基爾霍夫電路定律

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