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Multipole radiation

Multipole radiation refers to the electromagnetic radiation emitted by localized distributions of oscillating electric charges and currents, which can be decomposed into a series of multipole components through the expansion of the electromagnetic potentials in spherical harmonics.^[1] These components are classified as electric multipoles (EL) or magnetic multipoles (ML) of order L, where L = 1 corresponds to dipole radiation, L = 2 to quadrupole, and higher L to octupole and beyond; each multipole carries L units of orbital angular momentum per photon.^[1] Monopole radiation (L = 0) is absent for both electric and magnetic types due to charge conservation and the nonexistence of magnetic monopoles, respectively.^[1] The radiation fields in the far zone (radiation zone, where kr \gg 1) derive from the multipole expansion of the vector potential \vec{A}, with the leading term being the electric dipole for non-symmetric charge oscillations.^[2] For electric dipole radiation, the electric and magnetic fields are transverse and orthogonal, falling off as $1/r, with the time-averaged power per unit solid angle given by \frac{dP}{d\Omega} = \frac{\mu_0}{32\pi^2} k^4 c^3 |\vec{p}|^2 \sin^2[\theta](/page/Theta), where \vec{p} is the dipole moment, k = \omega/c, and [\theta](/page/Theta) is the angle from the dipole axis; this \omega^4 dependence explains phenomena like the blue color of the sky from higher-frequency scattering.^[2] Higher-order terms, such as magnetic dipole (L=1) and electric quadrupole (L=2), contribute when the dipole moment vanishes due to symmetry, with quadrupole power scaling as k^6 and featuring angular patterns like \sin^2[\theta](/page/Theta) \cos^2[\theta](/page/Theta).^[2] The total radiated energy is an incoherent sum over all multipoles, while the angular momentum flux reflects the m-quantum numbers of the modes.^[3] Multipole radiation is fundamental in describing transitions in atoms and nuclei, where selection rules determine the dominant multipole (e.g., electric dipole for allowed optical transitions, with decay rates suppressed by factors of $10^{-3} per higher order), as well as in engineering applications like antenna design for center-fed linear antennas.^[1]^[2] The hierarchy of radiation strengths typically follows electric dipole strongest, followed by magnetic dipole and electric quadrupole (comparable), and then higher multipoles, enabling precise modeling of radiative processes in quantum electrodynamics.^[1]

General Framework

Time-Dependent Sources and Potentials

In classical electromagnetism, the electromagnetic fields generated by time-dependent charge and current distributions serve as the fundamental sources. The charge density \rho(\mathbf{r}, t) describes the distribution of electric charge at position \mathbf{r} and time t, while the current density \mathbf{J}(\mathbf{r}, t) represents the flow of charge, both of which drive the time-varying fields according to Maxwell's equations.^[4] To account for the finite speed of light c, the scalar potential \phi(\mathbf{r}, t) and vector potential \mathbf{A}(\mathbf{r}, t) are expressed using retarded times, ensuring causality in the propagation of electromagnetic effects. The scalar potential is given by

\phi(\mathbf{r}, t) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\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 at which the source contributes to the field at the observation point. Similarly, the vector potential is

\mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}', t_r)}{|\mathbf{r} - \mathbf{r}'|} \, d^3\mathbf{r}'.

These retarded potentials satisfy the Lorentz gauge condition, \nabla \cdot \mathbf{A} + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0, which simplifies the equations of motion for the potentials while preserving the physical fields.^[5] Under the Lorentz gauge, the potentials obey inhomogeneous wave equations derived from Maxwell's equations:

\left( \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right) \phi = -\frac{\rho}{\epsilon_0},

\left( \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right) \mathbf{A} = -\mu_0 \mathbf{J}.

These equations highlight the wave-like nature of the potentials, with the sources \rho and \mathbf{J} acting as driving terms for electromagnetic disturbances that propagate at speed c.^[4]

Near-Field and Far-Field Expansions

The retarded scalar potential generated by a localized time-dependent charge distribution admits a multipole expansion derived from the expansion of the denominator in the integral expression for the potential. Specifically, for observation point \mathbf{r} with r \gg r' (where r' = |\mathbf{r}'| spans the source region), the scalar potential is approximated as

\phi(\mathbf{r}, t) \approx \frac{1}{4\pi\epsilon_0} \sum_{n=0}^{\infty} \frac{1}{r^{n+1}} \int (r')^n P_n(\cos\alpha) \, \rho(\mathbf{r}', t_r) \, d^3\mathbf{r}',

where P_n are the Legendre polynomials, \cos\alpha = \hat{\mathbf{r}} \cdot \hat{\mathbf{r}}' is the angle between the observation direction and the source point, and t_r = t - |\mathbf{r} - \mathbf{r}'|/c is the retarded time (approximated as t - r/c for the lowest-order terms when the source is compact).^[6]^[7] The vector potential \mathbf{A}(\mathbf{r}, t) undergoes a similar multipole expansion, expressed in terms of magnetic dipole and higher-order magnetic multipole moments, along with contributions from the time-varying electric moments via the Lorentz gauge condition \nabla \cdot \mathbf{A} = -\mu_0\epsilon_0 \partial\phi/\partial t. This expansion takes the form

