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Magnetic scalar potential

The magnetic scalar potential, denoted as \Phi_m or \psi, is a scalar field in classical electromagnetism that describes the magnetic field intensity \mathbf{H} in regions free of free currents, where \mathbf{H} = -\nabla \Phi_m.^[1]^[2]^[3] This potential is analogous to the electric scalar potential in electrostatics, enabling the representation of irrotational magnetic fields (\nabla \times \mathbf{H} = 0) as the negative gradient of a scalar function, which simplifies calculations in magnetostatics by reducing the problem to solving scalar differential equations rather than vector ones.^[1]^[2] In current-free regions, the magnetic scalar potential satisfies Laplace's equation \nabla^2 \Phi_m = 0, allowing for analytical or numerical solutions similar to those in electrostatics.^[1]^[3] When magnetic materials are present, such as in hard ferromagnets with magnetization \mathbf{M}, the potential obeys Poisson's equation \nabla^2 \Phi_m = -\rho_m / \mu_0, where \rho_m = -\nabla \cdot \mathbf{M} represents the volume magnetic charge density, and surface charges \sigma_m = \mathbf{M} \cdot \hat{\mathbf{n}} arise at boundaries.^[2]^[3] Boundary conditions for \Phi_m include continuity across interfaces except where currents cause discontinuities, and the potential may be multi-valued in multiply connected domains enclosing net currents, reflecting the topological nature of magnetic fields.^[1]^[2] This formulation is particularly useful in engineering applications, such as designing permanent magnets, solenoids, and magnetic circuits, where it facilitates the computation of \mathbf{H} and \mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M}) without directly solving for the magnetic vector potential \mathbf{A}.^[1]^[2] In time-harmonic or dynamic fields, extensions incorporate additional terms, but the scalar potential remains valuable for decoupling equations in regions with magnetic sources.^[4]

Fundamentals

Definition

The magnetic scalar potential, denoted as \phi_m, is a scalar quantity employed in classical electromagnetism to describe the magnetic field strength \mathbf{H} in regions devoid of free currents, where \nabla \times \mathbf{H} = 0. This condition allows \mathbf{H} to be expressed as the negative gradient of \phi_m, simplifying the analysis of magnetostatic fields by reducing vector problems to scalar ones analogous to those in electrostatics.^[5]^[1] Introduced in the 19th century by Siméon Denis Poisson in 1824, the concept drew direct analogy to the electric scalar potential, enabling the modeling of magnetic effects through hypothetical "magnetic poles" and surface/volume densities, much like electric charges.^[6] This framework was further refined in magnetostatics by William Thomson (Lord Kelvin) and others, who integrated it into broader electromagnetic theory during the mid-1800s.^[6] In SI units, \phi_m is measured in amperes (A), as it represents the line integral of \mathbf{H} along a path, yielding a unit consistent with the ampere definition in magnetostatics.^[5] The convention \mathbf{H} = -\nabla \phi_m incorporates the negative sign to align with the conservative nature of such fields, ensuring the potential decreases in the direction of \mathbf{H}, similar to gravitational or electric potentials.^[7] This scalar approach contrasts with the magnetic vector potential \mathbf{A}, which is necessary in regions with currents where \nabla \times \mathbf{H} \neq 0./09%3A_Magnetic_Potential/9.02%3A_The_Magnetic_Vector_Potential)

Mathematical Formulation

In magnetostatics, the magnetic scalar potential \phi_m is introduced in regions where the free current density \mathbf{J}_f = 0. From Ampère's law in the form \nabla \times \mathbf{H} = \mathbf{J}_f, the absence of free currents implies \nabla \times \mathbf{H} = 0, meaning \mathbf{H} is irrotational and can be expressed as the negative gradient of a scalar potential: \mathbf{H} = -\nabla \phi_m.^[8] Combining this with Gauss's law for magnetism, \nabla \cdot \mathbf{B} = 0, and the constitutive relation \mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M}) in materials with magnetization \mathbf{M}, yields \nabla \cdot (\mathbf{H} + \mathbf{M}) = 0, or \nabla \cdot \mathbf{H} = -\nabla \cdot \mathbf{M}. Substituting \mathbf{H} = -\nabla \phi_m gives the Poisson equation for the scalar potential:

\nabla^2 \phi_m = \nabla \cdot \mathbf{M}.

