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Leopoldo Nobili developed an astatic galvanometer for detecting small electric currents. A mechanically linked pair of nearly opposing magnetic dipoles reduces the torque from the approximately uniform geomagnetic field, while the coil geometry retains sensitivity to the current being measured. A many-turn coil increases the response and a scale makes the deflection observable. The preserved Nobili-pattern instrument used by Carlo Matteucci connects these design choices with measurements of weak physiological currents.^[ref]

A physicist associated with the astatic galvanometer.

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

In an external field that varies slowly across a localized magnetic dipole,

The magnetic dipole moment determines the coupling to the external magnetic field.

Electromagnetic duality replaces the electric dipole $p$ by $m/c$ for a magnetic dipole moment $m$. Therefore

This is the magnetic-dipole term of the multipole radiation expansion.

If a rigid body's inclined magnetic dipole rotates at angular speed $\Omega$, its radiated power is $C{\Omega }^4$. Conservation of rotational energy $I{\Omega }^2/2$ then gives

A magnetic monopole carries an isolated magnetic charge, producing nonzero net magnetic flux through a closed enclosing surface. In the usual macroscopic Maxwell equations without magnetic charges, Gauss's law for magnetism instead gives $\nabla \cdot \mathbf{B}=0$. Cutting a conventional bar magnet produces smaller magnetic dipoles, rather than separating its north and south poles into isolated charges.

A model radial field $\mathbf{B}=g\hat{\mathbf{r}}/r^2$ away from the origin has flux $4\pi g$ through every enclosing sphere. It can be described locally by a magnetic vector potential, but not by one potential smooth over the whole enclosing sphere: the Stokes theorem would otherwise force its total flux to vanish. The two-patch vector potential for a radial monopole makes that obstruction explicit. A classical monopole model, a monopole solution in a field theory, and a monopole-like excitation in a material require their own physical interpretation and evidence.

In the small-sphere oscillatory-field limit, surface current density excludes the changing magnetic field. Adding a central magnetic dipole to a uniform applied field and imposing zero normal field at the sphere gives $\mathbf{m}=-2\pi a^3{\mathbf{H}}_{\mathrm{inc}}$. The minus sign expresses diamagnetism; it does not make this dipole's backward radiation cancel the electric contribution.^[ref]

For a harmonic magnetic dipole observed with kr of order unity, its quasistatic, induction and radiation fields scale as $r^{-3}$, $k{r}^{-2}$ and $k^2{r}^{-1}$ and need not be cleanly separated experimentally. The radiated-power law remains an asymptotic flux statement, but $P/(4\pi r^2)$ alone does not specify the local detected field. Detectability further depends on directional pattern, plasma propagation, antenna coupling, bandwidth, integration and noise. A tiny nonzero vacuum radiation term does not by itself establish either a practical detection or an absolute impossibility.

NMR dipolar coupling is the interaction between the magnetic moments of atomic nuclei through their magnetic fields. It depends on internuclear separation and orientation, allowing nuclear magnetic resonance measurements to constrain spatial structure. It differs from NMR scalar coupling, whose mechanism involves intervening electrons.

For two point magnetic dipoles separated by $\mathbf{r}=r\hat{\mathbf{r}}$, the interaction energy isHere ${\mu }_0$ is the vacuum permeability. Write ${\mathbf{\mu }}_I={\gamma }_I\hslash \mathbf{I}$ and ${\mathbf{\mu }}_S={\gamma }_S\hslash \mathbf{S}$, where $\mathbf{I},\mathbf{S}$ are dimensionless spin angular momentum operators and each $\gamma$ is an angular gyromagnetic ratio in radians per second per tesla. A signed angular-frequency coupling scale is thenThis convention puts the geometrical factor separately; some texts absorb an extra minus sign or convert to cycles per second. Comparing numerical coupling constants requires matching those conventions.

For an unlike-spin pair with well-separated resonance frequencies in a strong field, neglecting rapidly oscillating terms gives the ideal secular Quantum Hamiltonianwhere $\theta$ is the angle between the internuclear vector and the field. The angular factor vanishes at $\theta =\arccos (1/\sqrt{3})$. Magic angle spinning exploits related orientation averaging, while dipolar recoupling deliberately restores sensitivity to interactions suppressed by spinning.

