# 電荷密度

## 古典電荷密度

$Q=\rho_0 V$

$\rho(\mathbf{r})=\sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i)$

## 量子電荷密度

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

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

$\psi_{nlm}(\mathbf{r})=R_{nl}(r)Y_l^m(\theta,\,\phi)$

## 電荷守恆的連續方程式

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

$\nabla \times \mathbf{B}=\mu_0\mathbf{J}+\mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}$

$\nabla\cdot(\nabla \times \mathbf{B}) =\mu_0\nabla\cdot\mathbf{J}+\mu_0\epsilon_0\frac{\partial }{\partial t}(\nabla\cdot\mathbf{E})$

$0=\nabla\cdot\mathbf{J}+\epsilon_0\frac{\partial }{\partial t}(\nabla\cdot\mathbf{E})=\nabla\cdot\mathbf{J}+\frac{\partial \rho}{\partial t}$

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

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

$Q=\int_\mathbb{V} \rho\ \mathrm{d}^3r$

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

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

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

## 電勢和電場

$\phi(\mathbf{r}) = \frac{1}{4 \pi \epsilon_0}\int_{\mathbb{V}'} \frac{\rho(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} \mathrm{d}^3{r}'$

$\mathbf{E}(\mathbf{r})= - \boldsymbol{\nabla} \phi(\mathbf{r}) = \frac{1}{4 \pi \epsilon_0}\int_{\mathbb{V}'} \rho(\mathbf{r}')\frac{\mathbf{r} - \mathbf{r}'}{|\mathbf{r} - \mathbf{r}'|^3} \mathrm{d}^3{r}'$

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

$\boldsymbol{\nabla} \cdot \mathbf{E}(\mathbf{r})= - \nabla^2 \phi(\mathbf{r}) = \frac{1}{4 \pi \epsilon_0}\int_{\mathbb{V}'} \rho(\mathbf{r}')4\pi \delta(\mathbf{r} - \mathbf{r}') \mathrm{d}^3{r}'$

$\boldsymbol{\nabla} \cdot \mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})}{\epsilon_0}$
$\nabla^2 \phi(\mathbf{r}) = - \frac{\rho(\mathbf{r})}{\epsilon_0}$

## 參考文獻

1. ^ Cao, Tian Yu, Conceptual developments of 20th century field theories. reprint, illustrated, Cambridge University Press. 1998:  pp. 146-147, ISBN 9780521634205
2. ^ A. French (1968) Special Relativity, chapter 8 Relativity and electricity, pp 229–65, W. W. Norton.
3. ^ 3.0 3.1 Jackson, John David, Classical Electrodynamic. 3rd., USA: John Wiley & Sons, Inc.. 1999:  pp. 29-31, 237-239, ISBN 978-0-471-30932-1
4. ^ Griffiths, David J., Introduction to Electrodynamics (3rd ed.), Prentice Hall. 1998:  pp. 213, ISBN 0-13-805326-X
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