# 基爾霍夫電路定律

（重定向自基尔霍夫第一定律

## 基爾霍夫電流定律

${\displaystyle \sum _{k=1}^{n}i_{k}=0}$

### 導引

${\displaystyle i=\sum _{k=1}^{n}i_{k}}$

${\displaystyle q=\sum _{k=1}^{n}q_{k}}$

${\displaystyle \sum _{k=1}^{n}i_{k}=0}$

### 含時電荷密度

${\displaystyle \nabla \cdot \mathbf {J} =-\epsilon _{0}\nabla \cdot {\frac {\partial \mathbf {E} }{\partial t}}=-{\frac {\partial \rho }{\partial t}}}$

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

## 基爾霍夫電壓定律

${\displaystyle \sum _{k=1}^{m}v_{k}=0}$

### 電場與電勢

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

${\displaystyle \oint _{\mathbb {C} }\mathbf {E} \cdot d\mathbf {l} =0}$

## 頻域

${\displaystyle \sum _{k=1}^{n}i_{k}=\sum _{k=1}^{n}I_{k}\cos(\omega t+\theta _{k})=\mathrm {Re} {\Big \{}\sum _{k=1}^{n}I_{k}e^{j(\omega t+\theta _{k})}{\Big \}}=\mathrm {Re} {\Big \{}\left(\sum _{k=1}^{n}I_{k}e^{j\theta _{k}}\right)e^{j\omega t}{\Big \}}=0}$

${\displaystyle \sum _{k=1}^{n}\mathbb {I} _{k}=0}$

${\displaystyle \sum _{k=1}^{m}\mathbb {V} _{k}=0}$

## 參考

1. Alexander, Charles; Sadiku, Matthew, Fundamentals of Electric Circuits 3, revised, McGraw-Hill: pp. 37–43, 2006, ISBN 9780073301150
2. ^ 普通物理学(修订版)（化学数学专业用）.汪昭义 主编.华东师范大学出版社.P320.9.3 基尔霍夫定律.ISBN 978-8-5617-0444-8
• Paul, Clayton R. Fundamentals of Electric Circuit Analysis. John Wiley & Sons. 2001. ISBN 978-0-471-37195-3.
• Serway, Raymond A.; Jewett, John W. Physics for Scientists and Engineers (6th ed.). Brooks/Cole. 2004. ISBN 978-0-534-40842-8.
• Tipler, Paul. Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. 2004. ISBN 978-0-7167-0810-0.