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Magnetic current

Magnetic current, more precisely termed magnetic current density in electromagnetism, is a hypothetical quantity representing the flow of magnetic charge per unit area per unit time, analogous to electric current density but arising from unconfirmed magnetic monopoles.^[1] It carries units of volts per square meter (V/m²) or equivalently webers per second per square meter (Wb/s/m²).^[1] Although no free magnetic monopoles exist in nature, the concept is mathematically useful for symmetrizing Maxwell's equations and modeling effects in magnetic materials.^[2] In the standard formulation of Maxwell's equations without magnetic sources, the curl of the electric field is given by ∇ × E = −∂B/∂t, reflecting Faraday's law of induction.^[1] To introduce symmetry with the electric case—where ∇ × H = J + ∂D/∂t includes the electric current density J—a magnetic current density J_m is added, yielding ∇ × E = −J_m − ∂B/∂t.^[2] This extension also incorporates a magnetic charge density ρ_m in ∇ · B = μ₀ ρ_m, ensuring consistency with the continuity equation ∇ · J_m + ∂ρ_m/∂t = 0.^[3] The approach highlights the duality principle in electromagnetism, where electric and magnetic quantities can be interchanged under certain transformations.^[4] Practically, magnetic current density serves as an equivalent source term in computational electromagnetics and antenna theory, particularly for analyzing apertures, slots, or time-varying magnetization in materials, where J_m = μ₀ ∂M/∂t and M is the magnetization vector.^[4] For instance, in problems involving perfect magnetic conductors or image theory, equivalent magnetic currents simplify boundary value solutions.^[4] This fictitious construct does not imply physical magnetic charges but aids in deriving boundary conditions and understanding polarization currents in dielectrics and magnetics.^[2] Ongoing searches for magnetic monopoles in particle physics continue to motivate theoretical explorations, though none have been detected.^[3]

Definition and Fundamentals

Definition

Magnetic current is a hypothetical construct in electromagnetism, representing the flow of magnetic monopoles analogous to the flow of electric charges that constitutes an electric current. Unlike electric current, which sources magnetic fields, magnetic current would source electric fields, embodying the duality principle in Maxwell's equations where electric and magnetic quantities are interchanged for theoretical symmetry. This concept arises because magnetic monopoles—isolated north or south magnetic charges—do not exist in nature, making magnetic current a mathematical tool rather than a physical reality, often used to simplify analyses in problems involving time-varying fields or material polarizations.^[4] In this duality, a magnetic current produces an electric field whose direction follows the left-hand rule, opposite to the right-hand rule governing the magnetic field produced by an electric current; specifically, the transformation in symmetric Maxwell's equations includes sign changes (e.g., \mathbf{H} \to -\mathbf{E}) that reverse the curl direction, altering the handedness.^[4]^[5] The total magnetic current is denoted by the symbol k, with units of volts (V), while the magnetic current density is denoted as \mathfrak{M}, with units of volts per square meter (V/m²). These units reflect the dual nature: just as electric current density \mathbf{J} has units of amperes per square meter (A/m²), magnetic current density carries the dual unit derived from electromagnetic symmetry.^[6]^[4] This formulation highlights the asymmetry in observed electromagnetism, where electric charges are ubiquitous but magnetic monopoles are hypothetical, yet introducing magnetic current enables elegant symmetrization of field equations, aiding in the study of phenomena like polarization currents in materials. For instance, equivalent magnetic current density can be expressed as \mathbf{J}_m = \mu_0 (\mu_r - 1) \partial \mathbf{H}/\partial t, linking it to time derivatives of the magnetic field in magnetized media, though only for varying fields.^[4]

Historical Development

The concept of magnetic current originated in the late 19th century amid efforts to address the apparent asymmetry in James Clerk Maxwell's original formulation of electromagnetism, which treated electric charges and currents as sources of magnetic fields but lacked analogous magnetic sources for electric fields. In 1885, Oliver Heaviside proposed symmetrizing the equations by introducing hypothetical magnetic charges and currents, thereby establishing a formal duality between electric and magnetic phenomena that mirrored the observed reciprocity in electromagnetic interactions.^[7] This innovation, detailed in Heaviside's reformulation of Maxwell's equations into their modern vector form, provided a theoretical framework for treating magnetic currents as the flow of such fictitious magnetic charges, motivated by the desire to unify the treatment of electric and magnetic fields.^[8] The theoretical underpinnings of magnetic currents advanced significantly in the 20th century through quantum mechanics. In 1931, Paul Dirac demonstrated that the existence of even a single magnetic monopole would impose quantization on electric charge, implying that magnetic charges—and by extension, magnetic currents arising from their motion—could reconcile quantum electrodynamics with the Dirac equation for electrons.^[9] Dirac's analysis, published in the Proceedings of the Royal Society, elevated the concept from a mathematical convenience to a physically plausible entity, albeit unobserved, linking magnetic currents to broader symmetries in fundamental physics. Post-World War II developments shifted focus toward engineering applications, where magnetic currents proved invaluable as auxiliary constructs. In 1961, Roger F. Harrington's seminal book Time-Harmonic Electromagnetic Fields formalized their use in solving time-harmonic boundary value problems, particularly through the equivalence principle, which allows complex structures to be modeled by equivalent electric and magnetic surface currents.^[10] This approach simplified computations for antennas and scattering problems by exploiting the duality symmetrized by Heaviside, enabling engineers to replace intricate geometries with distributed current sources without altering far-field radiation patterns. The concept persists in contemporary electromagnetics education and research, underscoring its enduring utility. For instance, Constantine A. Balanis' 2012 edition of Advanced Engineering Electromagnetics employs magnetic currents to analyze aperture antennas and slot radiators, illustrating their role in modeling equivalent sources for practical design.^[11] Overall, the evolution of magnetic currents reflects a sustained motivation to symmetrize Maxwell's inherently asymmetric equations and streamline the resolution of boundary conditions in electromagnetic theory via duality principles.^[12]

