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Hermann Anton Haus was an electrical engineer whose research ranged from quantum optics to optical communication systems and ultrashort laser pulses. His work on optical solitons helped connect nonlinear wave propagation with the transmission of information through optical fibers.^[ref]

With James R. Melcher, he wrote Electromagnetic Fields and Energy by Hermann A. Haus and James R. Melcher, a text that links field equations with concrete devices and demonstrations. Its magnetic circuit analysis, for example, shows how geometry and constitutive assumptions turn an electromagnetic field problem into an approximate engineering model.^[ref]

James R. Melcher was an electrical engineer whose work helped define continuum electromechanics: the interaction of electromagnetic fields with deformable materials and flowing fluids. His applications included energy conversion, flow measurement and the control of industrial processes.^[ref]

With Hermann Anton Haus, he wrote Electromagnetic Fields and Energy by Hermann A. Haus and James R. Melcher. A central engineering theme is the relation between a full field description and a justified approximation. In a magnetic circuit, for example, geometry and permeability assumptions determine when a small set of circuit-like variables can represent the spatial fields.^[ref]

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.

For reversible quasistatic magnetic circuits with flux linkages ${\lambda }_i$, currents $I_i$ and mechanical coordinate $q$, write $d{W}_B={\sum }_i{I}_{i}d{\lambda }_i-F_{q}dq$. The Legendre transform $W_B^{\prime }={\sum }_i{I}_i{\lambda }_i-W_B$ obeys $d{W}_B^{\prime }={\sum }_i{\lambda }_{i}d{I}_i+F_{q}dq$. Thus $F_q=(\partial W_B^{\prime }/\partial q{)}_{I_i}$, while $F_q=-(\partial W_B/\partial q{)}_{{\lambda }_i}$. In a linear reciprocal system $W_B^{\prime }=W_B=\frac{1}{2}{\mathbf{I}}^{\mathsf{T}}L(q)\mathbf{I}$ numerically, but the variables held fixed in their force derivatives differ. The coenergy formulation includes the effect of electrical sources maintaining the currents; it avoids reversing the sign of the magnetic force by differentiating stored energy under the wrong constraint.

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 $NI=\Phi {\sum }_i{\mathcal{R}}_i$.

Magnetic saturation is the regime in which increasing the applied magnetic field intensity produces little further increase in magnetization. The total magnetic flux density can still increase through its vacuum contribution, $\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M})$. Consequently a constant large magnetic permeability cannot describe a core throughout this regime; ideal magnetic circuit predictions based on that assumption overestimate field concentration.

The microscopic magnetic response becomes a macroscopic field problem through the magnetization $\mathbf{M}$, the magnetic dipole moment per unit volume. Definein magnetostatics. Bound currents are represented by ${\mathbf{J}}_b=\nabla \times \mathbf{M}$ and surface current ${\mathbf{K}}_b=\mathbf{M}\times \hat{\mathbf{n}}$. Normal $B$ is continuous across an interface, while $\hat{\mathbf{n}}\times ({\mathbf{H}}_2-{\mathbf{H}}_1)={\mathbf{K}}_{\mathrm{free}}$. It is generally incorrect to impose continuity of every component of both fields. A linear isotropic material has $\mathbf{M}=\chi \mathbf{H}$ and ${\mu }_r=1+\chi$, but a ferromagnet usually requires a nonlinear, history-dependent constitutive relation.

Internal field differs from externally applied field. For uniform magnetization along a principal axis of an ellipsoid, the demagnetizing field is $-N_{d}M$ and $H_{\mathrm{int}}=H_{\mathrm{applied}}-N_{d}M$. A linear specimen therefore has apparent susceptibility $\chi /(1+N_d\chi )$ relative to the applied field. Shape is especially important for large susceptibility. A long closed magnetic circuit reduces stray fields; a bar magnet still has demagnetizing fields despite the absence of magnetic monopoles. Equivalent magnetic-pole descriptions are bookkeeping for magnetization divergence, not free magnetic charge.^[ref]

In paramagnetism, permanent localized moments are preferentially aligned by the field, against thermal disorder. For independent spin-one-half moments with projections $\pm \mu$, the two Zeeman energies givein the dilute weak-field limit with $B\simeq {\mu }_{0}H$. This is a Curie law; the response saturates at $n\mu$ when alignment dominates. More generally, spin $J$ gives $\chi ={\mu }_{0}n(g{\mu }_B{)}^{2}J(J+1)/(3k_{B}T)$. Interactions can produce a Curie–Weiss response $C/(T-\Theta )$ over a suitable range. Conduction-electron Pauli paramagnetism need not show the localized-moment $1/T$ law.

