Electrostatic Field

电偶极子

电势

$$\varphi (\mathbf{r})=\frac{1}{4\pi {\epsilon }_0}\frac{\mathbf{p}\cdot \mathbf{r}}{r^3}=\frac{1}{4\pi {\epsilon }_0}\frac{p\cos \theta }{r^2}$$

在球坐标系中的电场

$$\begin{array}{rl}{E}_r& =-\frac{\partial \varphi }{\partial r}=\frac{1}{4\pi {\epsilon }_0}\frac{2p\cos \theta }{r^3}\\ E_{\theta }& =-\frac{1}{r}\frac{\partial \varphi }{\partial \theta }=\frac{1}{4\pi {\epsilon }_0}\frac{p\sin \theta }{r^3}\\ E_{\phi }& =-\frac{1}{r\sin \theta }\frac{\partial \varphi }{\partial \phi }=0\end{array}$$

能量

$$E_p=-\mathbf{p}\cdot \mathbf{E}$$

受力矩

$$\mathbf{M}=-\frac{\partial E_p}{\partial \theta }\hat{\mathbf{\theta }}=-\mathbf{p}\times \mathbf{E}$$

受梯度力

$$\mathbf{F}=-\nabla E_p=\nabla (\mathbf{p}\cdot \mathbf{E})$$

静电能

离散体系

$$W_e=\frac{1}{2}\sum _i{q}_i{\phi }_i$$

连续体系

$$W_e=\frac{1}{2}{\int }_V{\rho }_e\phi \mathrm{d}V$$

或者

$$W_e={\int }_V{w}_e\mathrm{d}V=\frac{{\epsilon }_r{\epsilon }_0}{2}{\int }_V{E}^2\mathrm{d}V$$