Theoretical Foundations in Electromagnetism

Relation to Magnetic Monopoles

A magnetic monopole is a hypothetical elementary particle that carries an isolated magnetic charge, analogous to the electric charge of an electron but manifesting as a single north or south magnetic pole without a counterpart. Such a particle would possess a magnetic charge g, typically expressed in units of weber (Wb) or ampere-meter (A·m), which would source a radial magnetic field \mathbf{B} \propto g / r^2 diverging from or converging to the monopole's position. The concept, first theoretically proposed by Paul Dirac in 1931 to explain electric charge quantization, remains unobserved despite extensive searches. In the framework of particle physics, a magnetic current density \mathbf{J}_m naturally arises from the collective motion of these monopoles, defined as \mathbf{J}_m = \rho_m \mathbf{v}, where \rho_m is the magnetic charge density and \mathbf{v} is the velocity of the monopoles. This expression mirrors the definition of electric current density from moving electric charges, enabling a symmetric treatment of electric and magnetic sources in electromagnetism. If monopoles existed, their motion would generate time-varying magnetic fields, contributing to the propagation of electromagnetic disturbances in a manner dual to electric currents. The existence of magnetic monopoles would introduce a conservation law for magnetic charge, \partial \rho_m / \partial t + \nabla \cdot \mathbf{J}_m = 0, paralleling the continuity equation for electric charge and restoring symmetry to Maxwell's equations. Accelerating monopoles could radiate electromagnetic waves where the roles of the electric field \mathbf{E} and magnetic field \mathbf{B} are interchanged compared to waves from accelerating electric charges, potentially altering predictions for radiation patterns and energy transport in high-energy processes. Dirac's quantization condition links electric and magnetic charges through the relation e g = n \hbar c / 2, where e is the elementary electric charge, n is an integer, \hbar is the reduced Planck's constant, and c is the speed of light, ensuring single-valuedness of the quantum mechanical wavefunction in the presence of a monopole. This topological constraint implies that magnetic charges must be quantized in discrete units tied to electric charge, with the minimal monopole strength g_D = \hbar c / (2 e). In cosmology, grand unified theories (GUTs) predict the production of magnetic monopoles during symmetry-breaking phase transitions in the early universe, where the Higgs mechanism generates stable monopole configurations with masses around $10^{16} GeV. However, the observed monopole density is far lower than expected, a discrepancy known as the monopole problem, resolved by cosmic inflation which exponentially dilutes their abundance shortly after formation. As of November 2025, no magnetic monopoles have been detected in experiments such as MoEDAL at the LHC or neutrino observatories like IceCube, though searches continue. If monopoles were detected, they could influence baryogenesis mechanisms, potentially contributing to the observed matter-antimatter asymmetry through processes like monopole-catalyzed baryon number violation in GUT models.^[13]^[14]^[15]

Symmetrized Maxwell's Equations

The standard Maxwell's equations in differential form, applicable in media and without magnetic monopoles, are given by:

\begin{align} \nabla \cdot \mathbf{D} &= \rho_e, \\ \nabla \cdot \mathbf{B} &= 0, \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t}, \\ \nabla \times \mathbf{H} &= \mathbf{J}_e + \frac{\partial \mathbf{D}}{\partial t}, \end{align}

where \mathbf{D} is the electric displacement field, \mathbf{B} is the magnetic flux density, \mathbf{E} is the electric field, \mathbf{H} is the magnetic field strength, \rho_e is the electric charge density, and \mathbf{J}_e is the electric current density (in SI units).^[2] To incorporate hypothetical magnetic monopoles, the equations are symmetrized by introducing a magnetic charge density \rho_m and a magnetic current density \mathbf{J}_m, yielding:

\begin{align} \nabla \cdot \mathbf{D} &= \rho_e, \\ \nabla \cdot \mathbf{B} &= \rho_m, \\ \nabla \times \mathbf{E} &= -\mathbf{J}_m - \frac{\partial \mathbf{B}}{\partial t}, \\ \nabla \times \mathbf{H} &= \mathbf{J}_e + \frac{\partial \mathbf{D}}{\partial t}, \end{align}