This equation treats \nabla \cdot \mathbf{M} as an effective magnetic charge density source, analogous to \rho in electrostatics.^[8] In source-free regions where \mathbf{M} = 0 and the permeability \mu is constant (typically \mu = \mu_0 in vacuum), the equation simplifies to Laplace's equation:

\nabla^2 \phi_m = 0.

Solutions to this equation in such regions describe the potential in a manner similar to electrostatics, with \phi_m harmonic and determined by boundary values. For non-constant \mu, the more general form is \nabla \cdot (\mu \nabla \phi_m) = 0.^[9] Boundary conditions for \phi_m arise from the continuity of the tangential component of \mathbf{H} and the normal component of \mathbf{B} across interfaces in current-free regions. The tangential continuity implies \phi_m is continuous: \phi_{m1} = \phi_{m2}. The normal continuity of \mathbf{B} requires \mu_1 \frac{\partial \phi_{m1}}{\partial n} = \mu_2 \frac{\partial \phi_{m2}}{\partial n}, where n is the normal direction; in the absence of surface magnetization charges, there is no jump in the normal derivative. These conditions ensure the potential and its derivative match appropriately at material boundaries or interfaces.^[8]^[9]

Relation to Magnetic Fields

In Current-Free Regions

In regions devoid of free currents, the magnetic field intensity \mathbf{H} satisfies \nabla \times \mathbf{H} = 0, making \mathbf{H} irrotational and expressible as the negative gradient of a magnetic scalar potential \phi_m, such that \mathbf{H} = -\nabla \phi_m.^[1] This condition holds because Ampère's law in the absence of free currents (\mathbf{J}_f = 0) implies the curl-free nature of \mathbf{H}.^[9] Consequently, \phi_m satisfies Laplace's equation \nabla^2 \phi_m = 0 in vacuum or regions of uniform permeability, derived from \nabla \cdot \mathbf{B} = 0 and \mathbf{B} = \mu \mathbf{H}.^[1]^[2] To compute \mathbf{H}, one solves Laplace's equation for \phi_m subject to appropriate boundary conditions, such as specified values of \phi_m or its normal derivative on surfaces enclosing the region.^[9] These boundary conditions typically arise from the continuity of the tangential component of \mathbf{H} (ensuring \phi_m is continuous) and the normal component of \mathbf{B} across interfaces.^[9] Once \phi_m is determined, \mathbf{H} follows directly from the gradient operation.^[1] A representative example is the interior of a long solenoid, where the magnetic field is approximately uniform and directed along the axis (z-direction), \mathbf{H} = H_0 \hat{z}, far from the current-carrying windings.^[10] In this case, the scalar potential takes the simple form \phi_m = -H_0 z, satisfying \mathbf{H} = -\nabla \phi_m and Laplace's equation in the current-free interior.^[11] This linear potential reflects the uniformity of the field, analogous to electrostatic potentials in uniform fields.^[10] The use of the magnetic scalar potential offers significant advantages by transforming vector field problems into scalar ones, specifically solving Laplace's (or Poisson's in more general cases) equations rather than full vector formulations.^[2] This simplification facilitates analytical solutions in symmetric geometries and enhances efficiency in numerical methods, such as the finite element method, where scalar variables reduce computational complexity compared to vector potentials.^[2] In regions with varying permeability, such as magnetizable materials, the formulation extends to a more general equation \nabla \cdot (\mu \nabla \phi_m) = 0, but the core principles remain rooted in current-free conditions.^[9]