\mathbf{A}(\mathbf{r}, t) \approx \frac{\mu_0}{4\pi} \sum_{n=0}^{\infty} \frac{1}{r^{n+1}} \int (r')^n P_n(\cos\alpha) \, \mathbf{J}(\mathbf{r}', t_r) \, d^3\mathbf{r}',

with \mathbf{J} the current density, though higher terms involve vector spherical harmonics for full angular dependence.^[6]^[7] In the near-field regime, where kr \ll 1 with k = \omega/c the wavenumber (corresponding to distances much smaller than the wavelength), the expansion terms resemble the static multipole fields, with the n-th electric $2^n-pole contribution decaying as $1/r^{n+1} and exhibiting quasi-static behavior dominated by the source moments without significant propagation effects.^[6] Conversely, in the far-field regime where kr \gg 1 (distances much larger than the wavelength), the leading radiative contributions arise from the highest time derivatives of the multipole moments in the expansions, resulting in fields that decay as $1/r and propagate as transverse electromagnetic waves. These far-field terms satisfy the Sommerfeld radiation condition, ensuring outgoing waves with no incoming components and net energy flux away from the source.^[6]

Properties of Multipole Radiation

Linearity of Moments

In multipole radiation, the moments serve as linear functionals of the source charge and current distributions, allowing for straightforward superposition in computing the radiated fields. The electric multipole moments, which characterize the spatial distribution of charge, are defined in Cartesian tensor form for order n as

Q_{i_1 i_2 \dots i_n}(t) = \int r_{i_1} r_{i_2} \dots r_{i_n} \rho(\mathbf{r}, t) \, d^3\mathbf{r},

where \rho(\mathbf{r}, t) is the charge density, and the integral is over all space. This definition demonstrates the linearity of the moments with respect to \rho, as the tensor Q scales directly with any proportional change in the charge distribution.^[8] Similarly, the magnetic multipole moments are linear in the current density \mathbf{J}(\mathbf{r}, t). For the magnetic dipole moment, a fundamental example, it is given by

\mathbf{m}(t) = \frac{1}{2} \int (\mathbf{r} \times \mathbf{J}(\mathbf{r}, t)) \, d^3\mathbf{r}.

Higher-order magnetic moments follow analogous integral forms, ensuring their additivity for composite sources. This linearity extends to discrete systems; for instance, the electric dipole moment of multiple point charges is \mathbf{p}(t) = \sum_i q_i \mathbf{r}_i(t), where each charge contributes proportionally without cross terms.^[9] The superposition principle arises from the linearity of Maxwell's equations, permitting the total radiation field from a system to be expressed as the vector sum of fields generated independently by each multipole moment. This additive property simplifies calculations for complex sources by decomposing them into contributions from individual moments. As a consequence of this linearity, the values of multipole moments depend on the choice of expansion origin.^[10]

Origin Dependence of Moments

The multipole moments in electromagnetism are defined relative to a chosen coordinate origin, and while the monopole moment—corresponding to the total charge q = \sum_i q_i—remains invariant under any shift of the origin, higher-order moments generally transform in a manner that depends on the displacement vector \boldsymbol{\delta}. This invariance of the monopole arises because the total charge is an intrinsic property of the source distribution, unaffected by the reference point.^[11] For the electric dipole moment \mathbf{p}, a shift in the origin by \boldsymbol{\delta} results in the transformed moment \mathbf{p}' = \mathbf{p} - q \boldsymbol{\delta}, where q is the total charge. This transformation highlights that the dipole moment is origin-independent only if the total charge vanishes (q = 0), as in neutral systems; otherwise, it mixes with the monopole contribution under translation.^[12] The electric quadrupole moment tensor Q_{ij} exhibits even greater sensitivity to the origin choice, transforming as Q'_{ij} = Q_{ij} - \delta_i p_j - \delta_j p_i + q \delta_i \delta_j. Here, the new tensor incorporates contributions from both the original quadrupole and the dipole moment, as well as a term proportional to the total charge and the shift components. This mixing demonstrates that higher multipole moments depend on the reference point unless all lower-order moments (monopole and dipole) are zero, underscoring the non-uniqueness of the expansion beyond the leading term.^[12] Despite this origin dependence of individual moments, the far-field radiation patterns remain independent of the choice of origin, as shifts correspond to gauge transformations that preserve the transverse components of the fields responsible for radiation. This invariance ensures that physical observables, such as radiated power, are well-defined regardless of the reference point, provided the source location is adjusted accordingly.^[11] To mitigate the effects of origin dependence and simplify the multipole expansion—particularly for radiation calculations—the origin is often selected at the center of charge, defined as \boldsymbol{r}_c = \frac{1}{q} \sum_i q_i \mathbf{r}_i for charged systems, or the center of energy for time-varying sources, which minimizes contributions from higher-order moments and isolates the dominant radiation mechanism.^[13]

Distance Dependence of Fields

In the vicinity of a time-varying multipole source, the electromagnetic fields can be decomposed into static (quasi-electrostatic and quasi-magnetostatic), inductive, and radiative contributions, each characterized by specific radial fall-off behaviors. For an electric 2^l-pole, the leading near-field term for the electric field scales as E \sim 1/r^{l+2}, reflecting the quasi-static dominance where retardation effects are negligible. The magnetic field in this regime follows a similar scaling but is suppressed by a factor involving the wavenumber k = \omega / c.^[14] As the observation distance increases, an intermediate inductive term emerges, where E \sim 1/r^{l+1}, arising from the next-order correction in the expansion of the retarded potentials that accounts for the finite propagation speed of electromagnetic disturbances. The magnetic field exhibits analogous $1/r^{l+1} scaling in this region, with the inductive contributions linking the near-field reactive energy storage to the far-field propagation. For magnetic 2^l-poles, the roles of electric and magnetic fields are interchanged, but the radial dependencies remain equivalent.^[3] In the far field, both electric and magnetic fields from any 2^l-pole scale uniformly as $1/r, independent of l, enabling efficient radiation transport over large distances. Here, the fields are transverse and related by \mathbf{B} \approx (1/c) \hat{\mathbf{n}} \times \mathbf{E}, where \hat{\mathbf{n}} is the unit vector in the radial direction and c is the speed of light, ensuring the wave impedance matches that of free space. Although the full field expressions incorporate angular variations through spherical harmonics Y_{lm}(\theta, \phi), the radial fall-off governs the transition between regimes.^[14] These behaviors manifest across well-defined spatial regimes relative to the wavelength \lambda = 2\pi c / \omega. The static regime occurs for r \ll \lambda / 2\pi, where phase differences across the source are minimal, and fields resemble those of stationary multipoles. The inductive regime prevails around r \sim \lambda / 2\pi, blending reactive and propagating characteristics. Finally, the radiative regime applies for r \gg \lambda / 2\pi, where the fields form outgoing spherical waves with minimal spherical spreading beyond the 1/r decay.

