Multipole radiation explained

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.

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.

Properties of multipole radiation

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

2 ^{\ell} -pole scales as 1 /r ^{\ell+2} . That is, the electric field from the electric monopole moment scales as inverse distance squared. Likewise, the electric dipole moment creates a field that scales as inverse distance cubed, and so on. As distance increases, the contribution of high-order moments becomes much smaller than the contribution from low-order moments, so high-order moments can be ignored to simplify calculations.

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

r , must scale as 1 /r ^{2} . As a spherical wave expands, the fixed energy of the wave must spread out over an expanding sphere of surface area 4 \pi r ^{2} . Accordingly, every time-dependent multipole moment must contribute radiant energy density that scales as 1 /r ^{2} , regardless of the order of the moment. Hence, high-order moments cannot be discarded as easily as in static case. Even so, the multipole coefficients of a system generally diminish with increasing order, usually as 1 /( 2 \ell +1 )!! , so radiation fields can still be approximated by truncating high-order moments.

Time-dependent electromagnetic fields

Sources

Time-dependent source distributions can be expressed using Fourier analysis. This allows separate frequencies to be analyzed independently. Charge density is given by$$\backslash rho(\backslash mathbf,t)\; =\; \backslash int\_^\backslash infty\; d\backslash omega\; \backslash ,\; \backslash hat(\backslash mathbf,\backslash omega)\; e^$$and current density by$$\backslash mathbf(\backslash mathbf,t)\; =\; \backslash int\_^\backslash infty\; d\backslash omega\; \backslash ,\; \backslash hat(\backslash mathbf,\backslash omega)\; e^.$$For convenience, only a single angular frequency ω is considered from this point forward; thus$$\backslash rho(\backslash mathbf,t)\; =\; \backslash rho(\backslash mathbf)\; e^$$$$\backslash mathbf(\backslash mathbf,t)\; =\; \backslash mathbf(\backslash mathbf)\; e^$$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.

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

M ( x ,t) would have to be taken into account. For simplicity however, these effects will be deferred to the discussion of generalized multipole radiation.

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.

$$\backslash phi(\backslash mathbf,t)\; =\; \backslash frac\; \backslash int\; d^3\backslash mathbf\backslash int\; dt\text{'}\; \backslash frac\backslash delta\backslash left(t\text{'}\; -\; \backslash left(t-\backslash frac\backslash right)\backslash right)$$$$\backslash mathbf(\backslash mathbf,t)=\backslash frac\backslash int\; d^3\backslash mathbf\backslash int\; dt\text{'}\backslash frac\backslash delta\backslash left(t\text{'}-\backslash left(t-\backslash frac\backslash right)\backslash right)$$

In these formulas c is the speed of light in vacuum,

\delta is the Dirac delta function, and $\backslash |\backslash mathbf-\backslash mathbf\backslash |\_2$ is the Euclidean distance from the source point x′ to the evaluation point x. Integrating the time-dependent source distributions above yields

$$\backslash phi(\backslash mathbf,t)\; =\; \backslash frac\; e^\; \backslash int\; d^3\backslash mathbf\backslash rho(\backslash mathbf)\backslash frac$$$$\backslash mathbf(\backslash mathbf,t)\; =\; \backslash frac\; e^\; \backslash int\; d^3\backslash mathbf\backslash mathbf(\backslash mathbf)\backslash frac$$where . 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

r = \| x \| _{2} is much smaller than the radiation wavelength λ=2 \pi/k , then k r\ll 1 . As a result, the exponential can be approximated in this region as:$$e^\; =\; 1\; +\; \backslash mathcal\; O(k\; r)$$

See Taylor expansion. By using this approximation, the remaining 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

1 /\| x - x' \| _{2} is expanded using spherical harmonics which are integrated separately to obtain spherical multipole coefficients.