in the symmetric formulation emphasizing duality (with \mathbf{J}_m in V/m² and \rho_m in Wb/m³).^[1] The magnetic charge density \rho_m satisfies a continuity equation \frac{\partial \rho_m}{\partial t} + \nabla \cdot \mathbf{J}_m = 0, analogous to the electric continuity equation.^[2] The magnetic current density \mathbf{J}_m plays a crucial role by serving as a source for the curl of the electric field \mathbf{E}, directly analogous to how the electric current density \mathbf{J}_e sources the curl of the magnetic field strength \mathbf{H}. This term arises from the motion of magnetic charges and restores balance in the dynamical equations, allowing magnetic currents to induce electric fields just as electric currents induce magnetic fields.^[16] The symmetrization is motivated by a derivation outline that adds magnetic source terms to the original equations, restoring electromagnetic duality (where electric and magnetic quantities are interchanged with scaling by the speed of light c and impedance of free space Z_0 = \sqrt{\mu_0 / \epsilon_0} to preserve SI unit consistency) and ensuring full Lorentz invariance of the theory. This extension, first proposed in the context of quantized singularities, aligns the equations with special relativity by treating electric and magnetic quantities on equal footing.^[16] These symmetrized equations remain consistent with known physics, leading to a modified wave equation for the fields that includes magnetic source terms while preserving propagation at the speed of light c = 1 / \sqrt{\mu_0 \epsilon_0} in vacuum; for example, taking the curl of Faraday's law and substituting Ampère's law yields \nabla^2 \mathbf{E} - \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = -\nabla \times \mathbf{J}_m - \frac{1}{\epsilon_0} \nabla \rho_e + \cdots, confirming electromagnetic waves sourced by both electric and magnetic currents.^[16]

Mathematical Description

Magnetic Current Density

Magnetic current density, often denoted as \mathbf{J}_m or \mathfrak{M}, is a vector field in theoretical electromagnetism that describes the flow of magnetic charge, serving as a source term in the symmetrized Maxwell's equations for scenarios involving magnetic monopoles.^[4] This quantity is analogous to electric current density \mathbf{J}_e, but pertains to magnetic charges rather than electric ones, enabling a dual symmetry in the equations.^[17] In the SI system, \mathbf{J}_m has units of volts per square meter (V/m²), consistent with the units of \nabla \times \mathbf{E} and engineering applications such as volume densities in antenna modeling.^[4] The total effective magnetic source in Faraday's law comprises the conduction component \mathbf{J}_m (or \mathfrak{M}^i), arising from external sources like moving magnetic monopoles, and the displacement term \partial \mathbf{B}/\partial t. This decomposition allows separation of true source contributions from induced effects in field calculations. It appears in the \nabla \times \mathbf{E} equation of the symmetrized Maxwell's equations as -\mathbf{J}_m - \partial \mathbf{B}/\partial t.^[18] Physically, a steady magnetic current density generates static electric fields that form closed loops encircling the current, mirroring the way a steady electric current produces looping magnetic fields via Ampère's law.^[19] This interpretation underscores the duality between electric and magnetic phenomena, where magnetic currents act as drivers for electric field circulation in monopole-inclusive theories.^[20] At boundaries or interfaces, the normal component of the magnetic current density influences discontinuities in the tangential electric field, analogous to how the tangential electric current affects the magnetic field in standard electromagnetism. Specifically, the jump in tangential \mathbf{E} across a surface is proportional to the normal \mathbf{J}_m, providing a boundary condition for solving field problems.^[21] In Schelkunoff's equivalence principle formulation, magnetic current densities are employed to create equivalent surface sources that replicate the external fields of scattering or aperture problems, combining with electric currents to fully characterize radiation from enclosed volumes.^[17] This approach simplifies analysis in diffraction and antenna design by replacing complex volume distributions with boundary currents.

Magnetic Displacement Current

The magnetic displacement current refers to the term involving the time derivative of the magnetic flux density, \partial \mathbf{B}/\partial t, which acts as an effective source in the context of hypothetical magnetic monopoles and symmetrized electromagnetism. In SI units, it is expressed as \partial \mathbf{B}/\partial t within Faraday's law, contributing to the total source alongside the conduction magnetic current density \mathbf{J}_m. This term ensures that a time-varying magnetic field behaves analogously to a conduction current in sourcing an electric field.^[22] In the modified Faraday's law, \nabla \times \mathbf{E} = -\mathbf{J}_m - \partial \mathbf{B}/\partial t, the magnetic displacement current \partial \mathbf{B}/\partial t accounts for the induced electric field due to changing magnetic flux, even in the absence of conduction currents. This formulation maintains continuity by allowing the curl of the electric field to be sourced by dynamic magnetic fields, mirroring the structure of Ampère's law with electric currents. Without this term, the equations would violate charge conservation for magnetic charges.^[23] The concept draws a direct analogy to the electric displacement current \partial \mathbf{D}/\partial t in the Ampère-Maxwell law, \nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t, where the time-varying electric field generates a magnetic field. Similarly, \partial \mathbf{B}/\partial t completes the symmetry, enabling the propagation of electromagnetic waves in vacuum by coupling changing electric and magnetic fields. In regions without monopoles, the displacement term alone drives wave dynamics, as seen in standard Maxwell's equations where \mathbf{J}_m = 0. With monopoles present, it supplements the conduction term, preserving wave consistency.^[24] This displacement current is crucial for electromagnetic wave propagation, where in vacuum it solely sources the curl of \mathbf{E}, essential for transverse waves traveling at the speed of light. For instance, the plane wave solutions to Maxwell's equations rely on \partial \mathbf{B}/\partial t inducing \mathbf{E}, and vice versa, ensuring energy transport without net charge or monopole flow.^[23] The term arises from ensuring consistency with magnetic charge conservation. Taking the divergence of Faraday's law, \nabla \cdot (\nabla \times \mathbf{E}) = 0 = -\partial (\nabla \cdot \mathbf{B})/\partial t - \nabla \cdot \mathbf{J}_m, and substituting Gauss's law for magnetism, \nabla \cdot \mathbf{B} = \rho_m (where \rho_m is magnetic charge density in Wb/m³), yields $0 = -\partial \rho_m / \partial t - \nabla \cdot \mathbf{J}_m. This gives the continuity equation \partial \rho_m / \partial t + \nabla \cdot \mathbf{J}_m = 0, confirming that the displacement term enforces conservation without monopoles while accommodating them if present.^[22]

