שדה ופוטנציאל של קוודרופול חשמלי

V_{quad}\left(\vec{r}\right)=V_{dip}\left(\vec{r}\right)-V_{dip}\left(\vec{r}-d\hat{z}\right)=-d\frac{d}{dz}\left(\frac{qdz}{4\pi\varepsilon_{0}r^{3}}\right)=-\frac{qd^{2}}{4\pi\varepsilon_{0}}\left(\frac{1}{r^{3}}-\frac{3z\cdot z/r}{r^{4}}\right)=\frac{qd^{2}}{4\pi\varepsilon_{0}}\frac{3\cos^{2}\theta-1}{r^{3}}

E_{quad,r}=-\frac{\partial V_{quad}}{\partial r}=\frac{3qd^{2}}{4\pi\varepsilon_{0}}\frac{3\cos^{2}\theta-1}{r^{4}}

E_{quad,\theta}=-\frac{1}{r}\frac{\partial V_{quad}}{\partial\theta}=\frac{3qd^{2}}{2\pi\varepsilon_{0}}\frac{\cos\theta\sin\theta}{r^{4}}

\begin{align*} \vec{E}_{quad}\left(r\right) & =-d\cdot\frac{d}{dz}\vec{E}_{dip}=-\frac{qd^{2}}{4\pi\varepsilon_{0}}\frac{d}{dz}\left(\frac{\hat{z}}{r^{3}}-\frac{3z\vec{r}}{r^{5}}\right)=-\frac{qd^{2}}{4\pi\varepsilon_{0}}\left(-\frac{3z\hat{z}}{r^{5}}-\frac{3\vec{r}}{r^{5}}-\frac{3z\hat{z}}{r^{5}}+\frac{15z^{2}\vec{r}}{r^{7}}\right)\\ & =\frac{3qd^{2}}{4\pi\varepsilon_{0}}\left(\frac{2z\hat{z}}{r^{5}}+\frac{\vec{r}\left(1-5\cos^{2}\theta\right)}{r^{5}}\right)=\frac{3qd^{2}}{4\pi\varepsilon_{0}}\left(\frac{\left(1-5\cos^{2}\theta\right)\hat{r}+2\cos\theta\hat{z}}{r^{4}}\right) \end{align*}