静电场边值问题
$$\varphi\overset{点电荷}{=}\frac{1}{4\pi\varepsilon_0}\sum_i\frac{q_i}{R_i}\\
\frac{1}{R}=\frac{1}{r}+\boldsymbol{r}’\cdot\nabla’\frac{1}{R}|_{r’=0}+\frac{1}{2}\boldsymbol{r}’\boldsymbol{r}’:\nabla’\nabla’\frac{1}{R}|_{r’=0}+\cdots\\
=\frac{1}{r}-\boldsymbol{r}’\cdot\nabla\frac{1}{r}+\frac{1}{2}\boldsymbol{r}’\boldsymbol{r}’:\nabla\nabla\frac{1}{r}+\cdots\\
\varphi\overset{r>>r’}{=}\frac{1}{4\pi\varepsilon_0}\left[(\sum_i q_i)\frac{1}{r}+(-\sum_iq_i\boldsymbol{r}_i)\cdot\nabla\frac{1}{r}+(\frac{1}{2}\sum_iq_i\boldsymbol{r}_i\boldsymbol{r}_i):\nabla\nabla\frac{1}{r}\right]\\=\frac{1}{4\pi\varepsilon_0}\left[(\sum_i q_i)\frac{1}{r}+(\sum_iq_i\boldsymbol{r}_i)\cdot\frac{\boldsymbol{r}}{r^3}+(\frac{1}{2}\sum_iq_i\boldsymbol{r}_i\boldsymbol{r}_i):\frac{3\boldsymbol{rr}-r^2\mathbb{I}}{r^5}\right]\\
=\frac{1}{4\pi\varepsilon_0}\left[\frac{\sum q_i}{r}+\frac{\sum q_i\boldsymbol{r}_i\cdot\boldsymbol{r}}{r^3}+\frac{1}{2}\frac{\sum q_i[3(\boldsymbol{r}_i\cdot\boldsymbol{r})^2-r_i^2r^2]}{r^5}\right]$$