Associated Potentials and Formulations

Electric Vector Potential

The electric vector potential serves as the dual counterpart to the magnetic vector potential in formulations of electromagnetism that incorporate magnetic currents, enabling a symmetric description of fields sourced by magnetic monopoles or currents. It arises naturally in the two-potential formalism for solving the symmetrized Maxwell equations, where both electric and magnetic sources are present.^[25] In the time domain, the electric vector potential \mathbf{F}(\mathbf{r}, t) is expressed via the retarded potential integral over the magnetic current density \mathbf{J}_m:

\mathbf{F}(\mathbf{r}, t) = \frac{\epsilon_0}{4\pi} \int \frac{\mathbf{J}_m(\mathbf{r}', t_r)}{|\mathbf{r} - \mathbf{r}'|} \, d^3\mathbf{r}'

where t_r = t - |\mathbf{r} - \mathbf{r}'|/c is the retarded time and c is the speed of light. Within this dual framework, the electromagnetic fields incorporate contributions from \mathbf{F} such that the magnetic field is \mathbf{H} = -\nabla \phi_m - \partial \mathbf{F}/\partial t, with \phi_m the magnetic scalar potential, while the electric displacement satisfies \mathbf{D} = \nabla \times \mathbf{F} in regions free of electric charges. This mirrors the standard expressions \mathbf{E} = -\nabla \phi_e - \partial \mathbf{A}/\partial t and \mathbf{B} = \nabla \times \mathbf{A}, where \phi_e and \mathbf{A} are the electric scalar and magnetic vector potentials, respectively.^[25] The electric vector potential possesses gauge freedom analogous to \mathbf{A}, with the Lorenz gauge condition given by \nabla \cdot \mathbf{F} = -(1/c^2) \partial \phi_m / \partial t, ensuring the potentials satisfy decoupled wave equations.^[25] For static magnetic currents (time-independent \mathbf{J}_m), the expression simplifies to \mathbf{F}(\mathbf{r}) = \frac{\epsilon_0}{4\pi} \int \frac{\mathbf{J}_m(\mathbf{r}') }{ |\mathbf{r} - \mathbf{r}'| } \, d^3\mathbf{r}'. This potential formulation offers advantages in solving the inhomogeneous wave equations sourced by magnetic currents, facilitating computational and analytical treatments of fields in symmetric dual systems without altering the differential structure of Maxwell's equations.^[25]

Phasor Representation

In the phasor representation, electromagnetic fields and sources are analyzed under the time-harmonic assumption, where all quantities vary as e^{j \omega t}, with the real physical fields obtained by taking the real part of the complex phasors. This convention simplifies the analysis of steady-state sinusoidal excitations by replacing time derivatives with multiplication by j \omega, where \omega is the angular frequency. The magnetic current density is thus expressed as \mathfrak{M}^i(\mathbf{r}, t) = \Re \left[ \mathfrak{M}^i(\mathbf{r}) e^{j \omega t} \right], enabling frequency-domain formulations that are particularly useful for harmonic problems in electromagnetism.^[26] The electric vector potential \mathbf{F} in the phasor domain, sourced by the magnetic current density \mathfrak{M}^i, takes the form

\mathbf{F}(\mathbf{r}) = \frac{\varepsilon_0}{4\pi} \int \frac{\mathfrak{M}^i(\mathbf{r}') e^{-j k |\mathbf{r} - \mathbf{r}'|}}{|\mathbf{r} - \mathbf{r}'|} d^3 \mathbf{r}',

where k = \omega / c is the wavenumber, c is the speed of light in vacuum, and \varepsilon_0 is the permittivity of free space. This integral solution incorporates the retarded Green's function for the Helmholtz equation, accounting for wave propagation effects in the frequency domain. The corresponding phasor forms of Maxwell's equations, incorporating magnetic current and the displacement current through frequency-domain terms, are

\nabla \times \mathbf{E} = -j \omega \mathbf{B} - \mathfrak{M}^i, \quad \nabla \times \mathbf{H} = j \omega \mathbf{D} + \mathbf{J}^i,

where the electric displacement \mathbf{D} = \varepsilon_0 \mathbf{E} and magnetic induction \mathbf{B} = \mu_0 \mathbf{H} in free space, with \mathbf{J}^i the electric current density phasor. These equations maintain the duality between electric and magnetic sources while embedding the displacement current in the j \omega terms.^[26]^[6] This phasor framework is widely applied in computational electromagnetics, such as the method of moments (MoM), where integral equations involving \mathbf{F} and \mathfrak{M}^i are discretized to solve for currents and fields in harmonic structures like antennas and scatterers. For plane wave interactions, a phasor magnetic current simplifies to dual polarization states, where the transverse components of \mathfrak{M}^i generate orthogonal electric and magnetic field polarizations, analogous to electric current sources but interchanged in duality. This leads to balanced treatments of TE and TM modes in waveguides or aperture problems.^[26]