Absence of Electric Monopole Radiation

Theoretical Derivation

In classical electrodynamics, the electric monopole moment is defined as the total charge Q(t) = \int \rho(\mathbf{r}, t) \, d^3\mathbf{r}, where \rho(\mathbf{r}, t) is the charge density. This moment is constant in time due to the conservation of charge, as expressed by the continuity equation \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0, where \mathbf{J} is the current density. Integrating the continuity equation over all space yields \frac{dQ}{dt} = \int \frac{\partial \rho}{\partial t} \, d^3\mathbf{r} = - \int \nabla \cdot \mathbf{J} \, d^3\mathbf{r} = 0, assuming the current has compact support so that surface terms vanish by the divergence theorem. Consequently, Q(t) = Q, a constant, implying that higher-order time derivatives such as \ddot{Q} are also zero. In the far field, the scalar potential for a monopole source is given by the static Coulomb form \phi(\mathbf{r}, t) \approx \frac{Q}{4\pi\epsilon_0 r}, with no retardation effects because Q does not depend on time. This potential produces only a static electric field \mathbf{E} = -\nabla \phi, which falls off as $1/r^2 and carries no energy to infinity, as there are no time-varying components to support a radiating $1/r term in the expansion. The vector potential for a pure monopole makes no contribution, as \mathbf{A}_\text{monopole} = 0; it arises from the current density via the retarded integral, but a monopole configuration with no net motion or current (\mathbf{J} = 0) yields zero. In the multipole expansion of the radiation fields, the leading term for electric monopole radiation would involve \ddot{Q} in the Poynting vector or power flux, but since \ddot{Q} = 0, there is no such term and thus no monopole radiation. This result follows directly from the far-field approximations of the retarded potentials under the assumption of localized sources.

Physical Interpretation

The absence of electric monopole radiation in classical electromagnetism stems directly from the conservation of electric charge, a fundamental principle encapsulated in Maxwell's equations. The electric monopole moment corresponds to the total charge within a localized source, which remains constant over time because charge cannot be created or annihilated in isolated systems.^[6] Consequently, an oscillating monopole moment—necessary for radiating electromagnetic waves at the lowest order—would require net charge creation or destruction, violating this conservation law and rendering such radiation impossible.^[15] The mathematical derivation from the multipole expansion confirms this, as the first time derivative of the monopole moment vanishes identically due to charge conservation.^[16] In contrast, gravitational radiation also lacks a monopole term because the total mass-energy is conserved, analogous to electric charge conservation; the leading contribution arises at the quadrupole order, as dipole radiation is forbidden by conservation of total momentum (center-of-mass motion).^[17] Unlike electromagnetism, where the monopole is forbidden, gravity's universal attractive nature allows static monopole fields, but dynamic radiation from a pure monopole is absent in general relativity.^[17] This absence implies that the lowest-order electric radiation must originate from the dipole term, which involves spatial separation and oscillation of positive and negative charges, such as in accelerating dipoles.^[6]

Electric Dipole Radiation

Potential

The electric dipole moment \vec{p}(t) for a localized charge distribution is defined as \vec{p}(t) = \int \vec{r} \, \rho(\vec{r}, t) \, d^3\vec{r}, where \rho(\vec{r}, t) is the charge density. This vector characterizes the leading non-uniformity in the charge distribution when the net charge (monopole) is zero or separately accounted for.^[18] In the far field (radiation zone, where kr \gg 1 with k = \omega/c), the potentials for electric dipole radiation are obtained from the multipole expansion of the retarded potentials. The scalar potential \phi is \phi(\vec{r}, t) \approx \frac{1}{4\pi \epsilon_0} \frac{\hat{n} \cdot \dot{\vec{p}}(t_r)}{c r}, and the vector potential \vec{A} is \vec{A}(\vec{r}, t) \approx \frac{\mu_0}{4\pi r} \dot{\vec{p}}(t_r), where t_r = t - r/c is the retarded time, \hat{n} is the unit vector in the observation direction, and the dot denotes the time derivative. These expressions arise from the first-order terms in the expansion of the sources, with the $1/r dependence ensuring radiative behavior. For systems with explicit currents, \dot{\vec{p}} relates to the integral of the current density, but for pure charge oscillations, it captures the displacement current effects.^[18]^[19] Under harmonic time dependence, \vec{p}(t) = \mathrm{Re} [\vec{p}_0 e^{-i \omega t}], the derivatives become \dot{\vec{p}}(t_r) = -i \omega \vec{p}(t_r), yielding potentials proportional to k / r (with k = \omega / c), consistent with the O(k^2) scaling of dipole radiation in the power spectrum. This form is crucial for analyzing oscillatory sources like atomic transitions or small antennas.^[20]

