${\displaystyle \mathbf {E} =\rho \mathbf {J} }$
${\displaystyle \mathbf {J} =\sigma \mathbf {E} }$

## 歷史

${\displaystyle \Delta I=m\ln \left(1+{\frac {x}{a}}\right)}$

${\displaystyle X={\frac {a}{b+x}}}$

${\displaystyle I={\frac {V}{R}}}$

1839年，法國物理學家，克勞德·普雷特Claude Pouillet）確定歐姆的實驗結果。同時，歐姆成為柏林科學院的院士。在英國，查爾斯·惠斯通又重新核對了歐姆的實驗結果。1841年，歐姆被選為皇家學會的外籍會員。1852年，歐姆榮膺為慕尼黑大學的物理學系主任。

## 熱力學類比

${\displaystyle \mathbf {J} =-\sigma \nabla V}$

${\displaystyle {\boldsymbol {\Gamma }}=-k\nabla T}$

## 電路分析

### 串聯電阻電路

${\displaystyle R_{\mathrm {total} }=R_{1}+R_{2}+\cdots +R_{n}}$

### 並聯電阻電路

${\displaystyle {1 \over R_{\mathrm {total} }}={1 \over R_{1}}+{1 \over R_{2}}+\cdots +{1 \over R_{n}}}$

### 週期性激發

${\displaystyle \sin {\omega t}={\frac {1}{2j}}(e^{j\omega t}-e^{-j\omega t})}$
${\displaystyle \cos {\omega t}={\frac {1}{2}}(e^{j\omega t}+e^{-j\omega t})}$

${\displaystyle {Z}=R}$

${\displaystyle {Z}=j\omega L}$

${\displaystyle {Z}=1/j\omega C}$

${\displaystyle {V}={I}{Z}}$

${\displaystyle {I}={I}_{0}e^{j\omega t}}$
${\displaystyle {V}={V}_{0}e^{j\omega t}}$

### 線性近似

${\displaystyle {\mathfrak {r}}\ {\stackrel {def}{=}}\ {\frac {\mathrm {d} V}{\mathrm {d} I}}}$

## 其它版本的歐姆定律

${\displaystyle \mathbf {E} =\rho \mathbf {J} }$
${\displaystyle \mathbf {J} =\sigma \mathbf {E} }$

${\displaystyle V_{gh}{\stackrel {def}{=}}\ {\frac {\mathrm {d} w}{\mathrm {d} q}}=\int _{g}^{h}{\mathbf {E} \cdot \mathrm {d} \mathbf {l} }=\rho \int _{g}^{h}{\mathbf {J} \cdot \mathrm {d} \mathbf {l} }}$

${\displaystyle \mathrm {d} \mathbf {l} =\mathrm {d} l{\hat {\mathbf {l} }}}$

${\displaystyle V_{gh}=J\rho l}$

${\displaystyle J=I/a}$

${\displaystyle V=V_{gh}=I\rho l/a}$

${\displaystyle R=\rho l/a}$

## 經典微觀表述

${\displaystyle \mathbf {F} _{ave}=-e\mathbf {E} _{ave}}$

${\displaystyle \mathbf {v} _{d}=-e\mathbf {E} _{ave}\tau /m}$

${\displaystyle \mathbf {J} =-ne\mathbf {v} _{d}=ne^{2}\tau \mathbf {E} _{ave}/m}$

${\displaystyle \rho ={\frac {m}{ne^{2}\tau }}}$

${\displaystyle \mathbf {E} =\rho \mathbf {J} }$

${\displaystyle {\frac {1}{2}}mv_{t}^{2}={\frac {3}{2}}k_{B}T}$

### 磁效應

${\displaystyle \mathbf {F} _{ave}=-e(\mathbf {E} _{ave}+\mathbf {v} _{0}\times \mathbf {B} _{ave})}$

${\displaystyle \mathbf {v} _{d}=-e(\mathbf {E} _{ave}+\mathbf {v} _{0}\times \mathbf {B} _{ave})\tau /m}$

${\displaystyle \mathbf {J} =-ne\mathbf {v} _{d}=ne^{2}\tau (\mathbf {E} _{ave}+\mathbf {v} _{0}\times \mathbf {B} _{ave})/m}$

${\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} =\rho \mathbf {J} }$

## 參考文獻

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