שדה ופוטנציאל של דיפול חשמלי

V_{dip}\left(\vec{r}\right)=V\left(\vec{r}-d\hat{z}\right)-V\left(\vec{r}\right)\approx-d\cdot\frac{d}{dz}V=-\frac{qd}{4\pi\varepsilon_{0}}\frac{dr}{dz}\frac{d}{dr}\left(\frac{1}{r}\right)=\frac{p\cos\theta}{4\pi\varepsilon_{0}r^{2}}

\begin{align*} \vec{E}_{dip}\left(r\right) & =-d\cdot\frac{d}{dz}\vec{E}=-\frac{qd}{4\pi\varepsilon_{0}}\frac{d}{dz}\left(\frac{\vec{r}}{r^{3}}\right)=-\frac{qd}{4\pi\varepsilon_{0}}\left(\frac{d\vec{r}}{dz}\cdot\frac{1}{r^{3}}+\vec{r}\frac{dr}{dz}\frac{d}{dr}\left(\frac{1}{r^{3}}\right)\right)\\ & =-\frac{qd}{4\pi\varepsilon_{0}}\left(\frac{\hat{z}}{r^{3}}-\frac{3\vec{r}}{r^{4}}\cos\theta\right)=\frac{3\left(\vec{p}\cdot\hat{r}\right)\hat{r}-\vec{p}}{4\pi\varepsilon_{0}r^{3}} \end{align*}