# 電荷密度

## 古典電荷密度

${\displaystyle Q=\rho _{0}V}$

${\displaystyle \rho (\mathbf {r} )=\sum _{i=1}^{N}\ q_{i}\delta (\mathbf {r} -\mathbf {r} _{i})}$

## 量子電荷密度

${\displaystyle \rho (\mathbf {r} )=q\cdot |\psi (\mathbf {r} )|^{2}}$

${\displaystyle \int _{all\ space}|\psi (\mathbf {r} )|^{2}\mathrm {d} ^{3}{r}=1}$

${\displaystyle \psi _{nlm}(\mathbf {r} )=R_{nl}(r)Y_{l}^{m}(\theta ,\,\phi )}$

## 電荷守恆的連續方程式

${\displaystyle {\frac {\partial \rho (\mathbf {r} ,\,t)}{\partial t}}+{\boldsymbol {\nabla }}\cdot \mathbf {J} (\mathbf {r} ,\,t)=0}$

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$

${\displaystyle \nabla \cdot (\nabla \times \mathbf {B} )=\mu _{0}\nabla \cdot \mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial }{\partial t}}(\nabla \cdot \mathbf {E} )}$

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

${\displaystyle I=-\oint _{\mathbb {S} }\mathbf {J} \cdot \mathrm {d} ^{2}\mathbf {r} }$

${\displaystyle I=-\int _{\mathbb {V} }\nabla \cdot \mathbf {J} \ \mathrm {d} ^{3}r}$

${\displaystyle Q=\int _{\mathbb {V} }\rho \ \mathrm {d} ^{3}r}$

${\displaystyle {\frac {\mathrm {d} Q}{\mathrm {d} t}}=I=\int _{\mathbb {V} }{\frac {\partial \rho }{\partial t}}\ \mathrm {d} ^{3}r}$

${\displaystyle \int _{\mathbb {V} }{\frac {\partial \rho }{\partial t}}+\mathbf {\nabla } \cdot \mathbf {J} \ \mathrm {d} ^{3}r=0}$

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0}$

## 電勢和電場

${\displaystyle \phi (\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\int _{\mathbb {V} '}{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}{r}'}$

${\displaystyle \mathbf {E} (\mathbf {r} )=-{\boldsymbol {\nabla }}\phi (\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\int _{\mathbb {V} '}\rho (\mathbf {r} '){\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}\mathrm {d} ^{3}{r}'}$

${\displaystyle {\boldsymbol {\nabla }}\cdot {\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}=4\pi \delta (\mathbf {r} -\mathbf {r} ')}$

${\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {E} (\mathbf {r} )=-\nabla ^{2}\phi (\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\int _{\mathbb {V} '}\rho (\mathbf {r} ')4\pi \delta (\mathbf {r} -\mathbf {r} ')\mathrm {d} ^{3}{r}'}$

${\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} )}{\epsilon _{0}}}}$
${\displaystyle \nabla ^{2}\phi (\mathbf {r} )=-{\frac {\rho (\mathbf {r} )}{\epsilon _{0}}}}$

## 參考文獻

1. ^ Cao, Tian Yu, Conceptual developments of 20th century field theories reprint, illustrated, Cambridge University Press: pp. 146–147, 1998, ISBN 9780521634205
2. ^ A. French (1968) Special Relativity, chapter 8 Relativity and electricity, pp 229–65, W. W. Norton.
3. Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc.: pp. 29–31, 237–239, 1999, ISBN 978-0-471-30932-1
4. ^ Griffiths, David J., Introduction to Electrodynamics (3rd ed.), Prentice Hall: pp. 213, 1998, ISBN 0-13-805326-X