## Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

## magnetic dipole mo

Any system possessing a net magnetic dipole moment m will produce a dipolar magnetic field (described below) in the space surrounding the system. While the net magnetic field produced by the system can also have higher-order multipole components, those will drop off with distance more rapidly, so that only the dipolar component will dominate the magnetic field of the system at distances far away from it.

The vector potential of magnetic field produced by magnetic moment m is

${mathbf{A}}({mathbf{r}})=frac{mu_{0}}{4pi}frac{{mathbf{m}} imes{mathbf{r}}}{r^{3}},$
$mathbf{B}({mathbf{r}})= abla imes{mathbf{A}}=frac{mu_{0}}{4pi}left(frac{3mathbf{r}(mathbf{m}cdotmathbf{r})}{r^{5}}-frac{{mathbf{m}}}{r^{3}} ight).$

Alternatively one can obtain the scalar potential first from the magnetic pole perspective,

$psi({mathbf{r}})=frac{{mathbf{m}}cdot{mathbf{r}}}{4pi r^{3}},$

and hence magnetic field strength is

${mathbf{H}}({mathbf{r}})=- ablapsi=frac{1}{4pi}left(frac{3mathbf{r}(mathbf{m}cdotmathbf{r})}{r^{5}}-frac{{mathbf{m}}}{r^{3}} ight).$

The magnetic field of an ideal magnetic dipole is depicted on the left.

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