# 克希荷夫電路定律

## 克希荷夫電流定律

$\sum_{k=1}^n i_k =0$

### 導引

$i=\sum_{k=1}^n i_k$

$q=\sum_{k=1}^n q_k$

$\sum_{k=1}^n i_k =0$

### 含時電荷密度

$\nabla \cdot \mathbf{J} = -\epsilon_0\nabla \cdot \frac{\partial \mathbf{E}}{\partial t} = -\frac{\partial \rho}{\partial t}$

$\oint_{\mathbb{S}}\mathbf{J}\cdot \mathrm{d}\mathbf{a} = -\frac{ \mathrm{d} Q}{ \mathrm{d} t}$

## 克希荷夫電壓定律

$\sum_{k=1}^m v_k = 0$

### 電場與電勢

$\phi(\mathbf{r})\stackrel{def}{=} - \int_\mathbb{L} \mathbf{E} \cdot \mathrm{d} \boldsymbol{\ell}\,\!$

$\oint_{\mathbb{C}} \mathbf{E} \cdot d\mathbf{l} = 0$

## 頻域

$\sum_{k=1}^n i_k =\sum_{k=1}^n I_k\cos(\omega t+\theta_k)=\mathrm{Re}\Big\{\sum_{k=1}^n I_k e^{j(\omega t + \theta_k)} \Big\}=\mathrm{Re}\Big\{\left(\sum_{k=1}^n I_k e^{j\theta_k} \right)e^{j\omega t}\Big\}=0$

$\sum_{k=1}^n\mathbb{I}_k =0$

$\sum_{k=1}^m \mathbb{V}_k = 0$

## 參考

1. ^ 1.0 1.1 Alexander, Charles; Sadiku, Matthew, Fundamentals of Electric Circuits 3, revised, McGraw-Hill, pp. 37–43, 2006, ISBN 9780073301150
2. ^ 普通物理學(修訂版)（化學數學專業用）.汪昭義 主編.華東師範大學出版社.P320.9.3 克希荷夫定律.ISBN 978-8-5617-0444-8/N·018
• Paul, Clayton R. Fundamentals of Electric Circuit Analysis. John Wiley & Sons. 2001. ISBN 0-471-37195-5.
• Serway, Raymond A.; Jewett, John W. Physics for Scientists and Engineers (6th ed.). Brooks/Cole. 2004. ISBN 0-534-40842-7.
• Tipler, Paul. Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. 2004. ISBN 0-7167-0810-8.

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