An original scaling comparison is especially useful: doubling $r$ reduces $|d_{IS}|$ by a factor of eight. Thus a coupling strength can constrain distance, but orientation averaging, motion and additional spins must be modeled before turning a measured signal change into a unique distance. REDOR supplies an experimental route to that information. Its early carbon-13–nitrogen-15 alanine measurements compared a full rotational echo with a dephased echo under timed pulses.^[ref]

A point magnetic dipole $\mathbf{\mu }$ producesFor $|\mu |\sim {\mu }_B$ and neighbour separation $r\sim 0.2\mathrm{nm}$, its characteristic magnitude isThe axial field is twice this scale; the exact orientation and crystal geometry are unspecified. The corresponding pair magnetic dipole interaction has energy scale $U_{\mathrm{dip}}\sim {\mu }_B{B}_{\mathrm{dip}}\simeq 1.08\times 10^{-24}\mathrm{J}$, givingAn axial orientation changes this to about $0.16\mathrm{K}$. A real ordering temperature depends on a lattice sum, symmetry and fluctuations, so this is a nearest-neighbour energy estimate, not a complete ordering calculation. It establishes that ordinary dipolar coupling is far too weak to explain a thousand-kelvin ferromagnetic transition.

Choose the loop axis as $+x$, with positive electric current circulating by the right-hand rule. The Biot-Savart law isFor a loop element and an axial observation point, $r=\sqrt{a^2+x^2}$. The transverse components cancel between opposite elements. The axial component of $d\mathbf{\ell }\times \mathbf{r}$ is $ad\ell$, soThusReversing the electric current reverses the field. At the centre $B={\mu }_{0}I/(2a)$, while the far field decreases as $|x{|}^{-3}$, as expected for a magnetic dipole.

For a circular current loop, the Biot-Savart law iswhere $\mathbf{s}$ points from a current element to the field point. Choose the loop in the $yz$ plane, with observation point $x\hat{\mathbf{x}}$. A parametrization consistent with positive axial magnetic field isThen $s=\sqrt{a^2+x^2}$ andThe transverse contributions cancel around the circle, leaving the on-axis magnetic field of a circular current loopThe right-hand rule fixes its direction; reversing the electric current reverses the magnetic field. At the centre this becomes ${\mu }_{0}I/(2a)$, and far away it decays as $|x{|}^{-3}$, as expected for a magnetic dipole.

Use the source orientation, anticlockwise viewed from $+z$. The position and oriented line element areThe Lorentz force on the electric current element is $d\mathbf{F}=Id\mathbf{l}\times \mathbf{B}$. Evaluating the cross product givesEach component integrates to zero around $0\le \theta <2\pi$, so ${\mathbf{F}}_{\mathrm{net}}=0$. Equivalently, uniform magnetic flux density permits $I\oint d\mathbf{l}\times \mathbf{B}=I(\oint d\mathbf{l})\times \mathbf{B}=0$. Uniformity is essential; a nonuniform magnetic field can exert a net force on a magnetic dipole.

A magnetic dipole is the leading magnetic source moment of a localized steady current distribution. For a small planar loop, $\mathbf{m}=I\mathbf{A}$ with area orientation set by the right-hand rule. A fixed permanent moment in a slowly varying external field has energy $U=-\mathbf{m}\cdot \mathbf{B}$ and force $\mathbf{F}=\nabla (\mathbf{m}\cdot \mathbf{B})$.

For the given purely azimuthal vector potential, the spherical curl givesThese are the magnetic dipole field components outside the localized source.

The conducting shell has $Q=4\pi {ϵ}_{0}aV$ and surface charge density $\sigma =Q/(4\pi a^2)$. A latitude strip of area $2\pi a^2\sin \theta d\theta$ carries current $dI=\sigma \omega a^2\sin \theta d\theta$. Its loop area is $\pi a^2{\sin }^2\theta$, givingIn vector form the magnetic moment of a rotating charged spherical shell isThe positive-z expression in the question takes $\mathbf{\omega }=\omega \hat{\mathbf{z}}$. “Clockwise” needs a viewing direction: this rotation looks clockwise when viewed looking along $+z$ from below, and counterclockwise when viewed from the positive-z side. Looking from the other end reverses the verbal label, not the vector relation.

Since the positive charge and the loop current circulate in the same stated sense, their dipole moments are parallel when judged from a common viewpoint. At $h\gg R$, the loop's axial field magnitude is $B\simeq {\mu }_{0}I{R}^2/(2h^3)$. With the sphere small compared with that variation scale,The magnetic force is attractive, pulling the sphere toward the loop. Reversing only one circulation would reverse that magnetic force. The calculation treats the rotating charge as a prescribed moment and uses the stated separated length scales; it does not add unrelated induction or electric interactions to the dipole model.