Multipole expansion in far field: Multipole radiation

At large distances from a high frequency source,

λ \ll r , the following approximations hold:

$$\backslash frac\; =\; \backslash frac\; +\; \backslash mathcal\; O(1/r^2)$$$$e^=e^=e^(1\; +\; \backslash mathcal\; O(1/r))$$

Since only the first-order term in

1 /r is significant at large distances, the expansions combine to give

$$\backslash frac=\backslash frac\backslash left(1\; -\; i\; k(\backslash mathbf\; \backslash cdot\; \backslash mathbf)\; +\; \backslash frac(\backslash mathbf\backslash cdot\backslash mathbf)^2\; +\; \backslash cdots\backslash right)\; +\; \backslash mathcal\; O(1/r^2)$$

Each power of

n ⋅ x' corresponds to a different multipole moment. The first few moments are evaluated directly below.

Electric monopole radiation, nonexistence

The zeroth order term,

(e ^{i k \| x - x' \| _{2}})/(\| x - x' \| _{2}) → (e ^{i})/(r) , applied to the scalar potential gives$$\backslash phi\_(\backslash mathbf,t)\; =\; \backslash frac\; \backslash frac\; \backslash int\; d^3\backslash mathbf\backslash rho(\backslash mathbf)=\backslash frac\; q$$where the total charge $q\; =\; \backslash int\; d^3\backslash mathbf\; \backslash rho(\backslash mathbf)$ is the electric monopole moment oscillating at frequency . Conservation of charge requires since$$q(t)\; =\; \backslash int\; d^3\backslash mathbf\backslash rho(\backslash mathbf,t)=\backslash int\; d^3\backslash mathbf\backslash rho(\backslash mathbf)e^\; =\; q\; e^.$$

If the system is closed then the total charge cannot fluctuate which means the oscillation amplitude q must be zero. Hence,

\phi _{Electricmonopole} ( x ,t) = 0 . The corresponding fields and radiant power must also be zero.

Electric dipole radiation

Electric dipole potential

Electric dipole radiation can be derived by applying the zeroth-order term to the vector potential.

$$\backslash mathbf\_(\backslash mathbf,t)\; =\; \backslash frac\; \backslash frac\; \backslash int\; d^3\backslash mathbf\backslash mathbf(\backslash mathbf)$$

Integration by parts yields^[1]

$$\backslash int\; d^3\backslash mathbf\backslash mathbf(\backslash mathbf)=-\backslash int\; d^3\backslash mathbf\; \backslash mathbf(\backslash boldsymbol\backslash cdot\backslash mathbf(\backslash mathbf)).$$

and the charge continuity equation shows

$$\backslash frac\; +\; \backslash boldsymbol\backslash cdot\backslash mathbf(\backslash mathbf,t)=\backslash left(-i\backslash omega\backslash rho(\backslash mathbf)\; +\; \backslash boldsymbol\backslash cdot\backslash mathbf(\backslash mathbf)\backslash right)e^=0.$$

It follows that

$$\backslash mathbf\_(\backslash mathbf,t)\; =\; \backslash frac\; \backslash frac\; \backslash int\; d^3\backslash mathbf\backslash mathbf\backslash rho(\backslash mathbf)$$

Similar results can be obtained by applying the first-order term,

(e ^{i k \| x - x' \| _{2}})/(\| x - x' \| _{2}) → (e ^{i})/(r) ( - i k)( n ⋅ x' ) to the scalar potential. The amplitude of the electric dipole moment of the system is $\backslash mathbf\; =\; \backslash int\; d^3\backslash mathbf\backslash mathbf\backslash rho(\backslash mathbf)$, which allows the potentials to be expressed as

$$\backslash phi\_(\backslash mathbf,t)\; =\; \backslash frac\; \backslash frac\; \backslash mathbf\backslash cdot\backslash mathbf$$$$\backslash mathbf\_(\backslash mathbf,t)\; =\backslash frac\; \backslash frac\; \backslash mathbf$$