Fields

The electric and magnetic fields for electric dipole radiation in the radiation zone are derived from the potentials using Maxwell's equations, resulting in transverse waves with $1/r falloff. For a dipole aligned along the z-axis, in spherical coordinates, the fields at large kr \gg 1 are dominated by the transverse components.^[18] The electric field has a \theta-component: E_\theta = -\frac{\mu_0}{4\pi r} \ddot{p}(t_r) \sin\theta, with E_r and E_\phi negligible in the far field (O(1/r^2)). The magnetic field has an azimuthal component: B_\phi = -\frac{1}{c} E_\theta, perpendicular to both \vec{E} and \hat{n}, satisfying |\vec{B}| = |\vec{E}| / c. More generally, the fields can be expressed as \vec{E} = \frac{\mu_0}{4\pi r} [\ddot{\vec{p}}(t_r) \times \hat{n}] \times \hat{n}, \vec{B} = \frac{1}{c} \hat{n} \times \vec{E}. These arise from \vec{E} \approx -\partial \vec{A}/\partial t (since \nabla \phi is higher order) and \vec{B} = \nabla \times \vec{A} in the far-field approximation.^[19]^[20] For monochromatic excitation, \vec{p}(t) = p_0 \cos(\omega t) \hat{z}, the fields become E_\theta = \frac{\mu_0 p_0 \omega^2 \sin\theta}{4\pi r} \sin(kr - \omega t), B_\phi = \frac{\mu_0 p_0 \omega^2 \sin\theta}{4\pi c r} \sin(kr - \omega t), showing the \omega^2 (or k^2) dependence and \sin^2\theta angular pattern from the transverse projection. The fields carry one unit of angular momentum per photon, with a dipolar radiation pattern featuring two lobes symmetric about the dipole axis. The amplitude scales as k^2 a relative to static fields, where a is the source size, making dipole radiation dominant for compact, asymmetric oscillators.^[20]

Radiated Power

The instantaneous power radiated by an electric dipole is given by the Larmor formula generalized for dipoles: P(t) = \frac{\mu_0}{6\pi c} |\ddot{\vec{p}}(t)|^2, where the double dot is the second time derivative. This expression is obtained by integrating the Poynting vector over a large sphere enclosing the source.^[18] For a harmonically oscillating dipole \vec{p}(t) = p_0 \cos(\omega t) \hat{z}, the time-averaged power is \langle P \rangle = \frac{\mu_0 p_0^2 \omega^4}{12 \pi c}, reflecting the \omega^4 dependence due to the two derivatives on p and the energy flux scaling. The time-averaged power per unit solid angle is \frac{d\langle P \rangle}{d\Omega} = \frac{\mu_0 p_0^2 \omega^4}{32 \pi^2 c} \sin^2 \theta, with the \sin^2 \theta pattern peaking perpendicular to the dipole axis. Integrating over all directions yields the total power above.^[20]^[19] Compared to higher multipoles, dipole radiation is the strongest for non-symmetric sources, suppressed only by symmetry (e.g., in parity-forbidden transitions), and is crucial for applications like optical scattering and antenna efficiency, where the power scales as (k a)^2 times monopole (absent) contributions.^[18]

Magnetic Dipole Radiation

Potential

The magnetic dipole moment \vec{m}(t) characterizes the distribution of localized oscillating currents beyond the monopole (absent due to \nabla \cdot \vec{J} = 0) and is defined as \vec{m}(t) = \frac{1}{2} \int \vec{r}' \times \vec{J}(\vec{r}', t) \, d^3\vec{r}', where \vec{J}(\vec{r}', t) is the current density.^[19] This vector captures the leading magnetic response for systems with no net electric dipole, such as a small current loop oscillating in time.^[21] In the far field (radiation zone, kr \gg 1), the scalar potential \phi vanishes to leading order for pure magnetic dipole sources (no explicit charge density variations), while the vector potential \vec{A} dominates and is given by \vec{A}(\vec{r}, t) = \frac{\mu_0}{4\pi} \frac{k^2}{r} \left( \vec{m}(t_r) \times \hat{n} \right) e^{i(kr - \omega t)}, under the assumption of harmonic time dependence \vec{m}(t) = \mathrm{Re} [\vec{m}_0 e^{-i \omega t}], with k = \omega / c, t_r = t - r/c the retarded time, and \hat{n} the unit vector toward the observation point.^[19] This expression derives from the multipole expansion of the retarded vector potential, where the magnetic dipole term arises from the first-order magnetic contribution, ensuring the $1/r falloff and transverse nature characteristic of radiation fields. The factor k^2 reflects the second time derivative scaling, analogous to the electric dipole but with a cross product for the magnetic origin.^[21] For a pure magnetic dipole source, higher-order corrections to \vec{A} (e.g., electric quadrupole) are suppressed, and the near-field terms decay faster than $1/r. In the time domain, the general form is \vec{A} \approx \frac{\mu_0}{4\pi r} [\ddot{\vec{m}}(t_r) \times \hat{n}], highlighting the acceleration of the magnetic moment as the source of radiation.^[18]

