Multipole radiation is a theoretical framework for the description of electromagnetic or gravitational radiation from time-dependent distributions of distant sources. These tools are applied to physical phenomena which occur at a variety of length scales - from gravitational waves due to galaxy collisions to gamma radiation resulting from nuclear decay. Multipole radiation is analyzed using similar multipole expansion techniques that describe fields from static sources, however there are important differences in the details of the analysis because multipole radiation fields behave quite differently from static fields. This article is primarily concerned with electromagnetic multipole radiation, although the treatment of gravitational waves is similar.
Contents
- Linearity of moments
- Origin dependence of multipole moments
- Field dependence on distance
- Sources
- Potentials
- Multipole expansion in near field
- Multipole expansion in far field Multipole radiation
- Electric monopole radiation nonexistence
- Electric dipole potential
- Electric dipole fields
- Pure electric dipole power
- Magnetic dipole potential
- Magnetic dipole fields
- Pure magnetic dipole power
- Electric quadrupole potential
- Electric quadrupole fields
- Pure electric quadrupole power
- Generalized multipole radiation
- Solutions of the wave equation
- Electric multipole fields
- Magnetic multipole fields
- General solution
- References
Electromagnetic radiation depends on structural details of the source system of electric charge and electric current. Direct analysis can be intractable if the structure is unknown or complicated. Multipole analysis offers a way to separate the radiation into moments of increasing complexity. Since the electromagnetic field depends more heavily on lower-order moments than on higher-order moments, the electromagnetic field can be approximated without knowing the structure in detail.
Linearity of moments
Since Maxwell's equations are linear, the electric field and magnetic field depend linearly on source distributions. Linearity allows the fields from various multipole moments to be calculated independently and added together to give the total field of the system. This is the well-known principle of superposition.
Origin dependence of multipole moments
Multipole moments are calculated with respect to a fixed expansion point which is taken to be the origin of a given coordinate system. Translating the origin changes the multipole moments of the system with the exception of the first non-vanishing moment. For example, the monopole moment of charge is simply the total charge in the system. Changing the origin will never change this moment. If the monopole moment is zero then the dipole moment of the system will be translation invariant. If both the monopole and dipole moments are zero then the quadrupole moment is translation invariant, and so forth. Because higher-order moments depend on the position of the origin, they cannot be regarded as invariant properties of the system.
Field dependence on distance
The field from a multipole moment depends on both the distance from the origin and the angular orientation of the evaluation point with respect to the coordinate system. In particular, the radial dependence of the electromagnetic field from a stationary
The radial dependence of radiation waves is different from static fields because these waves carry energy away from the system. Since energy must be conserved, simple geometric analysis shows that the energy density of spherical radiation, radius
Sources
Time-dependent source distributions can be expressed using Fourier analysis. This allows separate frequencies to be analyzed independently. Charge density is given by
and current density by
For convenience, only a single angular frequency ω is considered from this point forward; thus
The superposition principle may be applied to generalize results for multiple frequencies. Vector quantities appear in bold. The standard convention of taking the real part of complex quantities to represent physical quantities is used.
It should be noted that the intrinsic angular momentum of elementary particles (see Spin (physics)) may also affect electromagnetic radiation from some source materials. To account for these effects, the intrinsic magnetization of the system
Potentials
The source distributions can be integrated to yield the time-dependent electric potential and magnetic potential φ and A respectively. Formulas are expressed in the Lorenz Gauge in SI units.
In these formulas c is the speed of light in vacuum,
where k=ω/c. These formulas provide the basis for analyzing multipole radiation.
Multipole expansion in near field
The near field is the region around a source where the electromagnetic field can be evaluated quasi-statically. If target distance from the multipole origin
See Taylor expansion. By using this approximation, the remaining x′ dependence is the same as it is for a static system, the same analysis applies. Essentially, the potentials can be evaluated in the near field at a given instant by simply taking a snapshot of the system and treating it as though it were static - hence it is called quasi-static. See near and far field and multipole expansion. In particular, the inverse distance
Multipole expansion in far field: Multipole radiation
At large distances from a high frequency source,
Since only the first-order term in
Each power of
Electric monopole radiation, nonexistence
The zeroth order term,
where the total charge
If the system is closed then the total charge cannot fluctuate which means the oscillation amplitude q must be zero. Hence,
Electric dipole potential
Electric dipole radiation can be derived by applying the zeroth-order term to the vector potential.
Integration by parts yields
and the charge continuity equation shows
It follows that
Similar results can be obtained by applying the first-order term,
Electric dipole fields
Once the time-dependent potentials are understood, the time-dependent electric field and magnetic field can be calculated in the usual way. Namely,
or, in a source-free region of space, the relationship between the magnetic field and the electric field can be used to obtain
where
which is consistent with spherical radiation waves.
Pure electric dipole power
The power density, energy per unit area per unit time, is expressed by the Poynting vector
The dot product with
where θ is measured with respect to
Magnetic dipole potential
The first-order term,
The integrand can be separated into symmetric and anti-symmetric parts in n and x′
The second term contains the effective magnetization due to the current
Notice that
on previous results yields magnetic dipole results.
Magnetic dipole fields
Pure magnetic dipole power
The average power radiated per unit solid angle by a magnetic dipole is
where θ is measured with respect to the magnetic dipole
Electric quadrupole potential
The symmetric portion of the integrand from the previous section can be resolved by applying integration by parts and the charge continuity equation as was done for electric dipole radiation.
This corresponds to the traceless electric quadrupole moment tensor
Electric quadrupole fields
The resulting magnetic and electric fields are:
Pure electric quadrupole power
The average power radiated per unit solid angle by an electric quadrupole is
where θ is measured with respect to the magnetic dipole
Generalized multipole radiation
As the multipole moment of a source distribution increases, the direct calculations employed so far become too cumbersome to continue. Analysis of higher moments requires more general theoretical machinery. Just as before, a single source frequency
respectively. The resulting electric and magnetic fields share the same time-dependence as the sources.
Using these definitions and the continuity equation allows Maxwell's equations to be written as
These equations can be combined by taking the curl of the last equations and applying the identity
Solutions of the wave equation
The homogeneous wave equations that describes electromagnetic radiation with frequency
The wave function
Where
then the solution is:
The Green function can be expressed in vector spherical harmonics.
Note that
Electric multipole fields
Applying the above solution to the electric multipole wave equation
gives the solution for the magnetic field:
The electric field is:
The forumula can be simplified by applying the identities
to the integrand, which results in
Green's theorem and integration by parts manipulates the formula into
The spherical bessel function
Retaining only the lowest order terms results in the simplified form for the electric multipole coefficients:
Magnetic multipole fields
Applying the above solution to the magnetic multipole wave equation
gives the solution for the electric field:
The magnetic field is:
As before, the forumula simplifies to:
Retaining only the lowest order terms results in the simplified form for the magnetic multipole coefficients:
General solution
The electric and magnetic multipole fields combine to give the total fields:
Note that the radial function
Thus the radial dependence of radiation is recovered.