In Magnetizable Materials

In regions containing magnetizable materials, the magnetic scalar potential \phi_m is defined such that the magnetic field intensity \mathbf{H} = -\nabla \phi_m, analogous to current-free regions but now accounting for magnetization \mathbf{M}. From Maxwell's equation \nabla \cdot \mathbf{B} = 0 and the constitutive relation \mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M}), it follows that \nabla \cdot \mathbf{H} = -\nabla \cdot \mathbf{M}, leading to Poisson's equation \nabla^2 \phi_m = \nabla \cdot \mathbf{M}. Here, \nabla \cdot \mathbf{M} acts as an effective source term, analogous to magnetic charge density \rho_m = -\nabla \cdot \mathbf{M}, which arises from the divergence of magnetization within the material.^[12]^[13] The magnetization \mathbf{M} also gives rise to bound currents, with volume bound current density \mathbf{J}_b = \nabla \times \mathbf{M} and surface bound current density \mathbf{K}_b = \mathbf{M} \times \hat{n}, where \hat{n} is the outward normal. These bound currents produce effects equivalent to the magnetic charges in sourcing the scalar potential, particularly in regions without free currents, leading to discontinuities or jumps in \phi_m or its derivatives at material interfaces due to surface bound charges \sigma_m = \mathbf{M} \cdot \hat{n}. In the scalar potential formulation, these are incorporated through the source terms rather than directly via Ampère's law.^[13]^[12] For linear isotropic media, where \mathbf{M} = \chi_m \mathbf{H} and permeability \mu = \mu_0 (1 + \chi_m) = \mu_r \mu_0 with relative permeability \mu_r, the relation simplifies to \mathbf{B} = \mu \mathbf{H}. Substituting into \nabla \cdot \mathbf{B} = 0 yields the governing equation \nabla \cdot (\mu \nabla \phi_m) = 0 in regions of constant \mu, reducing to Laplace's equation \nabla^2 \phi_m = 0 within uniform material domains. At interfaces between materials with permeabilities \mu_1 and \mu_2, continuity of the normal component of \mathbf{B} implies \mu_1 \frac{\partial \phi_m}{\partial n}\big|_1 = \mu_2 \frac{\partial \phi_m}{\partial n}\big|_2, while \phi_m itself remains continuous to ensure tangential \mathbf{H} continuity in the absence of free surface currents.^[12]^[14] A representative example is a sphere of linear isotropic ferromagnetic material with permeability \mu placed in a uniform applied field \mathbf{H}_0 = H_0 \hat{z}. Inside the sphere, the field is uniform, with the scalar potential \phi_m^\text{in} = -H_\text{in} r \cos\theta, where H_\text{in} = \frac{3 H_0}{\mu_r + 2} and \mu_r = \mu / \mu_0. This interior potential is thus proportional to the applied field strength H_0, reflecting the material's enhancement of the field lines. Outside, the potential includes a dipole term perturbing the uniform applied potential \phi_m^\text{out} = -H_0 r \cos\theta + \frac{A}{r^2} \cos\theta, with A determined by boundary conditions to match the interior solution.^[14]^[12]

Applications

Magnetostatics Problems

In magnetostatics, the magnetic scalar potential \phi_m is employed to solve problems in current-free regions (\mathbf{J} = 0) by defining \mathbf{H} = -\nabla \phi_m, which satisfies \nabla \times \mathbf{H} = 0 and leads to Laplace's equation \nabla^2 \phi_m = 0 in regions of uniform permeability. The typical workflow begins by identifying current-free domains, such as air gaps or non-conducting materials, where the potential can be applied. Boundary conditions are then established from known \mathbf{H} or \mathbf{B} fields on the surfaces, often derived from Ampère's law around current-carrying elements. The equation is solved analytically or numerically for \phi_m, after which \mathbf{H} = -\nabla \phi_m and \mathbf{B} = \mu \mathbf{H} (with \mu the permeability) are computed to obtain the fields. This approach simplifies computations by reducing the vector problem to a scalar one, particularly useful in linear media.^[2]^[15] A representative application occurs in the air gaps of electromagnetic devices like transformers and relays, where the scalar potential models fringing fields that extend beyond the ideal gap boundaries. In such setups, currents in windings produce a magnetomotive force that drives the field across the gap, but fringing causes non-uniform \mathbf{H} distributions at the edges due to the high contrast in permeability between the ferromagnetic core (\mu \gg \mu_0) and air (\mu = \mu_0). For instance, in a transformer core with an air gap, \phi_m satisfies Laplace's equation in the gap, with boundary conditions matching the tangential \mathbf{H} continuity and normal \mathbf{B} continuity at the core-air interface; solutions reveal field lines bulging outward, decreasing effective reluctance by up to 10-20% compared to uniform approximations. This modeling aids in predicting leakage flux and optimizing gap lengths to prevent saturation. Similar fringing analysis applies to relays, where the potential jump across the gap determines actuation forces.^[16] For complex geometries, numerical methods integrate the scalar potential into boundary element methods (BEM) and finite element analysis (FEA) to handle 2D or 3D simulations. In BEM, the potential is formulated via surface integrals over boundaries, ideal for infinite domains like exterior fields, while FEA discretizes the volume in current-free regions, solving the weak form of Laplace's equation with Galerkin methods. Hybrid FEM-BEM couplings are common, using FEA inside bounded regions (e.g., device interiors) and BEM for unbounded exteriors, ensuring continuity of \phi_m and its normal derivative at interfaces; this reduces computational cost for large-scale magnetostatic problems in engineering designs. These techniques have been applied to simulate fields in permanent magnet devices with accuracies below 1% error in benchmark tests.^[17]^[18] The scalar potential formulation is limited in regions with free currents, as \nabla \times \mathbf{H} \neq 0 invalidates \mathbf{H} = -\nabla \phi_m; it cannot cross current sheets without discontinuities. Near currents, hybrid approaches combine the scalar potential in current-free zones with the magnetic vector potential \mathbf{A} (where \mathbf{B} = \nabla \times \mathbf{A}) in conducting regions, matching fields at boundaries via multi-region formulations. This transition ensures global solutions but increases complexity for overall simulations.^[2]^[19]