Engineering Applications

Role in Antenna Theory

In antenna theory, magnetic currents provide a powerful mathematical tool for modeling and analyzing radiation and scattering problems through equivalence principles. Love's equivalence principle, a key application, allows the replacement of physical apertures or slots in antennas with equivalent surface currents that produce identical fields in the region of interest. Specifically, an aperture can be modeled using a magnetic surface current density \mathbf{M}_s = \mathbf{E} \times \hat{\mathbf{n}}, where \mathbf{E} is the total electric field in the aperture and \hat{\mathbf{n}} is the outward unit normal to the surface, while setting the interior fields to zero.^[27] This formulation simplifies the analysis of open structures like slot antennas by converting volumetric problems into surface integrals, enabling efficient computation of far-field patterns without solving the full interior domain.^[28] Duality principles further extend the utility of magnetic currents in antenna design, establishing equivalence between electric and magnetic sources. An infinitesimal electric dipole antenna, driven by an electric current, produces fields that are dual to those of a small magnetic loop antenna, where the loop acts as a magnetic dipole excited by a magnetic current. This duality swaps electric and magnetic fields (\mathbf{E} \leftrightarrow \mathbf{H}, \mathbf{H} \leftrightarrow -\mathbf{E}) and permittivities with permeabilities, allowing radiation patterns from one to be directly mapped to the other.^[29] Similarly, magnetic currents effectively model the behavior of open-ended waveguides, treating the aperture as a source of magnetic current to predict radiation into free space, which is particularly useful for horn or waveguide-fed antennas.^[30] The incorporation of magnetic currents offers significant benefits in numerical methods for antenna analysis, particularly in reducing computational complexity. In hybrid finite element method (FEM) formulations combined with method of moments (MoM), magnetic surface currents on apertures couple interior volume discretizations with exterior surface integrals, limiting the FEM domain to bounded regions and avoiding infinite meshes for open radiation problems.^[31] This approach decreases the number of unknowns and matrix size compared to pure volume FEM, enhancing efficiency for large-scale simulations. Additionally, the linearity of Maxwell's equations permits superposition of fields from electric and magnetic sources, facilitating the modeling of composite antennas where both types contribute, such as in aperture-coupled patches.^[32] Practical examples illustrate these concepts in antenna modeling. Finite-diameter wire antennas, such as monopoles, can be analyzed by approximating the feed gap or coaxial transition as a sheet of magnetic current, which captures the excitation and computes accurate radiation patterns without detailed volumetric meshing of the feed structure.^[33] Likewise, transformers or baluns in antenna feeds are modeled using distributed sheet magnetic currents to represent the azimuthal magnetic field discontinuity, enabling prediction of input impedance and pattern distortions due to finite size effects.^[33] Roger F. Harrington's foundational work on integral equations prominently features magnetic currents for scattering and radiation analysis. In his formulation, the electric field integral equation (EFIE) is supplemented by the magnetic field integral equation (MFIE), where magnetic currents arise naturally in the boundary conditions for apertures and slots, allowing unified treatment of both electric and magnetic sources in moment-method solutions.^[34] This dual-source approach improves convergence and accuracy for thin-wire and surface scatterers, forming the basis for modern computational electromagnetics in antenna design.^[34]

Magnetic Frill Generator

The magnetic frill generator is an equivalent circuit model that employs a surface of magnetic current density distributed across the annular aperture formed by a coaxial transmission line flush-mounted to a ground plane, thereby simulating the voltage excitation of an antenna without introducing physical connecting wires. This technique is commonly applied to model the feed for wire antennas, such as monopoles or dipoles, by treating the aperture as an idealized source of electromagnetic radiation. The magnetic surface current density \mathbf{M}_s for the frill is derived from the azimuthal electric field in the coaxial TEM mode across the annular region defined by the inner radius a (corresponding to the antenna wire radius) and outer radius b (corresponding to the coaxial shield radius):

\mathbf{M}_s = -\frac{V_0}{\ln(b/a)} \frac{1}{\rho} \hat{\phi}, \quad a \leq \rho \leq b,

where V_0 represents the incident voltage at the feed and \hat{\phi} is the azimuthal unit vector. This expression arises from \mathbf{M}_s = \mathbf{E} \times \hat{\mathbf{n}}, with E_\phi = \frac{V_0}{\ln(b/a)} \frac{1}{\rho}, ensuring that the line integral of the associated electric field across the aperture yields the applied voltage V_0, maintaining equivalence to the physical coaxial feed.^[35] Radiation fields from the magnetic frill generator are computed by integrating \mathbf{M}_s to obtain the electric vector potential \mathbf{F}, from which the far-field electric and magnetic fields are derived for structures like monopoles or dipoles. For instance, the far-field pattern can be expressed using the Fourier transform of the current distribution on the aperture, enabling predictions of directivity and gain patterns. This integration approach leverages the symmetry of the annular source to simplify calculations in spherical coordinates. Key advantages of the magnetic frill generator include the elimination of feed-point singularities that plague simpler models like the delta-gap source, as well as enhanced accuracy for broadband antenna designs where feed geometry influences impedance bandwidth and pattern stability. It facilitates realistic simulations of flush-mounted feeds in method-of-moments or finite-element solvers without mesh discontinuities at the excitation. The model was developed in the 1960s to support early numerical electromagnetics codes for antenna analysis and remains a foundational tool, prominently featured in authoritative texts such as Balanis' Antenna Theory: Analysis and Design. Its adoption stems from seminal work on aperture equivalence principles, enabling precise input impedance and radiation computations for practical wire antenna configurations.

