Magnetostatics
Magnetostatics is the theory of the fields produced by steady electric currents: currents themselves, the Biot–Savart law, the vector potential, and magnetic media. It stands to Electrostatics as the second static limit of the full Maxwell system assembled in The Maxwell Equations, and it is where electromagnetism became a single subject: before 1820 electricity and magnetism were separate sciences, and this chapter's first task is the experiment that joined them. Its second task is a structural asymmetry with electrostatics that has survived every later refinement of the theory: there are electric charges, and there are—as far as every search to date can tell—no magnetic ones. That absence is not an assumption but an experimental verdict, and it is treated as such in Section 59.6.
Magnetostatics: all derivations of this chapter are pending.
Steady currents
Current density and charge conservation
[Reserved: the current density \(\vect{J}\), the current through a surface, the continuity equation \(\pp\rho/\pp t+ \vect{\nabla}\cdot\vect{J}=0\), and the magnetostatic regime \(\vect{\nabla}\cdot\vect{J}=0\) as its steady limit; surface and line currents; the standard treatment is [Jackson:1999]. The dynamical role of the continuity equation—it is what forces the displacement current—is taken up in The Maxwell Equations.]
The SI unit of current
[Reserved: the ampere, historically defined through the force per unit length between parallel currents and since 2019 fixed by the elementary charge [BIPM:2019]; how current is realised and measured, with the metrological framework of Measurement, SI Units, and the Theory of Errors.]
The discovery of electromagnetism
Magnetism before 1820
[Reserved: lodestones, compass needles, and Gilbert's De Magnete, which established the Earth itself as a magnet and separated magnetic from electric attraction [Gilbert:1600]; the phenomenology of poles, the impossibility of isolating one by cutting a magnet, and the inverse-square law between pole strengths, put on an absolute quantitative footing by Gauss's measurement of the terrestrial field [Gauss:1841].]
Every magnet has two poles of opposite character, and no operation performed upon it isolates one of them. A bar magnet broken in two yields two complete magnets, each with its own north and south pole, and the subdivision may be repeated to the limit of what can be handled without ever producing a body that acts upon a compass needle as a single pole would. The Earth itself behaves as one such magnet, which is why the needle points at all [Gilbert:1600], and the strength of its field was reduced to absolute measure — to mechanical units, and so ultimately to the SI — by Gauss [Gauss:1841].
Inseparability of magnetic poles: derive, from the Amperian representation of a magnetized body by bound currents, that every subdivision of such a body again carries a complete circulating current and is therefore again a dipole, and connect the statement to the vanishing divergence of the magnetic field.
Ørsted's experiment
[Reserved: the deflection of a compass needle by a nearby current-carrying wire, announced by Ørsted in July 1820 [Oersted:1820]—the first observed connection between electricity and magnetism, and the observation from which the whole of electrodynamics descends. The circular geometry of the effect (the needle sets tangentially around the wire, not toward it) already encodes the curl structure of the field equations. Full experimental detail belongs to Experiments: Electromagnetism.]
A magnetic needle placed near a straight conducting wire is deflected the instant a galvanic current is sent through the wire, and it comes to rest across the wire rather than along the line joining the two: the needle sets tangentially to the circle that passes through it and has the wire for its axis. The sense of the deflection reverses with the sense of the current, and reverses again when the needle is carried to the opposite side of the wire [Oersted:1820]. This is the first observed action of electricity upon magnetism, and the first force in physics whose direction is not along the line joining its sources.
Derivation. The derivation runs opposite to the history: it assumes the law for which this observation was the first evidence, namely the Biot–Savart law of Section 59.2.3, by which an element \(I\,\dd\vect{l}\) of a steady current contributes
where \(\vect{r}\) runs from the element to the field point.
Take the wire along the \(z\)-axis and the field point at perpendicular distance \(s\) in the plane \(z=0\). Each element \(\dd\vect{l}=\dd z\,\hat{\vect{z}}\) contributes along \(\hat{\vect{z}}\times\hat{\vect{r}}\), which is \(\hat{\vect{\varphi}}\) for every \(z\); the contributions therefore do not compete in direction and the total field is purely azimuthal. Writing \(z=s\tan\alpha\), so that \(r=s\sec\alpha\) and \(\dd z=s\sec^{2}\alpha\,\dd\alpha\), and noting that the sine of the angle between \(\dd\vect{l}\) and \(\hat{\vect{r}}\) is \(\cos\alpha\),
The field circles the wire. A compass needle is a magnetic dipole \(\vect{m}\) and feels the torque \(\vect{m}\times\vect{B}\) (Section 59.4.4), which vanishes only when \(\vect{m}\) lies along \(\vect{B}\); the needle therefore comes to rest tangentially to that circle, reverses with \(I\), and reverses again on the far side of the wire, where \(\hat{\vect{\varphi}}\) points the other way.