Fields

The electric and magnetic fields for magnetic dipole radiation in the radiation zone (kr \gg 1) are derived from the vector potential \vec{A} using Maxwell's equations, yielding transverse fields perpendicular to \hat{n} and to each other, with $1/r dependence.^[19] The magnetic field has a form \vec{B}(\vec{r}, t) = \frac{\mu_0 k^2}{4\pi r} \left( \hat{n} \times \vec{m}(t_r) \right) e^{i(kr - \omega t)}, while the electric field is \vec{E}(\vec{r}, t) = -Z_0 \hat{n} \times \vec{B}(\vec{r}, t), where Z_0 = \sqrt{\mu_0 / \epsilon_0} is the impedance of free space.^[19] For a magnetic moment aligned along the z-axis, \vec{m} = m \hat{z}, the fields in spherical coordinates simplify: the electric field is purely azimuthal, E_\phi = -\frac{\mu_0 k^2 m}{4\pi r} \sin\theta \, e^{i(kr - \omega t)}, with E_r = E_\theta = 0 in the far field, and the magnetic field has \theta and r components but the transverse part dominates radiation. The \sin\theta angular dependence arises from the cross product projection, yielding a dipolar pattern with no radiation along the dipole axis.^[21] More generally, \vec{E} = \frac{\mu_0 k^2}{4\pi r} [\hat{n} \times (\vec{m} \times \hat{n})] e^{i(kr - \omega t)}, ensuring transversality. The fields scale as k^2 times the moment, comparable to electric dipole radiation but suppressed by (v/c)^2 for non-relativistic sources, where v is the characteristic velocity. The radiation pattern is identical to that of electric dipole, with power concentrated in the equatorial plane.^[18]

Radiated Power

The time-averaged power radiated by a magnetic dipole under harmonic oscillation is P = \frac{\mu_0 \omega^4}{12 \pi c^3} |\vec{m}_0|^2, or equivalently P = \frac{\mu_0 k^4 |\vec{m}_0|^2}{12 \pi c}, where \vec{m}_0 is the complex amplitude.^[19] This follows from the Poynting vector integrated over a sphere in the far field, analogous to the electric dipole formula but with |\vec{m}_0|^2 / c^2 replacing | \vec{p}_0 |^2. The differential power per unit solid angle is \frac{dP}{d\Omega} = \frac{\mu_0 k^4}{32 \pi^2 c} |\vec{m}_0|^2 \sin^2 \theta, exhibiting the characteristic \sin^2 \theta distribution, with \theta the angle from the magnetic moment axis.^[19] Compared to electric dipole radiation, magnetic dipole power is typically weaker by a factor of (ka)^2 or (v/c)^2, where a is the source size, making it relevant when electric dipole symmetry forbids radiation, such as in neutral atoms with spin-flip transitions or small loop antennas.^[21]

Electric Quadrupole Radiation

Potential

The electric quadrupole moment tensor characterizes the distribution of charge in a localized system beyond the dipole approximation and is defined as Q_{ij}(t) = \int \left( 3 x_i x_j - r^2 \delta_{ij} \right) \rho(\mathbf{r}, t) \, d^3\mathbf{r}, where \rho(\mathbf{r}, t) is the charge density, \mathbf{r} = (x, y, z), r = |\mathbf{r}|, and \delta_{ij} is the Kronecker delta.^[22] This symmetric, traceless tensor captures the leading non-spherical variation in the charge distribution for systems where the net charge and dipole moment vanish or are separately considered.^[19] In the far field (radiation zone), where the distance r from the source greatly exceeds both the wavelength and the source size, the vector potential \mathbf{A} for electric quadrupole radiation is dominated by the term \mathbf{A} \approx \frac{\mu_0}{4\pi} \frac{1}{6 c r} \hat{n}_i \hat{n}_j \dddot{Q}_{ij}(t_r) \hat{\mathbf{n}}, with the summation over repeated indices i, j = 1, 2, 3 implied, \hat{\mathbf{n}} the unit vector in the observation direction, t_r = t - r/c the retarded time, and \dddot{Q}_{ij} the third time derivative of the quadrupole tensor.^[19] This expression arises from the multipole expansion of the retarded vector potential, where the third derivative emerges from the combination of spatial expansion to second order and the Taylor series for the retarded time argument, ensuring the $1/r falloff characteristic of radiating fields.^[22] The factor of $1/6 follows from the normalization of the quadrupole tensor and the angular contraction \hat{n}_i \hat{n}_j Q_{ij}, which projects the tensor onto the observation direction. The transverse part of this potential contributes to the radiation fields after projection. For a pure electric quadrupole source with no explicit conduction currents (\mathbf{J} = 0), the leading $1/r term in the vector potential arises from the retardation effects and implicit displacement currents associated with the time-varying charge density; the scalar potential \phi provides only a correction of higher order in $1/r.^[19] In the near field, the scalar potential \phi dominates the electrostatic-like behavior, while the radiation in the far field stems from the second time derivative of the quadrupole tensor in the field expressions derived from \mathbf{A}, scaled by the appropriate powers of c.^[22] Under the common assumption of harmonic time dependence, Q_{ij}(t) = \mathrm{Re} [ Q_{ij,0} e^{-i \omega t} ], the third derivative becomes \dddot{Q}_{ij}(t_r) = -i \omega^3 Q_{ij}(t_r), yielding a far-field vector potential proportional to k^3 / r with k = \omega / c, consistent with the O(k^3) scaling of quadrupole radiation relative to the dipole's O(k^2).^[19] This frequency-domain form facilitates analysis of angular patterns and power spectra in oscillatory systems such as atomic transitions or antenna arrays.

