Magnetic field

The magnetic field contributes the velocity-dependent term $qv\times B$ to the Lorentz force.

A solenoid is a helical winding of conducting wire. For an ideal very long winding with $N$ turns per unit length and negligible pitch, Ampère's law gives a uniform axial magnetic field $B={\mu }_{0}NI$ inside and zero field outside. Finite-length end fields and the pitch correction are excluded by this approximation. The self-inductance of a length $l$ and cross-sectional area $A$ is $L={\mu }_0{N}^{2}Al$.

The magnetic field has zero divergence. The ideal magnetohydrodynamic induction equation preserves this constraint from initial data, since the divergence of its curl is zero.

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 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 $B_{\theta }=\mu \sin \theta /r^3$.

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.

A magnetostatic equilibrium is a time-independent configuration in which magnetic, pressure, gravitational, and any other body-force densities balance.

A plane-parallel magnetostatic equilibrium with constant downward gravitational acceleration $g$, horizontal magnetic field, constant isothermal sound speed $c_s$, and constant Alfvén speed $v_a$ hasIndeed the magnetohydrodynamic total pressure is $(c_s^2+v_a^2/2)\rho$, and its vertical derivative is $-g\rho$. The magnetic pressure therefore increases the scale height above that of an unmagnetized atmosphere.

A cylindrical magnetostatic equilibrium is an axisymmetric vector field and pressure configuration independent of the axial coordinate. Its radial Lorentz force density balances the pressure gradient, as expressed by cylindrical magnetostatic pressure balance.

In a Z-pinch, an axial electric current produces a toroidal magnetic field whose inward magnetic tension confines a plasma. A cylindrical magnetostatic equilibrium balances the resulting Lorentz force density against a pressure gradient.

For an axisymmetric vector field independent of the axial coordinate and regular on the axis, Gauss's law for magnetism gives $B_R=0$. The magnetostatic Ampère-Maxwell equation then gives $j_R=0$, $j_{\varphi }=-c{B}_z^{\prime }/(4\pi )$, and $j_z=c(R{B}_{\varphi }{)}^{\prime }/(4\pi R)$ in Gaussian units. Radial magnetostatic equilibrium isFor enclosed axial electric current $I(R)$, the magnetostatic Ampère-Maxwell equation gives $B_{\varphi }=2I/(cR)$, henceThe first term contains pressure and axial magnetic pressure; the second combines toroidal magnetic pressure with the hoop force from magnetic tension.

For a regular cylindrical flux tube with $\mathbf{B}=(0,B_{\varphi }(s),B_z(s))$, vanishing field beyond radius $s_0$, and exterior pressure $p_e$, define $⟨h⟩=2s_0^{-2}{\int }_0^{s_0}sh(s)ds$ and $B_p^2=2{\mu }_0{p}_e$. Integrating cylindrical magnetostatic pressure balance against $s^2$ gives the displayed identity. The toroidal magnetic pressure and hoop magnetic tension cancel in this cross-sectional axial-field average. Nonnegative gas pressure gives $⟨B_z^2⟩\le B_p^2$, with strict inequality precisely when $⟨p⟩>0$.

For a flux tube with $B_{\varphi }=\gamma (s/s_0)B_z$, regularity at the axis and cylindrical magnetostatic pressure balance give $B_z(0{)}^2=B_p^2-2{\mu }_{0}p(0)+{\gamma }^2⟨B_z^2⟩$. The flux-tube axial field virial identity and $p\ge 0$ give the bound. A twisted toroidal magnetic field can support a central axial field above the untwisted magnetic pressure limit even though its cross-sectional mean obeys that limit.

For constant axial magnetic field and $p=p_0(1+R^2/a^2{)}^{-k}$ with $p_0>0$ and $k>0$, cylindrical magnetostatic pressure balance gives, with $x=R^2/a^2$,The nontrivial electric current is finite at infinity exactly when $k>1$, in which case $I^2(\infty )=2\pi c^2{p}_0{a}^2/(k-1)$. Either sign of $I$ is possible because the pressure balance fixes its square.

With no axial electric current, cylindrical magnetostatic pressure balance reduces to constant $p+B_z^2/(8\pi )$. For $p=p_0{e}^{-R^2/a^2}$ and $B_z(0)=0$, its radial solutions areAs $R\downarrow 0$, $B_z\sim \sigma \sqrt{8\pi p_0}R/a$ and $j_{\varphi }\to -\sigma c\sqrt{8\pi p_0}/(4\pi a)$. These finite scalar limits do not give a smooth function of Cartesian position: $R=\sqrt{x^2+y^2}$ has a cusp at the axis, and the azimuthal unit vector has no unique limit there. Thus these data describe radial equilibrium away from the axis, but cannot satisfy smooth vector-field regularity on the axis without modifying the data.

A conductor or plasma moving with nonrelativistic velocity $\mathbf{u}$ through a magnetic field experiences the motional electric field $\mathbf{E}=-\mathbf{u}\times \mathbf{B}/c$ in Gaussian units when the electric field vanishes in its rest frame.

The London equation relates supercurrent to the electromagnetic vector potential. In a gauge with constant condensate phase it has the form $\mathbf{J}=-\mathbf{A}/({\mu }_0{\lambda }_L^2)$ and implies exponential magnetic-field screening.

The Meissner effect is the expulsion of magnetic field from the bulk of a superconductor. A static field decays from its surface over the London penetration depth.

The magnetic flux through an oriented surface $S$ is ${\Phi }_B={\int }_{S}B\cdot dS$.