Geomagnetism and Earth's Field

In the region exterior to Earth's core, where electric currents are negligible (approximating \mathbf{J} \approx 0), the geomagnetic field \mathbf{B} is irrotational and divergence-free, allowing it to be expressed as the negative gradient of a magnetic scalar potential \phi_m that satisfies Laplace's equation \nabla^2 \phi_m = 0. This formulation is particularly useful for modeling the main geomagnetic field, which originates from dynamo processes in the fluid outer core and propagates outward through current-free mantle and crustal regions via boundary conditions at the core-mantle interface.^[20] The scalar potential is typically expanded in spherical harmonics to represent the field's spatial variation, given by

\phi_m(r, \theta, \phi, t) = a \sum_{\ell=1}^{N} \sum_{m=0}^{\ell} \left( \frac{a}{r} \right)^{\ell+1} \left[ g_\ell^m(t) \cos(m\phi) + h_\ell^m(t) \sin(m\phi) \right] P_\ell^m (\cos \theta),

where a = 6371.2 km is Earth's reference radius, P_\ell^m are the associated Legendre functions, and the Gauss coefficients g_\ell^m(t) and h_\ell^m(t) are time-dependent, determined from global magnetic measurements; the expansion excludes positive powers of r due to the internal source nature of the field. The dominant term is the axial dipole (\ell=1, m=0), which accounts for approximately 90% of the field strength at Earth's surface, with higher-order multipoles capturing deviations from a perfect dipole.^[21] A prominent example is the International Geomagnetic Reference Field (IGRF), a series of mathematical models that parameterize the main field using scalar potential coefficients up to degree and order 13, updated every five years to reflect secular variation. The latest iteration, IGRF-14, released in November 2024, provides definitive coefficients from 1900 to 2020 and predictive values through 2030, enabling precise computation of field components for navigation, satellite operations, and geophysical studies.^[21] Crustal magnetic anomalies, arising from magnetized rocks in the lithosphere, are incorporated as high-degree perturbations to the scalar potential in advanced models, often extending spherical harmonics to degrees beyond 720 to resolve features down to wavelengths of about 50 km. These anomalies, mapped via satellite missions like Swarm and CHAMP, aid mineral exploration by identifying iron ore deposits and other magnetic mineral concentrations through targeted airborne surveys that interpret potential variations.^[22]^[23]

Comparisons and Extensions

With Electric Scalar Potential

The magnetic scalar potential \phi_m and the electric scalar potential V share a fundamental analogy as scalar functions used to describe conservative fields in electromagnetism. In electrostatics, the irrotational nature of the electric field, given by \nabla \times \mathbf{E} = 0, allows the definition \mathbf{E} = -\nabla V, where V simplifies the computation of field lines and energies in charge distributions. Similarly, in regions free of currents, \nabla \times \mathbf{H} = 0 enables \mathbf{H} = -\nabla \phi_m, providing a parallel scalar representation for the magnetic field intensity \mathbf{H}, which aids in solving boundary value problems without vector complications.^[24]^[9] A key distinction arises from the underlying source terms dictated by Maxwell's equations. For the electric potential, Gauss's law \nabla \cdot \mathbf{E} = \rho / \epsilon_0 leads to Poisson's equation \nabla^2 V = -\rho / \epsilon_0, incorporating charge density \rho as the source, which enables direct computation of potentials from localized charges. In contrast, the absence of magnetic monopoles, enshrined in Gauss's law for magnetism \nabla \cdot \mathbf{B} = 0 (and thus \nabla \cdot \mathbf{H} = 0 in vacuum where \mathbf{B} = \mu_0 \mathbf{H}), results in Laplace's equation \nabla^2 \phi_m = 0 for the magnetic scalar potential, implying no analogous "magnetic charge" sources and requiring boundary conditions to determine the field. This difference underscores the topological distinction between electric fields, which can originate from point sources, and magnetic fields, which form closed loops.^[25]^[1] The units of these potentials reflect their physical origins: the electric potential V is measured in volts (joules per coulomb), linking directly to energy per unit charge in electrostatic interactions driven by charges. The magnetic scalar potential \phi_m, however, is expressed in amperes, consistent with \mathbf{H} deriving from current distributions or magnetization rather than monopolar sources, as the gradient operation yields amperes per meter for \mathbf{H}. This unit choice highlights the current-based nature of magnetism versus the charge-based electrostatics./05%3A_Electrostatics/5.15%3A_Poissons_and_Laplaces_Equations)^[26] Historically, both potentials emerged in the 19th century amid efforts to mathematize electromagnetism, with the electric scalar potential formalized by pioneers like Laplace and Poisson in the context of gravitational and electrostatic analogies. The magnetic scalar potential was introduced by Siméon-Denis Poisson in 1824, building on Ampère's current theories but incorporating fictitious magnetic poles to mimic electrostatic methods, while adhering to the monopole-free constraint of Gauss's law for magnetism established shortly thereafter. This development facilitated early calculations of magnetic fields in permanent magnets and currents, paralleling electrostatic advances without violating fundamental laws.^[27]^[28]

