שדה של דיפול מגנטי מתוך לולאה ריבועית

\vec{B}_{0\hat{n}}=\frac{\mu_{0}}{4\pi}\frac{Id\hat{n}\times\vec{r}}{r^{3}}

\vec{B}=\vec{B}_{0\hat{x}}\left(\vec{r}\right)-\vec{B}_{0\hat{x}}\left(\vec{r}-d_{y}\hat{y}\right)-\vec{B}_{0\hat{y}}\left(\vec{r}\right)+\vec{B}_{0\hat{y}}\left(\vec{r}+d_{x}\hat{x}\right)

\vec{B}=\frac{\mu_{0}Id_{x}d_{y}}{4\pi}\left(\hat{x}\times\frac{d}{dy}\frac{\vec{r}}{r^{3}}-\hat{y}\times\frac{d}{dx}\frac{\vec{r}}{r^{3}}\right)=\frac{\mu_{0}Id_{x}d_{y}}{4\pi}\left(\hat{x}\times\left(\frac{\hat{y}}{r^{3}}-\frac{3y\vec{r}}{r^{5}}\right)-\hat{y}\times\left(\frac{\hat{x}}{r^{3}}-\frac{3x\vec{r}}{r^{5}}\right)\right)

\vec{B}=\frac{\mu_{0}m}{4\pi}\left(\frac{2\hat{z}}{r^{3}}-\frac{3y\left(y\hat{z}-z\hat{y}\right)}{r^{5}}+\frac{3x\left(z\hat{x}-x\hat{z}\right)}{r^{5}}\right)=\frac{\mu_{0}m}{4\pi}\left(\frac{\left(2r^{2}-3y^{2}-3x^{2}\right)\hat{z}+3z\left(x\hat{x}+y\hat{y}\right)}{r^{5}}\right)

\vec{B}=\frac{\mu_{0}m}{4\pi}\left(\frac{-r^{2}\hat{z}+3z\vec{r}}{r^{5}}\right)=\frac{\mu_{0}}{4\pi}\frac{3\left(\vec{m}\cdot\hat{r}\right)\hat{r}-\vec{m}}{r^{3}}