# 電流密度

## 定義

$J = \lim\limits_{A \rightarrow 0}\frac{I(A)}{A}$

$q=\int_{t_1}^{t_2}\iint_S \bold{J}\cdot\bold{\hat{n}}{\rm d}A{\rm d}t$

## 計算電流密度

### 自由電流

$\mathbf{J}(\mathbf{r}, t) = qn(\mathbf{r},t)\; \mathbf{v}_d(\mathbf{r},t) = \rho(\mathbf{r},t) \; \mathbf{v}_d(\mathbf{r},t)$

$\mathbf{J}=\sigma\mathbf{E}$

$\mathbf{J}(\mathbf{r}, t) = \int_{-\infty}^t \mathrm{d}t' \int \mathrm{d}^3 r' \; \sigma(\mathbf{r}-\mathbf{r}', t-t') \; \mathbf{E}(\mathbf{r}',\ t')$

$\mathbf{J}(\mathbf{r}, t) = \int_{-\infty}^{\infty} \mathrm{d}t' \int \mathrm{d}^3 r' \; \sigma(\mathbf{r}-\mathbf{r}', t-t') \; \mathbf{E}(\mathbf{r}',\ t')$

$\mathbf{J}(\mathbf{k}, \omega) = \sigma(\mathbf{k}, \omega) \; \mathbf{E}(\mathbf{k}, \omega)$

## 穿過曲面的電流

$I=\int_\mathbb{S}{ \mathbf{J} \cdot \mathrm{d}\mathbf{a}}$

## 連續方程式

$\int_\mathbb{S}{ \mathbf{J} \cdot \mathrm{d}\mathbf{a}} = -\frac{\mathrm{d}}{\mathrm{d}t} \int_\mathbb{V}{\rho \ \mathrm{d}r^3} = -\ \int_\mathbb{V}{\left( \frac{\partial \rho}{\partial t} \right) \mathrm{d}r^3}$

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

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

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

## 參考文獻

1. ^ Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1
2. ^ Richard P Martin, Electronic Structure:Basic theory and practical methods, Cambridge University Press, pp. 369ff, 2004, ISBN 0521782856
3. ^ Anthony C. Fischer-Cripps, The electronics companion, CRC Press, pp. 13, 2004, ISBN 9780750310123
4. ^ Jørgen Rammer, Quantum Field Theory of Non-equilibrium States, Cambridge University Press, pp. 158ff, 2007, ISBN 9780521874991
5. ^ Griffiths, D.J., Introduction to Electrodynamics 3rd Edition, Pearson/Addison-Wesley, pp. 213, 1999, ISBN 013805326X