Magnetic field line

A magnetic field line is an integral curve tangent to the magnetic field. In coordinates $x^i$, it obeys $d{x}^i/B^i=d{x}^j/B^j$ wherever the relevant components are nonzero.

A magnetic arcade is a family of arch-shaped magnetic field lines joining two regions of a boundary. The boundary intersections are its footpoints. In a simple line-tied family, paired footpoints can be labelled by Euler potentials. A decaying linear force-free magnetic arcade is one explicitly solvable planar example; a general arcade need not be linear or force-free.

The curvature vector of a magnetic field line parametrized by arc length. For circular lines, it points towards their center and has magnitude $1/R$. The curvature contribution to magnetic tension is $B^2\mathbf{\kappa }/(4\pi )$ in Gaussian units.

A flux tube is a bundle of magnetic field lines bounded by a surface everywhere tangent to the magnetic field. Gauss's law for magnetism makes the magnetic flux through any transverse cross-section constant along the tube. For a narrow tube with nearly uniform field magnitude $B$ and transverse area $A$, this gives $BA=\mathrm{constant}$.

The cross-sectional area of a narrow flux tube, measured normal to its magnetic field. Constancy of the magnetic flux gives $A|\mathbf{B}|=\mathrm{constant}$ along the tube. This area controls the mass transport when flow follows the field.

The magnetic-field-line equation states that the tangent to a field line is parallel to $\mathbf{B}$. In spherical coordinates for an axisymmetric poloidal field, it can be written $dr/(rd\theta )=B_r/B_{\theta }$.

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.