# 電流密度

## 定義

${\displaystyle J=\lim \limits _{A\rightarrow 0}{\frac {I(A)}{A}}}$

${\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}{\mathbf {J} }\cdot {\mathbf {\hat {n}} }{\rm {d}}A{\rm {d}}t}$

## 計算電流密度

### 自由電流

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

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

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

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

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

## 穿過曲面的電流

${\displaystyle I=\int _{\mathbb {S} }{\mathbf {J} \cdot \mathrm {d} \mathbf {a} }}$

## 連續方程式

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

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

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

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

## 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