∎The Biot–Savart law
[Reserved: Biot and Savart's quantitative measurement, within months of Ørsted's announcement, of the force exerted by a long straight current on a magnetic pole, falling off as the inverse distance [Biot:1820]; the modern differential statement \(\dd\vect{B}\propto I\,\dd\vect{l}\times\hat{\vect{r}}/r^{2}\) and its integral form for closed circuits; worked fields of the straight wire, the circular loop and the solenoid [Jackson:1999].]
The magnetic action of a long straight steady current, measured by Biot and Savart within weeks of Ørsted's announcement from the oscillation period of a short magnetic needle suspended at a series of distances from a vertical conducting wire, is proportional to the current \(I\) and falls off as the inverse first power of the perpendicular distance \(s\) from the wire, not as the inverse square [Biot:1820]:
Since the SI revision of 2019 the magnetic constant \(\mu_{0}\) appearing here is a measured quantity rather than a defined one [BIPM:2019].
Derivation. This is Equation (59.2), already obtained from the Biot–Savart law Equation (59.1) for exactly this configuration. The inverse first power, against the inverse square of each separate element's contribution, is the signature of a line source of unbounded length: the length of wire lying within a given range of the angle \(\alpha\) grows in proportion to \(s\), which cancels one power of the distance.
∎Ampère's electrodynamics
[Reserved: Ampère's demonstration that two currents attract or repel one another, his force law between current elements, and the theorem that a closed current loop is magnetically equivalent to a sheet of magnetic dipoles, all deduced from his null and balance experiments in the Théorie des phénomènes électro-dynamiques [Ampere:1826]; the hypothesis of molecular currents as the origin of all magnetism, vindicated a century later in Magnetism in Matter.]
Two long straight parallel wires carrying steady currents act on one another mechanically, with no magnets present anywhere: they attract when the currents run the same way and repel when they run opposite ways, and the force per unit length is proportional to each current and inversely proportional to their separation \(d\) [Ampere:1820] [Ampere:1826],
For \(I_{1}=I_{2}=1\,\mathrm{A}\) at \(d=1\,\mathrm{m}\) this is \(2\times 10^{-7}\,\mathrm{N}/\mathrm{m}\), the number through which the ampere was defined until the SI revision of 2019 replaced that definition by a fixed value of the elementary charge [BIPM:2019].
Derivation. Wire \(1\) produces at the location of wire \(2\) the azimuthal field Equation (59.2) with \(s=d\), of magnitude \(B_{1}=\mu_{0}I_{1}/2\pi d\) and directed perpendicular both to wire \(1\) and to the line joining the wires. A length \(L\) of wire \(2\), carrying \(I_{2}\) along the unit vector \(\hat{\vect{l}}_{2}\), feels the Ampère force \(\vect{F}=I_{2}L\,\hat{\vect{l}}_{2}\times\vect{B}_{1}\) (Section 59.4.4). Since \(\hat{\vect{l}}_{2}\) is parallel to the wires and \(\vect{B}_{1}\) is perpendicular to them, the magnitudes multiply without an angular factor and \(F/L=I_{2}B_{1}=\mu_{0}I_{1}I_{2}/2\pi d\), which is Equation (59.4). Taking \(\hat{\vect{l}}_{2}\) parallel to \(\hat{\vect{l}}_{1}\) makes \(\hat{\vect{l}}_{2}\times\vect{B}_{1}\) point from wire \(2\) towards wire \(1\) — attraction — and reversing either current reverses it. With \(\mu_{0}=4\pi\times10^{-7}\,\mathrm{N}/\mathrm{A}^{2}\), as it was by definition before 2019, the numerical value quoted follows by substitution.
∎The field equations of magnetostatics
Gauss's law for the magnetic field
[Reserved: \(\vect{\nabla}\cdot\vect{B}=0\) and its integral form, the vanishing of the magnetic flux through every closed surface [Maxwell:1873]; its reading as the statement that magnetic field lines do not end, and its empirical status as the absence of magnetic charge, examined against experiment in Section 59.6.]