Fields

The electric and magnetic fields associated with electric quadrupole radiation are obtained by taking appropriate derivatives of the scalar and vector potentials in the radiation zone, where the distance r satisfies kr \gg 1 with k = \omega / c the wavenumber.^[23] In the far field, these fields exhibit a $1/r dependence and are transverse to the propagation direction \hat{\mathbf{n}}, with the leading contributions arising from the second time derivative of the quadrupole moment tensor evaluated at retarded time.^[21] The quadrupole moment tensor Q_{ij} is defined as the traceless symmetric tensor

Q_{ij}(t) = \int \left( 3 x_i' x_j' - r'^2 \delta_{ij} \right) \rho(\mathbf{r}', t_r) \, d^3 r',

where \rho(\mathbf{r}', t_r) is the charge density at retarded time t_r = t - r/c, and the integral is over the source volume.^[23] For monochromatic sources, Q_{ij}(\omega) is complex, and the fields are expressed in terms of projections like Q_{ij} n_i n_j, where \mathbf{n} is the unit vector toward the observation point. In spherical coordinates, the electric field components for electric quadrupole radiation take the general form involving the traceless tensor projections. The radial component E_r is negligible in the far field compared to the transverse components, but near the radiation zone, it includes higher-order terms. The transverse components are

E_\theta = \frac{1}{4\pi \epsilon_0 r} \frac{k^3}{6} \left[ \hat{e}_\theta \cdot (\mathbf{n} \times (\mathbf{n} \times \mathbf{Q})) \right] e^{i(kr - \omega t)},

E_\phi = \frac{1}{4\pi \epsilon_0 r} \frac{k^3}{6} \left[ \hat{e}_\phi \cdot (\mathbf{n} \times (\mathbf{n} \times \mathbf{Q})) \right] e^{i(kr - \omega t)},

where \mathbf{Q} = Q_{ij} \hat{e}_i \hat{e}_j is the tensor contracted with the basis vectors, and the double cross product ensures transversality.^[21] For a specific axisymmetric case with Q_{zz} - Q_{xx} = Q_0 (and Q_{yy} = -Q_0/2 to maintain tracelessness, assuming alignment along z), the E_\theta component simplifies to

E_\theta \approx \frac{1}{4\pi \epsilon_0 r} \frac{k^3}{6} Q_0 \sin\theta \cos\theta \, e^{i(kr - \omega t)},

with E_\phi = 0 and E_r of order $1/r^2. This illustrates the involvement of tensor projections like Q_{zz} - Q_{xx}.^[23] The magnetic field in the far field has only an azimuthal component for this symmetry,

B_\phi = -\frac{1}{c} E_\theta,

which satisfies |\mathbf{B}| = |\mathbf{E}| / c (in SI units), and is perpendicular to both \mathbf{E} and \mathbf{n}. More generally,

\mathbf{B} = -\frac{i c k^3 \mu_0}{24\pi r} e^{i(kr - \omega t)} \, \hat{\mathbf{n}} \times \mathbf{Q}(\hat{\mathbf{n}}),

where \mathbf{Q}(\hat{\mathbf{n}}) = Q_{ij} n_j \hat{\mathbf{e}}_i, confirming the transverse nature and $1/r falloff.^[23] The angular dependence of these fields corresponds to second-order Legendre polynomials P_2(\cos\theta), such as (3\cos^2\theta - 1)/2 or \sin\theta \cos\theta, yielding a quadrupolar radiation pattern with four lobes symmetric about the source axes. Unlike dipole radiation, the quadrupole fields incorporate higher angular complexity, with contributions from both even and odd parity terms arising from the tensor's off-diagonal elements, leading to asymmetric patterns for non-axisymmetric sources.^[21] The amplitude of these fields scales as (k a)^2 times the dipole field strength, where a is the characteristic source size, reflecting the higher-order nature of the multipole.^[23]

Radiated Power

The time-averaged radiated power in the electric quadrupole approximation is given by P = \frac{1}{4\pi \epsilon_0} \frac{1}{180 c^5} \left< \dddot{Q}_{ij}(t) \dddot{Q}_{ij}(t) \right>, where the repeated indices imply summation over i, j = 1, 2, 3, \dddot{Q}_{ij} denotes the third time derivative of the traceless symmetric quadrupole tensor Q_{ij}, c is the speed of light, and \left< \cdot \right> denotes time averaging.^[24] For a harmonically oscillating source with Q_{ij}(t) = \mathrm{Re} [Q_{ij} e^{-i \omega t}], the time-averaged radiated power becomes P = \frac{1}{4\pi \epsilon_0} \frac{\omega^6}{360 c^5} \sum_{i,j} |Q_{ij}|^2, where the sum is over the independent components of the complex amplitude tensor, accounting for the time average of the oscillating terms.^[24] The differential power distribution dP/d\Omega for electric quadrupole radiation is more intricate than the \sin^2 \theta pattern of dipole radiation, involving quadratic forms of the quadrupole tensor projected onto the propagation direction \mathbf{n}, such as [ (Q^* \cdot \mathbf{n}) \cdot (Q \cdot \mathbf{n}) - | \mathbf{n} \cdot Q \cdot \mathbf{n} |^2 ]; for general orientations, this can produce azimuthal variations including \cos(4\phi) terms, leading to four-lobed patterns.^[24] Compared to electric dipole radiation, the quadrupole power is suppressed by a factor of order (\omega a / c)^2, where a is the characteristic source size, since the quadrupole moment scales as the dipole moment times a while involving two higher powers of \omega / c; this makes quadrupole radiation negligible for small, low-frequency sources but relevant in atomic transitions or compact astrophysical systems where dipole contributions vanish.^[25]

Generalized Multipole Radiation

Solutions of the Wave Equation

The electromagnetic fields in source-free regions of space obey the vector Helmholtz equation,

