Magnetostatics

Magnetostatics studies time-independent magnetic fields and steady currents. In vacuum its field equations are $\nabla \cdot \mathbf{B}=0$ and $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$.

A uniform electric current density outside a parallel cylindrical cavity can be constructed by the superposition principle: add a full cylinder carrying that density and a displaced cylinder carrying its negative. At an exterior point, each contribution is the external magnetic flux density of its own net electric current. The combined magnetic flux density generally lacks rotational symmetry even though Ampère's law fixes its line integral around the whole wire.

Subtract a cylinder of current density $\mathbf{J}$ occupying the cavity from a complete current-carrying cylinder. Inside each cylinder, $\mathbf{B}={\mu }_0\mathbf{J}\times \mathbf{r}/2$. The difference is independent of observation point and equals ${\mu }_0\mathbf{J}\times \mathbf{b}/2$, where $\mathbf{b}$ is the offset of the cavity axis. This assumes parallel axes, uniform current density in the remaining metal, and negligible end effects.

A localized magnetic source described at leading nonvanishing dipole order by a magnetic dipole moment. A small current loop is one classical model. The source object, its moment vector and the resulting magnetic dipole field are distinct concepts. At large distance from a source with nonzero dipole moment, higher multipoles give smaller corrections; if the dipole moment vanishes, a higher-order term can lead instead.

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.

For a light nonmagnetic rod of length $l$ wound uniformly with $N$ thin turns of mean radius $a$, the magnetic dipole moment is $m_B=NI\pi a^2$. Wire of cross-sectional area $A_w$ and density $\rho$ has mass approximately $M=2\pi aN\rho A_w$. The transverse central moment of inertia is $J=M(l^2/12+a^2/2)$ in the thin-winding approximation. Consequently the oscillation period in a transverse magnetic field does not depend on the number of turns: wire inertia and magnetic moment both scale with $N$.

A generic instrument for force and torque uses a magnetically responsive bead in a controlled magnetic field, with optical tracking. Translational force requires a field gradient. Permanent or anisotropic moments can feel orienting torque; an isotropic induced moment parallel to the field has no such torque. This definition concerns the physical instrument, without biological applications.

At fixed linking number, local DNA unwinding causes compensating writhe. Its mechanical extension signal has opposite signs on positively and negatively supercoiled branches, distinguishing unwinding from simple shortening. The measured quantity is tether extension, converted through a torsional mechanical calibration, not direct optical separation of bases.^[ref]

For a tether with extension $\ell$ and tensile load $F$, a small transverse coordinate costs $F{x}^2/(2\ell )$. Equipartition gives $F=k_{B}T\ell /⟨x^2⟩$ for one Cartesian transverse coordinate. Using the sum of two transverse variances changes the factor of two. Geometry and measurement filtering must match the calibration model.

For an equilibrium overdamped harmonic coordinate with covariance $⟨x(t)x(0)⟩=V{e}^{-|t|/\tau }$, a camera exposure $\Delta$ records the average over that interval. Integrating the covariance twice gives $\mathrm{Var}(\bar{x})=2V{\Delta }^{-2}{\int }_0^{\Delta }(\Delta -u)e^{-u/\tau }du$. The displayed reduction follows with $s=\Delta /\tau$. In the tether calibration $F=k_{B}Th/V$, ignoring exposure averaging overestimates the inferred force. Independent localization-noise variance instead adds to the measured variance and biases it in the opposite direction. Other filters require their actual response functions.

For a linear isotropic unsaturated effective susceptibility contrast $\Delta \chi$, the induced-moment energy is $-V\Delta \chi B^2/(2{\mu }_0)$, giving $\mathbf{F}=V\Delta \chi \nabla B^2/(2{\mu }_0)$. The one-half differs from the fixed permanent-dipole energy because the applied field creates the moment. Demagnetization and saturation are absorbed into or replace the assumed response model.

An infinite solid cylinder with radius $R$, prescribed uniform volume charge density $\rho$, and angular velocity $\omega \hat{\mathbf{z}}$ has current density $\mathbf{J}=\rho \omega r\hat{\mathbf{\varphi }}$. Treat a shell at radius $a$ and thickness $da$ as a solenoid carrying surface current $\rho \omega ada$. With no applied background magnetic field and permeability $\mu$,The same result follows from Ampère's circuital law, since $-d{B}_z/dr=\mu \rho \omega r$ inside.