Every measurement of magnetic flux ever performed, over any closed surface and about any source — a current-carrying circuit, a permanent magnet, the Earth, a star — has given zero net flux:
equivalently \(\vect{\nabla}\cdot\vect{B}=0\) [Maxwell:1873]. The statement is not a mathematical necessity but an experimental one: it says that magnetic charge has never been found (Section 59.6.2, [Navas:2024]), and it is the one Maxwell equation whose empirical support is a null record rather than a positive measurement.
Derivation. For fields produced by steady currents the vanishing follows from the Biot–Savart law. Since \((\vect{x}-\vect{x}')/\abs{\vect{x}-\vect{x}'}^{3} =-\vect{\nabla}\left(1/\abs{\vect{x}-\vect{x}'}\right)\), where \(\vect{\nabla}\) acts on the field point, and since \(\vect{J}(\vect{x}')\) does not depend on the field point,
using \(\vect{\nabla}\times(f\vect{J})=(\vect{\nabla}f)\times\vect{J}\) for constant \(\vect{J}\). A curl is divergence-free, so \(\vect{\nabla}\cdot\vect{B}=0\) identically, and the divergence theorem turns that into Equation (59.5). What the derivation establishes is conditional: if every magnetic field has currents for its only source, the flux vanishes. The unconditional statement is the experimental one, and the bracketed expression is the vector potential of Section 59.4.
∎Ampère's circuital law
[Reserved: \(\vect{\nabla}\times\vect{B}=\mu_{0}\vect{J}\) in the steady regime and the integral statement that the circulation of \(\vect{B}\) around a closed curve measures the enclosed current [Ampere:1826] [Maxwell:1873]; its consistency requirement \(\vect{\nabla}\cdot\vect{J}=0\), whose failure for open circuits is the door through which the displacement current of The Maxwell Equations enters; applications to the solenoid and the torus.]
Boundary conditions
[Reserved: continuity of the normal component of \(\vect{B}\) and the jump of its tangential component across a surface current [Jackson:1999], in parallel with the electrostatic interface conditions of Electrostatics.]
The vector potential
Existence and gauge freedom
[Reserved: \(\vect{\nabla}\cdot\vect{B}=0\) implies \(\vect{B}=\vect{\nabla}\times\vect{A}\); the vector potential introduced by Neumann in his general law of induced currents [Neumann:1845]; the gauge ambiguity \(\vect{A}\to\vect{A}+\vect{\nabla}\chi\) and the Coulomb gauge \(\vect{\nabla}\cdot\vect{A}=0\); the systematic treatment of gauge freedom is deferred to The Maxwell Equations.]
The Biot–Savart law as the general solution
[Reserved: in Coulomb gauge each component of \(\vect{A}\) obeys a Poisson equation, solved by the same Green function as the electrostatic potential of Electrostatics (general theory in Partial Differential Equations); recovering the Biot–Savart field [Biot:1820] [Jackson:1999] by taking the curl.]
Multipole expansion and the magnetic dipole
[Reserved: the expansion of \(\vect{A}\) for a localised current distribution; the vanishing of the monopole term as the first appearance of Section 59.3.1 in an expansion; the magnetic dipole moment \(\vect{m}=\frac{1}{2}\int\vect{x}'\times\vect{J}\,\dd V'\) and its field [Jackson:1999]; the dipole–dipole interaction and the field of the Earth as leading-order dipole [Gauss:1841].]
Seen from far enough away, no steady current distribution and no magnetized body produces a field falling as \(1/r^{2}\), the magnetic counterpart of the field of a point charge. The leading behaviour is always the \(1/r^{3}\) field of a dipole. The Earth is the case measured first and best: it acts on a compass as a magnet [Gilbert:1600], its field strength was reduced to absolute mechanical units by Gauss [Gauss:1841], and to leading order in the distance that field is dipolar, with the higher multipoles appearing as corrections.
