Magnetic dipole field

Outside a localized magnetic dipole $\mathbf{\mu }$, the vacuum field in Gaussian conventions isFor a dipole along the polar axis, $B_r=2\mu \cos \theta /r^3$ and $B_{\theta }=\mu \sin \theta /r^3$.

For coaxial circular loops with radii $a_1,a_2\ll d$, the first loop's on-axis magnetic field of a circular current loop has far value $B_0\simeq {\mu }_0{I}_1{a}_1^2/(2d^3)$. Its near-axis radial magnetic field at the second loop is $B_{\rho }\simeq 3{\mu }_0{I}_1{a}_1^2{a}_2/(4d^4)$. Integrating the axial Lorentz force from $I_2{a}_{2}d\varphi \hat{\mathbf{\varphi }}\times B_{\rho }\hat{\mathbf{\rho }}$ gives the displayed magnitude. Equal current senses attract; opposite senses repel. This is the leading dipole limit and is not an exact close-coil force law.

For a centred magnetic dipole moment pointing toward geographic south, the field at northern latitude $\lambda$ has northward component $B_N=B_{\mathrm{eq}}\cos \lambda$ and downward component $B_D=2B_{\mathrm{eq}}\sin \lambda$. These follow by resolving the magnetic dipole field into local radial and meridional directions. The downward inclination is therefore $\tan \iota =B_D/B_N=2\tan \lambda$, while the magnitude is $B_{\mathrm{eq}}\sqrt{1+3{\sin }^2\lambda }$. This ideal aligned-dipole model has no eastward component and does not capture actual geographic departures of the geomagnetic field.

An axisymmetric dipole field line satisfies $r=C{\sin }^2\theta$, where $C$ is its equatorial crossing radius. This follows by integrating $dr/(rd\theta )=B_r/B_{\theta }=2\cot \theta$.

Near the polar axis of a magnetic dipole field, $B\propto r^{-3}$. Conservation of magnetic flux in a flux tube consequently gives $A(r)\propto r^3$. A dipole magnetic-field line intersecting a sphere of radius $R_{\star }$ at small polar angle ${\theta }_{\star }$ obeysThis area is for one polar tube. The approximation requires ${\theta }_{\star }^{2}r/R_{\star }\ll 1$; the full magnetic dipole field does not remain a narrow polar tube at arbitrarily large radius.