In diamagnetism, the field induces a magnetic moment opposing it, so the linear susceptibility is negative. The microscopic cause is the change in orbital electronic motion, and a closed-shell insulating material has no permanent atomic moment to orient. This weak response is often only mildly temperature dependent away from other transitions. It also exists as a background in materials with a stronger positive response. Perfect flux expulsion in a superconductor is a distinct collective phenomenon and should not be identified with ordinary weak atomic diamagnetism.

In ferromagnetism, exchange interactions stabilize spontaneous magnetic order below the Curie temperature. A specimen can have zero net moment while its magnetic domains remain locally magnetized in different directions. Domain-wall motion and moment rotation produce a large response; anisotropy and pinning lead to remanence, coercivity and hysteresis. Increasing field eventually approaches saturation. Heating above the ordering transition removes spontaneous order, although an applied field still induces a response. Soft magnetic cores favor easy reversal and small cyclic loss; hard permanent-magnet materials favor retained magnetization and resistance to reversal. These properties concern collective order and barriers, not merely independent moments all following a weak field.^[ref]

A practical distinction requires a controlled field sweep and a separate magnetic readout. A toroidal core with drive winding $N_1$, path length $l$ and current $I$ has approximately $H=N_{1}I/l$. A pickup winding measures ${\mathcal{E}}_2=-N_{2}AdB/dt$; integrating after calibration gives $B$, and $M=B/{\mu }_0-H$. Reversing the sweep reveals a loop whose positive loss area per volume is $\oint HdB$. An air-core/empty-holder reference, known geometry and turn count, offset correction, temperature control and slower sweeps test background, heating and rate-dependent eddy-current losses. A single force or susceptibility reading cannot establish spontaneous order or hysteresis. The original curves below are illustrative responses, not measured material data.

For a uniformly magnetized ring with a narrow gap, take the local direction of magnetization as positive around the magnetic circuit. Neglect leakage and fringing in the narrow gap, and take the cross section as uniform. Continuity of magnetic flux gives $H_g=H_r+M$. With no free current, Ampère's law gives $l{H}_r+a{H}_g=0$. HenceThe rod's field is a demagnetizing field opposite its magnetization; its magnitude is $aM/(l+a)$. The gap field points with the magnetization and $B_g={\mu }_0{H}_g$.

With uniform section and negligible magnetic fringing, constant magnetic flux gives equal magnetic flux densities in the core and gap:Here $L$ is the core arc length and $h$ the additional gap length. Applying the magnetostatic Ampère law around the complete magnetic circuit,Consequently,This tends to $NI/h$ when the gap dominates the magnetic reluctance, $\mu h\gg L$, and reduces to the no-gap result as $h\to 0$. The gap magnetic flux density is $B_{\mathrm{gap}}={\mu }_0{H}_{\mathrm{gap}}$.

In this magnetic circuit approximation, each circular path has air-gap length $d$, core length $2\pi r-d$, and the same normal magnetic flux density in air and core. Ampère's law then gives $B(r)\simeq {\mu }_0{\mu }_{r}NI/[2\pi r+({\mu }_r-1)d]$; integrating across the section gives the displayed mutual inductance. Even $d\ll a$ can strongly reduce inductance when ${\mu }_{r}d/a$ is appreciable. This is a path estimate, not an exact local field solution in the slit geometry. It neglects magnetic fringing and redistribution between paths. In particular, a literal field proportional to $\hat{\mathbf{\varphi }}/(r+q)$ everywhere in a current-free core is not curl-free; transverse corrections are required. Narrow-gap circuit approximations and their limitations are discussed in ^[ref].

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.