# 坡印亭定理

$\frac{\partial u}{\partial t} + \nabla\cdot\mathbf{S} = -\mathbf{J}\cdot\mathbf{E}$

$u = \frac{1}{2}\left(\epsilon_0 \mathbf{E}^2 + \frac{\mathbf{B}^2}{\mu_0}\right).$

$\frac{\partial}{\partial t} \int_V u \ dV + \oint_{\partial V}\mathbf{S} \ d\mathbf{A} = -\int_V\mathbf{J}\cdot\mathbf{E} \ dV$

$\nabla\cdot\mathbf{S} + \epsilon_0 \mathbf{E}\cdot\frac{\partial \mathbf{E}}{\partial t} + \frac{\mathbf{B}}{\mu_0}\cdot\frac{\partial\mathbf{B}}{\partial t} + \mathbf{J}\cdot\mathbf{E} = 0$

## 推导

$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$

$\mathbf{B} \cdot (\nabla \times \mathbf{E}) = - \mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t}$

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

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

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

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

$\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B}$

$\nabla\cdot\mathbf{S} + \epsilon_0 \mathbf{E}\cdot\frac{\partial \mathbf{E}}{\partial t} + \frac{\mathbf{B}}{\mu_0}\cdot\frac{\partial\mathbf{B}}{\partial t} + \mathbf{J}\cdot\mathbf{E} = 0$

## 参考文献

1. ^ Poynting, J. H. On the Transfer of Energy in the Electromagnetic Field. Phil. Trans. 1884, 175: 277.