Assume the greenstone belt is more magnetically susceptible than the host rocks and granite, its magnetization is mainly induced magnetization, and the body is roughly east–west elongated so that the south–north line crosses it. At southern magnetic latitudes the main field has negative magnetic inclination in a north-and-down coordinate convention. The total-field magnetic anomaly is then generally asymmetric: the principal negative lobe is on the southern side and the positive lobe is displaced northward for the simple induced source illustrated here. The exact lobe sizes and peak positions depend on body geometry and the actual magnetic inclination.^[ref]

An explicit sketching model is an induced magnetic dipole at depth $d$. With northward coordinate $x$, $\mathbf{r}=(x,-d)$ and unit field direction $\hat{\mathbf{F}}=(\cos I,\sin I)$, its normalized total-field magnetic anomaly isThe plot uses $I=-36^{\circ }$ only as an illustrative dipole-field inclination for a latitude near $20^{\circ }$ south, not a measured modern field at the site. A wide greenstone belt would broaden the response and can produce prominent edge anomalies. Remanent magnetization or a magnetic granite could change the sketch substantially; gravity alone cannot identify the magnetic source.

Lay out a surveyed grid extending beyond the possible burial area into unaffected ground. Use closely spaced parallel traverses with measurements along each line; spacing should resolve a car-scale magnetic anomaly, with a finer grid over any candidate. For a shallow car, a pilot survey with roughly metre-scale spacing is a sensible starting point, refined according to anomaly width rather than treated as a universal prescription. Record position, time, sensor height, repeat readings and any nearby interference. Cross-lines and repeat stations check reproducibility. Maintain a constant sensor height on a nonmagnetic staff and remove magnetic items from the operator; map fences, gates, pipes, buildings and vehicles before interpreting an anomaly.

Operate a synchronized stationary base-station magnetometer in quiet ground while the rover surveys, or make frequent base returns if a second instrument is unavailable. A total-field magnetic anomaly is the corrected field magnitude minus a specified background:For a field-sized target, a local background estimated away from the car is often sufficient. A larger regional survey may subtract the International Geomagnetic Reference Field and an independently justified regional trend. Do not remove the car's broad anomaly by overfitting that trend.

Correct rover observations for time-varying geomagnetic field using the base record, for example $F_c(\mathbf{x},t)=F_r(\mathbf{x},t)-[F_b(t)-F_b(t_0)]$. Check instrument offsets and tie-line differences, reject disturbed or poor-quality readings, and allow for differing sensor heights. Preserve localized target signals when interpolating a contour map or comparing with a magnetic dipole model. There are no compulsory gravity-style free-air or Bouguer gravity anomaly reductions for this magnetic search.

Postpone surveying during a geomagnetic storm or rapid irregular magnetic variation, when base correction may be inadequate. Avoid periods of moving farm machinery or electrical interference, and recognize that strong permanent cultural interference or highly magnetic host rock may make this particular method unsuitable. Ordinary cloud cover or darkness does not prevent a magnetic survey.^[ref]

The sketches assume the car can be approximated by a compact magnetic dipole whose induced magnetic moment is parallel to the local geomagnetic field. The host contributes negligible local contrast. Real steel also has remanent magnetization and shape-dependent magnetization, so the car's actual profile need not follow this idealization. A proton-precession magnetometer observeswhen $|\mathbf{b}|\ll |{\mathbf{B}}_0|$. This scalar total-field magnetic anomaly must not be confused with a fixed vertical field component. For induced $\mathbf{m}=m{\hat{\mathbf{B}}}_0$, the magnetic dipole field givesReversing both the inducing field and induced moment leaves this total-field magnetic anomaly unchanged, exactly for the magnitude definition as well as in the weak-anomaly approximation.^[ref]

A magnetic dipole of fixed magnitude $m_B$, free to rotate about an axis with moment of inertia $J$, has potential energy $-m_{B}B\cos \theta$ in a uniform magnetic field. Near stable alignment, $J\ddot{\theta }=-m_{B}B\theta$, so its small-angle period is $2\pi \sqrt{J/(m_{B}B)}$. This neglects damping, bearing friction and any other restoring torque.

A perpendicular magnetic dipole $m\hat{z}$ a height $h$ above an ideal flux-excluding plane has image $-m\hat{z}$ at $-h$. The exterior field has zero normal component at the plane. Images are a mathematical construction outside the physical solution domain.

Wilhelm Eduard Weber developed a description of magnetization in which the microscopic magnets already exist and respond to an applied magnetic field. James Clerk Maxwell's treatise contrasts this with a model in which the field induces molecular magnetism. The distinction matters because changing the orientation of persistent magnetic dipoles can change the bulk magnetization without creating each magnetic moment from zero. A material can therefore have a small net magnetic response even when its microscopic magnetic contributions are individually nonzero.^[ref]

Portrait source. Gottlieb Biermann; scanned by Michael Schönitzer in November 2008 from Die großen Deutschen im Bilde (1936). Public-domain artwork; faithful photographic reproduction, PD-Art / Public Domain Mark.