With Magnetic Vector Potential

The magnetic vector potential \mathbf{A} is defined such that the magnetic flux density \mathbf{B} = \nabla \times \mathbf{A}, providing a vectorial representation that is valid throughout space in magnetostatics, regardless of current distributions.^[29] In contrast, the magnetic scalar potential \phi_m offers a scalar alternative, where \mathbf{H} = -\nabla \phi_m, but it is only applicable in regions where \nabla \times \mathbf{H} = 0, such as current-free spaces.^[1]^[8] The choice between \phi_m and \mathbf{A} depends on the problem's geometry and sources: \phi_m is preferable in current-free regions for its simplicity, as it satisfies the scalar Laplace equation \nabla^2 \phi_m = 0, reducing computational complexity compared to the vector Poisson equation \nabla^2 \mathbf{A} = -\mu_0 \mathbf{J} for \mathbf{A} under the Coulomb gauge.^[1]^[30] Near currents where \nabla \times \mathbf{H} \neq 0, \mathbf{A} is essential, as \phi_m becomes ill-defined or multi-valued.^[8] Both potentials exhibit non-uniqueness, but differently: \mathbf{A} has gauge freedom, allowing \mathbf{A}' = \mathbf{A} + \nabla \psi for arbitrary scalar \psi without altering \mathbf{B}, often fixed by the Coulomb gauge \nabla \cdot \mathbf{A} = 0 to simplify equations.^[29]^[31] Meanwhile, \phi_m is unique up to an additive constant in simply connected domains.^[1] For complex geometries involving both current-free and current-laden regions, hybrid formulations combine \phi_m in non-conducting areas with \mathbf{A} in conducting ones, enabling efficient numerical solutions like finite element methods.^[32] A representative example is the infinite solenoid with uniform axial field \mathbf{B} = B \hat{z} inside: \mathbf{A} takes an azimuthal form \mathbf{A} = \frac{1}{2} B s \hat{\phi} (in cylindrical coordinates s, \phi, z) near the windings to capture the circulating currents, while \phi_m = -H z yields a uniform gradient inside, simplifying to a linear function.^[31]^[10] At the current sheet on the solenoid surface, \phi_m exhibits a discontinuity equal to the surface current times the azimuthal angle, requiring special boundary handling for continuity across the sheet.^[10]

References

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The vector potential A describes magnetic fields that possess curl wherever there is a current density J (r). In the space free of current, ...
[2]
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Therefore, one can use methods of solving differential equations to find the magnetic scalar potential and therefore the magnetic fields H and B. In hard ...
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(1.345) B → = - ∇ → Φ M . Applying the divergence equations leads to Laplace equations for the electric and magnetic scalar potential: (1.346) ...<|control11|><|separator|>
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Just as E = -∇V, we define the magnetic scalar potential Vm (in amperes) as related to H according to. H = - ∇ Vm if J = 0. (1.28).
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... magnetic scalar potential. Now, we know that. ∇ · B = µ0 ∇ · (H + M) = 0. (6.22). Equations (6.21) and (6.22) combine to give. ∇2φm = −ρm,. (6.23) where ρm ...
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[16]
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In this thesis a novel formulation using the magnetic scalar potential in the air region and magnetic vector potential in the conducting region will be used.