\nabla^2 \mathbf{E} + k^2 \mathbf{E} = 0,

where k = \omega / c is the wavenumber, with an identical equation for the magnetic field \mathbf{B}. This equation arises from Maxwell's equations in the frequency domain for time-harmonic fields with angular frequency \omega. Solutions to this equation in spherical coordinates (r, \theta, \phi) are obtained through separation of variables, yielding a complete orthogonal basis in terms of vector spherical harmonics. The scalar Helmholtz equation, \nabla^2 \psi + k^2 \psi = 0, provides the foundation for constructing vector solutions. Separation of variables in spherical coordinates separates the equation into radial and angular parts, with the angular solutions given by the spherical harmonics Y_{lm}(\theta, \phi), where l = 0, 1, 2, \dots and m = -l, \dots, l. The corresponding radial solutions are the spherical Bessel function j_l(kr) for regular (interior) waves and the spherical Hankel function of the first kind h_l^{(1)}(kr) for outgoing (exterior) waves. Thus, the scalar solutions take the form \psi_{lm}(r, \theta, \phi) = f_l(kr) Y_{lm}(\theta, \phi), where f_l(kr) is either j_l(kr) or h_l^{(1)}(kr). Vector solutions are generated from these scalar functions to satisfy the vector Helmholtz equation while being divergence-free. The two fundamental sets are the magnetic multipole functions \mathbf{M}_{lm} and the electric multipole functions \mathbf{N}_{lm}, defined as

\mathbf{M}_{lm}(\mathbf{r}) = \frac{1}{\sqrt{l(l+1)}} \nabla \times \left[ \mathbf{r} \, \psi_{lm}(r, \theta, \phi) \right]

and

\mathbf{N}_{lm}(\mathbf{r}) = \frac{1}{k} \nabla \times \mathbf{M}_{lm}(\mathbf{r}),

where \psi_{lm} = f_l(kr) Y_{lm}(\theta, \phi). These functions are transverse (\nabla \cdot \mathbf{M}_{lm} = 0 and \nabla \cdot \mathbf{N}_{lm} = 0) and form a complete basis for expanding solutions to the vector wave equation, normalized such that \int \mathbf{M}_{lm} \cdot \mathbf{M}_{l'm'}^* \, d\Omega = \delta_{ll'} \delta_{mm'}, and similarly for \mathbf{N}_{lm}. For outgoing radiation in the exterior region (r > r', where r' is the source radius), the Hankel functions h_l^{(1)}(kr) are used. The general solution for the electric field in the outgoing radiation zone is the superposition

\mathbf{E}(\mathbf{r}) = \sum_{l=1}^\infty \sum_{m=-l}^l \left[ b_{lm} \mathbf{M}_{lm}^{(h)}(\mathbf{r}) + a_{lm} \mathbf{N}_{lm}^{(h)}(\mathbf{r}) \right],

where the coefficients a_{lm} (for TM/electric) and b_{lm} (for TE/magnetic) are determined by source conditions, and the superscript (h) denotes the use of h_l^{(1)}(kr). The associated magnetic field follows from Maxwell's equations as

\mathbf{B}(\mathbf{r}) = \frac{1}{i \omega} \nabla \times \mathbf{E}(\mathbf{r}).

This expansion ensures the fields satisfy the radiation boundary conditions at infinity, with power radiating outward.

Electric Multipole Fields

The electric multipole fields describe the transverse magnetic (TM) modes in the spherical wave expansion of electromagnetic radiation, which arise from time-varying charge distributions and the longitudinal components of current densities. These fields form a complete basis for solutions to the wave equation in source-free regions outside the radiating system, ensuring divergence-free and curl-consistent behavior that satisfies Maxwell's equations. Unlike static fields, these dynamic modes propagate as outgoing spherical waves, capturing the radiative contributions from higher-order electric multipoles beyond the dipole approximation.^[3] The electric field for the electric multipole expansion is expressed as

\mathbf{E} = \sum_{l=1}^\infty \sum_{m=-l}^l a_{lm} \mathbf{N}_{lm}\bigl(h_l(kr)\bigr),

where \mathbf{N}_{lm} are the normalized TM vector spherical harmonics, defined in terms of the angular momentum operator applied to scalar spherical harmonics, and h_l(kr) are the spherical Hankel functions of the first kind, ensuring outgoing wave behavior in the radiation zone. The corresponding magnetic field is derived from Faraday's law as

\mathbf{B} = -\frac{i}{\omega} \nabla \times \mathbf{E},

which guarantees that the radial component B_r = 0, a hallmark of TM modes where the magnetic field is purely tangential. This structure links the fields directly to the sources, with the coefficients a_{lm} given by projection onto the sources, involving integrals of the current density \mathbf{J} and charge density \rho such as \int (\mathbf{J} \cdot \mathbf{N}_{lm}^* - i \omega \rho \psi_{lm}^*) \, d^3\mathbf{r}, emphasizing the role of both transverse currents and charge oscillations in exciting these modes. For small source regions (kr \ll 1), this reduces to near-zone electrostatic-like fields, transitioning to transverse radiation in the far zone (kr \gg 1).^[26]^[3] In the specific case of l=1, the expansion simplifies to the electric dipole fields, where the leading term involves the time derivative of the dipole moment \mathbf{p}(t), yielding the well-known radiation pattern with \mathbf{E} \propto [\mathbf{n} \times (\mathbf{n} \times \ddot{\mathbf{p}})] / r and \mathbf{B} = (1/c) \mathbf{n} \times \mathbf{E} in the far field, serving as the foundational example for atomic transitions and antenna radiation. Higher l introduce angular dependencies that suppress radiation along the source axis, with power scaling as \omega^{2l+2}. The parity transformation properties of these fields under spatial inversion are odd for odd l (changing sign) and even for even l (unchanged), which dictates selection rules in quantum mechanical applications such as forbidden transitions.^[4]^[3]

