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.
This article is primarily concerned with electromagnetic multipole radiation, although the treatment of gravitational waves is similar.
Multipole analysis offers a way to separate the radiation into moments of increasing complexity.
Multipole moments are calculated with respect to a fixed expansion point which is taken to be the origin of a given coordinate system.
Because higher-order moments depend on the position of the origin, they cannot be regarded as invariant properties of the system.
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.
Likewise, the electric dipole moment creates a field that scales as inverse distance cubed, and so on.
Accordingly, every time-dependent multipole moment must contribute radiant energy density that scales as
Even so, the multipole coefficients of a system generally diminish with increasing order, usually as
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.
The intrinsic angular momentum of elementary particles (see Spin (physics)) may also affect electromagnetic radiation from some source materials.
For simplicity however, these effects will be deferred to the discussion of generalized multipole radiation.
where k = ω / c. These formulas provide the basis for analyzing multipole radiation.
By using this approximation, the remaining x′ dependence is the same as it is for a static system, the same analysis applies.
[4][5] 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.
If the system is closed then the total charge cannot fluctuate which means the oscillation amplitude q must be zero.
[5] Electric dipole radiation can be derived by applying the zeroth-order term to the vector potential.
It follows that the time averaged power density per unit solid angle is given by
[5] The average power radiated per unit solid angle by a magnetic dipole is
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.
The average power radiated per unit solid angle by an electric quadrupole is
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.
The resulting electric and magnetic fields share the same time-dependence as the sources.
The homogeneous wave equations that describes electromagnetic radiation with frequency
To determine the other coefficients, the Green's function for the wave equation is applied.
Retaining only the lowest order terms results in the simplified form for the electric multipole coefficients:[5]
corresponds to an induced electric multipole moment from the intrinsic magnetization of the source material.
Retaining only the lowest order terms results in the simplified form for the magnetic multipole coefficients:[5]