Experimental Status and Analogues

Searches for Real Magnetic Monopoles

Early experimental efforts to detect magnetic monopoles date back to the 1930s, when Felix Ehrenhaft used induction coil detectors to search for monopole-like particles in colloids and gases, observing induced currents attributed to changing magnetic flux as hypothetical monopoles passed through matter.^[36] These claims, suggesting monopoles with charges around the Dirac unit, were highly controversial and later attributed to experimental artifacts such as charged dust particles.^[37] Despite the disputes, such induction techniques laid the groundwork for later searches by exploiting the electromagnetic duality, where a moving monopole induces an electric field akin to Faraday's law.^[38] In modern particle physics, the MoEDAL (Monopole and Exotics Detector at the LHC) experiment, operational since 2015, has conducted extensive searches for magnetic monopoles produced in proton-proton collisions at the Large Hadron Collider.^[39] Using plastic nuclear track detectors and aluminum trapping volumes, MoEDAL identifies highly ionizing particles, setting stringent limits on monopole production for magnetic charges from 1 to 10 times the Dirac unit (g_D = ħc / (2e)), with no detections reported.^[40] As of 2025, these limits exclude monopoles with masses up to several TeV for low charges and higher masses for multiply charged ones, while grand unified theory monopoles are predicted to have masses around 10^{16} GeV/c², far beyond current direct detection capabilities.^[41] Astrophysical searches complement collider efforts, with the IceCube Neutrino Observatory probing for magnetic monopoles through their potential catalysis of proton decay via the Rubakov-Callan effect.^[42] IceCube's deep-ice detectors, spanning over a cubic kilometer, analyze neutrino-like signals from monopole-induced baryon decays in surrounding ice or Earth matter, but no evidence has been found as of 2025.^[43] These analyses set upper limits on the monopole flux at around 10^{-18} cm^{-2} s^{-1} sr^{-1} for sub-relativistic speeds, tightening constraints on relic monopoles from the early universe.^[44] Non-observations across these experiments impose severe constraints on monopole abundance, with the cosmic density limited to below 10^{-18} cm^{-3} from flux measurements in cosmic rays and neutrino data.^[45] Dirac quantization, requiring the product of electric and magnetic charges to be an integer multiple of ħc/2, implies a minimum monopole mass of approximately 10^{16} GeV in grand unified theories, far beyond current direct detection capabilities and consistent with the lack of discoveries.^[46] Monopoles' catalyzing effects, where they could dramatically accelerate particle reactions like proton decay at rates up to 10^{20} times the non-catalyzed rate, have motivated dedicated searches in cosmic rays.^[45] Experiments such as those using scintillator arrays and air shower detectors have scanned for anomalous high-energy events induced by monopole-nucleus interactions in the atmosphere, yielding no confirmations and flux limits below 10^{-19} cm^{-2} s^{-1} sr^{-1} as of 2025.^[47] These indirect probes highlight monopoles' potential role in baryon number violation if present at low densities.^[48]

Quasiparticle and Emergent Phenomena

In spin ice materials, such as dysprosium titanate (Dy₂Ti₂O₇), emergent magnetic monopoles arise as quasiparticle excitations from the collective behavior of frustrated magnetic moments on a pyrochlore lattice, first theoretically proposed in the mid-2000s and experimentally observed through their charge and current signatures in muon spin relaxation experiments.^[49] These quasiparticles emerge when spin flips create defects analogous to proton disorder in water ice, leading to point-like magnetic charge carriers that propagate through the lattice, effectively mimicking magnetic currents (J_m) without violating the absence of free monopoles in classical electromagnetism.^[49] Detection of Dirac strings in spin ice provides direct evidence of these emergent phenomena, where the strings—topological defects terminating at monopole pairs—manifest as correlated spin excitations detectable via neutron scattering in Dy₂Ti₂O₇, inducing effective magnetic currents through sequential spin flips along the string paths. This process allows monopoles to move as quasiparticles, with their dynamics producing measurable signatures like relaxation rates in the spin ice system.^[50] Beyond spin ice, analogous emergent monopoles appear in other quantum materials, including thin-film topological insulators where external electric fields induce monopole-like excitations in the surface states, and at the edges of graphene structures hosting fractionalized quasiparticles with monopole characteristics due to topological band structures.^[51] In superconductors, vortex currents—arising from Abrikosov flux lines—exhibit emergent monopole behavior when interacting with chiral magnetic textures, where skyrmion-induced stray fields drive vortex motion equivalent to localized magnetic current loops.^[52] These quasiparticle systems serve as low-energy testbeds for studying monopole dynamics, enabling observations of fractional magnetic charge quantization (in units of the underlying dipole moment) and Coulomb-like interactions without the extreme energies required for fundamental particle searches.^[49] Recent overviews highlight advances in artificial spin ices and quantum materials, where room-temperature monopole quasiparticles in nanoscale arrays further probe these effects for potential applications in topological quantum computing.