Derivation. Expand the vector potential of Equation (59.6) for a current distribution confined to a bounded region, at a field point far outside it. The first term of \(1/\abs{\vect{x}-\vect{x}'}\) is \(1/r\) with \(r=\abs{\vect{x}}\), and it contributes
For a steady current this integral vanishes. Indeed \(\pp'_{j}\left(x'_{i}J_{j}\right) =J_{i}+x'_{i}\,\vect{\nabla}'\cdot\vect{J}=J_{i}\), because \(\vect{\nabla}\cdot\vect{J}=0\) in the magnetostatic regime (Section 59.1.1); integrating over a volume that encloses all the current, where \(\vect{J}\) vanishes on the boundary, the left-hand side integrates to zero by the divergence theorem, so
The monopole term Equation (59.7) therefore disappears identically, the expansion begins at the next order, and that order is the dipole \(\vect{m}=\tfrac{1}{2}\int\vect{x}'\times\vect{J}\,\dd V'\) whose potential falls as \(1/r^{2}\) and whose field falls as \(1/r^{3}\). This is the same absence, read in an expansion, that Phenomenon 59.5 reads as a flux.
∎Force and torque on a dipole
[Reserved: force \(\vect{\nabla}(\vect{m}\cdot\vect{B})\) and torque \(\vect{m}\times\vect{B}\) on a magnetic dipole in an external field; Ampère's equivalence of the current loop and the dipole sheet [Ampere:1826]; the contrast with the electric dipole of Electrostatics, and the relativistic reading of the magnetic force developed in Relativistic Dynamics.]
Magnetostatics in matter
Magnetization and bound currents
[Reserved: the magnetization field \(\vect{M}\) as dipole moment density; the equivalent bound currents \(\vect{J}_{b}=\vect{\nabla}\times\vect{M}\) in the volume and \(\vect{M}\times\hat{\vect{n}}\) on the surface, following the macroscopic construction of Maxwell's Treatise [Maxwell:1873]; the microscopic averaging that justifies it is carried out in Electrodynamics in Matter.]
The auxiliary field $\vect{H}$
[Reserved: \(\vect{H}=\vect{B}/\mu_{0}-\vect{M}\) and the circuital law \(\vect{\nabla}\times\vect{H}=\vect{J}_{\text{free}}\) sourced by free currents alone [Maxwell:1873]; interface conditions for \(\vect{H}\); linear media, permeability, and the magnetic scalar potential in current-free regions [Jackson:1999].]
Permanent magnets and boundary-value problems
[Reserved: the uniformly magnetized sphere and bar magnet; hysteresis named as the phenomenon that makes “the” permeability of a ferromagnet ill-defined, with its phenomenology and microscopic origin deferred to Electrodynamics in Matter and Magnetism in Matter; magnetic circuits and shielding [Jackson:1999].]
Magnetic monopoles: the empirical status
The Dirac quantization argument
[Reserved: Dirac's demonstration that a single magnetic charge \(g\) anywhere in the universe forces electric charge to be quantized, in units inversely proportional to \(g\) [Dirac:1931]—the one theoretical result that keeps the monopole question open, since charge quantization is observed; the argument requires the quantum mechanics of Part IX and is stated here only for its bearing on Section 59.3.1.]
Searches and the null record
[Reserved: the honest record of monopole searches. Cabrera's superconducting induction detector registered on 14 February 1982 a single candidate event of exactly one Dirac charge [Cabrera:1982]; the event was never repeated, and larger-exposure induction and track-etch searches, accelerator searches, and cosmic-ray searches have all been null—the current limits are collected in the Review of Particle Physics [Navas:2024]. Gauss's law for \(\vect{B}\) therefore stands as an empirical law: one unrepeated candidate against decades of null exposure, stated as such and not better.]
No magnetic charge has been observed. A superconducting induction detector registered, on 14 February 1982, a single flux jump of the magnitude a monopole of one Dirac charge would produce [Cabrera:1982]; the event was never repeated, in that apparatus or in any of the far larger induction, track-etch, accelerator and cosmic-ray exposures that followed, and every such search has returned a limit rather than a signal [Navas:2024]. The record is therefore one unrepeated candidate against decades of null exposure, and this is the whole of the evidence on which Phenomenon 59.5 rests. The question is kept open, rather than closed, by Dirac's demonstration that a single magnetic charge anywhere would force the quantization of electric charge [Dirac:1931] — and electric charge is observed to be quantized.
Monopole searches: derive the flux quantum that an induction detector would register for the passage of one Dirac charge through a superconducting loop, and the conversion of a null exposure into a flux limit per unit area, time and solid angle. The Dirac quantization argument itself requires the quantum mechanics of Part IX and is stated, not derived, here.