Magnetic Multipole Fields

Magnetic multipole fields in electromagnetic radiation correspond to transverse electric (TE) modes, which are sourced by transverse current distributions without net charge oscillation. These fields are expressed in the spherical coordinate basis using vector spherical harmonics, providing a complete orthogonal decomposition of the radiation pattern. The electric field for the magnetic multipole expansion takes the form

\mathbf{E} = \sum_{l=1}^\infty \sum_{m=-l}^l b_{lm} \mathbf{M}_{lm} \left( h_l^{(1)}(kr) \right),

where h_l^{(1)}(kr) is the spherical Hankel function of the first kind representing outgoing waves, \mathbf{M}_{lm} are the magnetic-type vector spherical harmonics defined as \mathbf{M}_{lm} = \frac{1}{\sqrt{l(l+1)}} \mathbf{L} Y_{lm}(\theta, \phi) with \mathbf{L} = -i \mathbf{r} \times \nabla, and the coefficients b_{lm} are determined by the source currents.^[27]^[28] The corresponding magnetic field is obtained from Maxwell's equations in the time-harmonic form as \mathbf{B} = \frac{1}{i \omega} \nabla \times \mathbf{E}, or equivalently in the far field approximation \mathbf{B} \approx \frac{i k}{\omega} \hat{r} \times \mathbf{E}, ensuring the fields satisfy the wave equation and radiation boundary conditions. A defining characteristic of these TE modes is the absence of a radial electric field component, E_r = 0, which distinguishes them from transverse magnetic (TM) modes and reflects their solenoidal nature derived from current sources.^[29]^[28] For the lowest order l=1, the magnetic multipole field reduces to the familiar magnetic dipole radiation pattern, where the fields exhibit a \sin \theta angular dependence and decay as $1/r in the far zone, analogous to the electric dipole but sourced by a magnetic moment. The orthogonality of the vector spherical harmonics ensures that the modes are decoupled in the spherical basis, with the inner product \int \mathbf{M}_{lm}^* \cdot \mathbf{M}_{l'm'} \, d\Omega = \delta_{ll'} \delta_{mm'}, allowing independent contributions from each (l, m) term without cross-talk in the expansion. This property facilitates the efficient computation of radiation from localized current distributions in applications such as antenna design and scattering theory.^[27]^[29]

General Superposition

In the far-field radiation zone, any arbitrary outgoing electromagnetic field generated by localized sources can be uniquely decomposed into a superposition of electric and magnetic multipole fields. This expansion leverages the complete and orthogonal set of vector spherical harmonics to represent the fields as a sum over multipole orders l and azimuthal indices m. Specifically, the electric field \mathbf{E} is expressed as

\mathbf{E}(\mathbf{r}) = \sum_{l=1}^{\infty} \sum_{m=-l}^{l} \left[ b_{lm} \mathbf{M}_{lm}(kr, \theta, \phi) + a_{lm} \mathbf{N}_{lm}(kr, \theta, \phi) \right],

where \mathbf{M}_{lm} denote the transverse electric (TE) modes corresponding to magnetic multipoles, and \mathbf{N}_{lm} denote the transverse magnetic (TM) modes for electric multipoles, with the radial dependence given by outgoing Hankel functions h_l^{(1)}(kr) for large r. The magnetic field \mathbf{H} follows analogously from Maxwell's equations, ensuring the decomposition satisfies the source-free wave equation outside the sources.^[28] The coefficients a_{lm} and b_{lm} are determined uniquely through the orthogonality properties of the vector spherical harmonics, which form a complete basis for expanding angular dependencies on the unit sphere. At a large radius r, these coefficients are obtained by projecting the observed field onto the dual basis functions:

a_{lm} = \int \mathbf{E} \cdot \mathbf{N}_{lm}^* \, d\Omega, \quad b_{lm} = \int \mathbf{E} \cdot \mathbf{M}_{lm}^* \, d\Omega,

where the integrals are over the solid angle d\Omega = \sin\theta \, d\theta \, d\phi, using the unit-normalized harmonics. This projection ensures that the decomposition is invariant under rotations and independent of the choice of origin, unlike Cartesian multipole moments which depend on the coordinate frame.^[28] The uniqueness of this multipole decomposition stems from the completeness of the vector spherical harmonics, guaranteeing that any transverse radiation field can be exactly represented without ambiguity, provided the sources are confined within a finite region. This framework extends naturally to higher multipole orders l > 2, encompassing arbitrary radiation patterns from complex sources or scattering processes. For instance, in scattering theory, the incident field excites these multipoles on the scatterer, allowing the far-field pattern to be analyzed via the coefficients. The total radiated power P generalizes the low-order cases, with each multipole contributing a term scaling as \omega^{2l+2} times the squared magnitude of the corresponding moment, quantified as P = \sum_{l,m} \frac{\mu_0 c k^{2l+2} l(l+1)}{4\pi (2l+1)} \left( |a_{lm}|^2 + |b_{lm}|^2 \right) for normalized coefficients; this formula accounts for the frequency scaling and degeneracy in m, facilitating efficient computation for high-l contributions in applications like antenna design or astrophysical radiation.^[28]^[4]

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