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Oliver Heaviside's electromagnetic theory - PMC - NIH
The year 2018 marks the 125th anniversary of the first of three published volumes on electromagnetic theory by the eminent Victorian electrical engineer, ...
[11]
Magnetic Monopoles - an overview | ScienceDirect Topics
Magnetic monopoles are defined as theoretical particles that carry a net magnetic charge, which can be understood through instanton solutions in non-Abelian ...
[12]
Taking the lead in the monopole hunt - CERN Courier
Jan 27, 2025 · Magnetic monopoles are hypothetical particles that would carry magnetic charge, a concept first proposed by Paul Dirac in 1931.
[13]
[PDF] Magnetic Monopoles in Pure Compact Lattice Quantum ...
Sep 22, 2014 · With the magnetic charge density ρM and magnetic current density jM, Maxwell's equations become v · E =ρE v · B =ρM. −v × E = ∂B. ∂t. + jM v × B ...
[14]
[PDF] Theoretical and experimental status of magnetic monopoles
Here ρe, je are the electric charge and current densities, and ρm, jm are the magnetic ... with speed v produce time-varying magnetic fields which induce signals ...
[15]
[PDF] Electric–Magnetic Duality and Magnetic Monopoles - UT Physics
As written, the electric-magnetic symmetry (2) works in empty space but not in presence of any electric charges or currents. However, we may repair this duality ...
[16]
[PDF] Magnetic Monopoles: Quantization and Quasiparticles - McGill Physics
Ellis. (Dated: May 6, 2013) Magnetic monopoles are proposed particles that carry magnetic charge. Their existence can be motivated by two observations: that ...
[17]
Dirac quantisation condition: a comprehensive review - ar5iv
A quantum-mechanical argument on these nodal lines led him to his celebrated quantisation condition: q g = n ℏ c / 2 q g n Planck-constant-over-2-pi c 2 ...
[18]
Dirac quantization condition for monopole in noncommutative space ...
Jun 29, 2009 · The Dirac quantization condition (DQC) μ e = N 2 ℏ c is a topological property, independent of space-time points, that tells us that the ...
[19]
[2509.23726] A non-inflationary solution to the monopole problem
Sep 28, 2025 · Magnetic monopoles are a long-standing prediction of Grand Unified Theories, yet their efficient production in early universe phase transitions ...
[20]
[PDF] GRAND UNIFIED THEORIES MAGNETIC MONOPOLE PROBLEM ...
Dec 5, 2022 · Linelike defects, which exist in some GUTs but not all. 3) Magnetic monopoles: Pointlike defects, which exist in all GUTs. Alan Guth.
[21]
On the origin of matter-antimatter asymmetry in the Universe
Jun 10, 2022 · This mechanism allows for a matter-antimatter asymmetry to appear separately in the DW and the aDW, while in the combined DW-aDW system no such asymmetry was ...
[22]
[PDF] Lorentz force with magnetics monopoles - arXiv
It is shown that it is possible to write an action integral from which one can derive, by least action principle, the symmetrized set of Maxwell's equations, ...
[23]
[PDF] Classical Electrodynamics - Duke Physics
but as far as the equations of classical electrodynamics are concerned the concept of “cause” is better expressed as one of interaction via a suitable ...
[24]
[PDF] Vector electromagnetic theory of transition and diffraction radiation
We have developed a novel method based on vector electromagnetic theory and. Schelkunoff's principles, to calculate the spectral and angular distributions ...
[25]
Electromagnetic Quantities
Magnetic current density (surface). Jms. volt/meter. -. Magnetic field. H. ampere/meter. A/m. Magnetic flux. Φ. weber. Wb. Magnetic flux density. B. tesla. T.
[26]
[PDF] Magnetic Monopoles - Department of Theoretical Physics
Feb 14, 2022 · In this method, a magnetic monopole passing through a ring induces in it a current of characteristic magnitude and furthermore, this current ...
[27]
On the Magnetic Current Density in the Maxwell Equations Based on ...
Nov 4, 2019 · Abstract page for arXiv paper 1911.08880: On the Magnetic Current Density in the Maxwell Equations Based on the Noether Theorem.
[28]
[PDF] surface equivalence theorem: huygens's principle 329
By the surface equivalence theorem, the fields outside an imaginary closed surface are obtained by placing, over the closed surface, suitable electric and.<|control11|><|separator|>
[29]
[PDF] Poynting's Theorem with Magnetic Monopoles 1 Problem 2 Solution
In the case of a loop of magnetic current density Jm, again in a static situation, the. Maxwell equation (27), V × E = −μ0. Jm, implies that, μ. 0 loop. Jm · dl ...Missing: J_m = | Show results with:J_m =
[30]
https://www.ece.mcmaster.ca/faculty/nikolova/antenna_dload/current_lectures/L03_RadIS.pdf
[31]
https://scholarsmine.mst.edu/cgi/viewcontent.cgi?article=1784&context=ele_comeng_facwork
[32]
[PDF] Two-Potential Formalism for Numerical Solution of the Maxwell ...
another pair of potentials, i.e., electric vector potential C and magnetic scalar potential ψ [6]:. = − ... magnetic monopoles. In contrast to the Maxwell ...
[33]
None
Below is a merged summary of the electric vector potential $ \mathbf{F} $ and magnetic current from *Antenna Theory: Analysis and Design* (3rd Edition) by Constantine A. Balanis. The information is synthesized from all provided segments, consolidating details into a comprehensive response. To handle the dense and varied information efficiently, I will use a table in CSV format for key details (e.g., formulas, conventions, and references), followed by a narrative summary to tie everything together. This ensures all information is retained while maintaining clarity and structure.
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### Summary of Love's Equivalence Principle and Magnetic Currents in Antenna Theory