A magnetic circuit approximates a narrow core and its gaps by sections carrying a common magnetic flux $\Phi$. With negligible leakage and approximately uniform magnetic flux density across each section, Gauss's law for magnetism gives $B=\Phi /A$. The magnetostatic Ampère law then gives $NI=\int Hdl=\Phi \int dl/({\mu }_0{\mu }_{r}A)$. This requires an appropriate material relation; constant magnetic permeability is an approximation for an actual magnetic core.

A rod of magnetic path length $l$ and tangential magnetization $M$ closed by a gap of width $a$ has constant flux in the one-dimensional magnetic circuit approximation. Equal cross sections and negligible leakage give $H_g=H_r+M$, while absence of free current gives $l{H}_r+a{H}_g=0$. Their solution is the displayed pair. Gap fringing or an unspecified cross-section change requires a more detailed field model.

A uniform-radius magnetic arc of length $L$ and radius $R$, two linear tapers of length $s$ to radius $r$, and an unfringed vacuum gap of length $h$ have the displayed gap field. Here ${\mu }_r=\mu /{\mu }_0$ and $B_g={\mu }_0{H}_g$. Tapering concentrates flux, but at fixed winding current the gap-dominated limit is $H_g\simeq NI/h$ regardless of pole area. Saturation and leakage prevent unrestricted field enhancement.

Magnetic fringing is the spreading of magnetic field lines near an air gap or pole edge, so the gap's effective magnetic flux area exceeds its geometrical pole-face area. A gap much shorter than its transverse dimensions permits a leading uniform-field approximation. High core magnetic permeability does not by itself justify neglecting magnetic fringing for a wide gap.

The magnetic reluctance of a linear section of magnetic circuit is the ratio of the winding's integrated magnetic field intensity to the magnetic flux, with units $\mathrm{A}\mathrm{W}{\mathrm{b}}^{-1}$. For uniform length $\ell$, cross-sectional area $A$, and relative magnetic permeability ${\mu }_r$, $\mathcal{R}=\ell /({\mu }_0{\mu }_{r}A)$. Series sections carrying the same magnetic flux have additive magnetic reluctances, and $NI=\Phi {\sum }_i{\mathcal{R}}_i$.

For a uniform core of path length $\ell$, a narrow air gap $d$, and $n$ winding turns, Ampère's circuital law gives $nI=\ell H_{\mathrm{core}}+dB/{\mu }_0$ when leakage and fringing are negligible. If a reversible branch has differential magnetic permeability ${\mu }_{\mathrm{core}}=dB/dH$, linearization gives $dB/dI=\kappa$. An offset in the operating branch changes the bias field, not this differential formula. For $B=B_m\tanh (H/H_m)$ at the origin, ${\mu }_{\mathrm{core}}=B_m/H_m$.

For a linear-radius taper from $R$ to $r$ over length $s$, constant absolute permeability $\mu$, and a common approximately axial flux, integrate $dx/[\mu \pi \{R-(R-r)x/s{\}}^2]$ to obtain the displayed reluctance. Leakage, fringing, nonuniform transverse fields and saturation limit this one-dimensional magnetic-circuit approximation.

In a simply connected current-free region of constant magnetic permeability, Ampère's circuital law makes the magnetic field curl-free, so it admits a scalar potential. The absence of magnetic monopoles then gives Laplace's equation for that potential. Some conventions define the potential for $\mathbf{H}$ instead of $\mathbf{B}$, differing by the constant permeability. Boundary conditions, including imposed fields at infinity, determine the potential up to an additive constant.

For an isolated razor-thin disk, rescale its exterior magnetic scalar potential to satisfy $\sqrt{4\pi G/{\mu }_0}\mathbf{B}=-\nabla {\Phi }_M$. Comparing the normal magnetic boundary derivative with ${\partial }_z\Phi =2\pi G\Sigma$ identifies the displayed equivalent signed surface density. Both exterior potentials satisfy Laplace's equation. The equivalent surface density is a representation of the magnetic boundary data, rather than physical negative mass.