[35]
[PDF] Some Equivalence Theorems of Electromagnetics and Their
Thus we arrive at the following Equivalence Principle discovered by. Love and Macdonald 8: a distribution of electric and magnetic currents on a given ...
[36]
[PDF] Small Loop Antenna and Duality Theorem
Such a source is called a magnetic current source. While magnetic current sources do not exist in nature, they can be useful mathematical tools because ...
[37]
[PDF] LECTURE 3: Radiation from Infinitesimal (Elementary) Sources
Radiation from Infinitesimal Magnetic Dipole (Electric-current Loop). 3.1 ... We denote the magnetic current, which is the dual of the electric current ...
[38]
[PDF] A Hybrid FEM/MOM Technique for Electromagnetic Scattering and ...
Nov 1, 1997 · Abstract—A hybrid formulation is presented, which combines the method of moments (MOM) with the edge-based vector finite element method (FEM) ...
[39]
[PDF] Chapter 10: Antennas and Radiation - MIT OpenCourseWare
An antenna is a device that couples currents to electromagnetic waves for purposes of radiation or reception. The process by which antennas radiate can be ...
[40]
[PDF] MIXED-POTENTIAL GREEN'S FUNCTION OF AN ... - PIER Journals
Abstract—An effective approach is suggested for calculating the. Green's function of an axially symmetric sheet magnetic current placed on a circular ...
[41]
Time-harmonic Electromagnetic Fields - Google Books
Time-harmonic Electromagnetic Fields. Front Cover. Roger F. Harrington. McGraw-Hill, 1961 - Mathematics - 480 pages. The IEEE Press Series on Electromagnetic ...
[42]
[PDF] ON THE USE OF THE FRILL GENERATOR IN INTEGRAL ... - URSI
For a frill-generator feed, certain fundamental mathematical properties of Hallén's and Pocklington's equations with the approximate kernel are developed.
[43]
[PDF] The Ehrenhaft-Mikhailov effect described as the behavior of a low ...
In some sixty papers, 1930-1951, Ehren- haft reported magnetic monopole-like behavior and magnetic currents in aerosols. These reports were disputed on the ...
[44]
The search for magnetic monopoles - Physics Today
Oct 1, 2016 · If magnetic monopoles existed, they would carry the magnetic equivalent of an electric charge, and they would restore the duality symmetry (see ...
[45]
[PDF] () MAGNETIC MONOPOLES SEARCHES
Induction coils are the only devices sensitive to poles of any velocity. As already stated, a moving monopole produces an electric field; thus an elec tromotive ...
[46]
MoEDAL zeroes in on magnetic monopoles - CERN
Apr 26, 2024 · For magnetic monopoles, the mass bounds were set for magnetic charges from 1 to 10 times the fundamental unit of magnetic charge, the Dirac ...
[47]
Search for Highly Ionizing Particles in Collisions during LHC Run 2 ...
This experiment searched for magnetic monopoles (MMs) and high electric charge objects (HECOs) with spins 0, 1/2, and 1, using the full MoEDAL detector.
[48]
[2311.06509] Search for Highly-Ionizing Particles in pp Collisions ...
Nov 11, 2023 · MoEDAL is the only LHC experiment capable of being directly calibrated for highly-ionizing particles using heavy ions and with a detector system ...
[49]
Search for Sub-Relativistic Magnetic Monopoles with the IceCube ...
Jul 8, 2025 · The Rubakov-Callan effect predicts that magnetic monopoles can catalyze nucleon decays, in particular the decay of protons. This results in a ...
[50]
Search for Sub-Relativistic Magnetic Monopoles with the IceCube ...
Sep 24, 2025 · The Rubakov-Callan effect predicts that magnetic monopoles can catalyze nucleon decays, in particular the decay of protons. This results in a ...
[51]
[PDF] Search for Sub-Relativistic Magnetic Monopoles with the IceCube ...
Jul 23, 2025 · Function as source and sink of magnetic fields. ▷ MMs symmetrize Maxwell's equations and motivate quantization of electric charge (electron).
[52]
[PDF] 95. Magnetic Monopoles - Particle Data Group
Dec 1, 2023 · 95.1 Theory of magnetic monopoles. The symmetry between electric and magnetic fields in the source-free Maxwell's equations.
[53]
[PDF] 95. Magnetic Monopoles - Particle Data Group
May 31, 2024 · For a unification scale of 1016 GeV, these monopoles would have a mass Mmon ∼ 1017 – 1018 GeV. In theories with several stages of symmetry ...
[54]
Searches for cosmic magnetic monopoles: past, present and future
Nov 11, 2019 · The minimum mass required for β = 10−3 gD downgoing MMs to reach a detector at ground level is M∼3 × 106 GeV; to reach underground ...
[55]
Magnetic monopole searches in the cosmic radiation - NASA ADS
... magnetic monopoles obtained by several experiments in direct searches, without the assumption of monopole induced nucleon decay catalysis (reprinted from [2]).
[56]
Measurement of the charge and current of magnetic monopoles in ...
Oct 15, 2009 · The spin ice family is predicted to support sharply defined magnetic monopole excitations and offers the possibility of exploring the general ...
[57]
Signature of magnetic monopole and Dirac string dynamics in spin ice
Mar 15, 2009 · We present an experimentally measurable signature of monopole dynamics. In particular, we show that previous magnetic relaxation measurements in the spin-ice ...
[58]
Witten effect in a crystalline topological insulator | Phys. Rev. B
Here we focus on a different type of emergent magnetic monopole that can arise in a thin-film STI placed in a uniform external electric field. The basic idea ...
[59]
Physics - A Chiral Magnet Induces Vortex Currents in Superconductors
Mar 17, 2021 · The stray magnetic field from the skyrmion induces vortices in the superconducting layer. These skyrmion-vortex pairs may host stable hybrid ...