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,

$$\backslash mathbf(\backslash mathbf,t)=-\backslash boldsymbol\backslash phi(\backslash mathbf,t)-\backslash frac$$$$\backslash mathbf(\backslash mathbf,t)=\backslash boldsymbol\backslash times\backslash mathbf(\backslash mathbf,t),$$

or, in a source-free region of space, the relationship between the magnetic field and the electric field can be used to obtain

$$\backslash mathbf(\backslash mathbf,t)=\backslash frac\backslash boldsymbol\backslash times\backslash mathbf(\backslash mathbf,t)$$$$\backslash mathbf(\backslash mathbf,t)=\backslash frac\backslash boldsymbol\backslash times\backslash mathbf(\backslash mathbf,t)$$

where $Z\_0\; =\; \backslash sqrt$ is the impedance of free space. The electric and magnetic fields that correspond to the potentials above are

$$\backslash mathbf\_(\backslash mathbf,t)=\backslash frac(\backslash mathbf\backslash times\backslash mathbf)\backslash frac$$$$\backslash mathbf\_(\backslash mathbf,t)=Z\_0(\backslash mathbf\_\backslash times\backslash mathbf)$$

which is consistent with spherical radiation waves.

Pure electric dipole power

S = E x H . It follows that the time averaged power density per unit solid angle is given by

$$\backslash frac=\backslash frac\backslash Re(\backslash mathbf\backslash cdot\backslash mathbf\backslash times\backslash mathbf).$$

The dot product with

n extracts the emission magnitude and the factor of 1/2 comes from averaging over time. As explained above, the r ^{2} cancels the radial dependence of radiation energy density. Application to a pure electric dipole gives

$$\backslash frac=\backslash frack^4\backslash |\backslash mathbf\backslash |\_2^2\backslash sin^2\backslash theta$$

where θ is measured with respect to

p . Integration over a sphere yields the total power radiated:

$$P\_=\backslash frack^4\backslash |\backslash mathbf\backslash |\_2^2$$

Magnetic dipole radiation

Magnetic dipole potential

The first-order term,

(e ^{i k \| x - x' \| _{2}})/(\| x - x' \| _{2}) → (e ^{i})/(r) ( - i k)( n ⋅ x' ) , applied to the vector potential gives magnetic dipole radiation and electric quadrupole radiation.

$$\backslash mathbf\_(\backslash mathbf,t)\; =\; \backslash frac\; \backslash frac(-i\; k)\backslash int\; d^3\backslash mathbf(\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf(\backslash mathbf)$$

The integrand can be separated into symmetric and anti-symmetric parts in J and x′

$$(\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf(\backslash mathbf)=\backslash frac\backslash left((\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf(\backslash mathbf)+(\backslash mathbf\backslash cdot\backslash mathbf(\backslash mathbf))\backslash mathbf\backslash right)+\backslash frac(\backslash mathbf\backslash times\backslash mathbf(\backslash mathbf))\backslash times\; \backslash mathbf$$

The second term contains the effective magnetization due to the current

M _{effective} ( x' ) =1 / 2 ( x' x J ( x' )) and integration gives the magnetic dipole moment.

$$\backslash int\; d^3\backslash mathbf\backslash mathbf\_(\backslash mathbf)\; =\; \backslash mathbf$$$$\backslash mathbf\_(\backslash mathbf,t)\; =\; \backslash frac\; \backslash frac\backslash mathbf\backslash times\backslash mathbf$$

Notice that

A _{Magneticdipole} has a similar form to H _{Electricdipole} . That means the magnetic field from a magnetic dipole behaves similarly to the electric field from an electric dipole. Likewise, the electric field from a magnetic dipole behaves like the magnetic field from an electric dipole. Taking the transformations

$$\backslash mathbf\_\; \backslash rightarrow\; Z\_0\backslash mathbf\_$$$$\backslash mathbf\_\; \backslash rightarrow\; \backslash frac\backslash mathbf\_$$$$\backslash mathbf\; \backslash rightarrow\; \backslash mathbf/c$$

on previous results yields magnetic dipole results.

Magnetic dipole fields

$$\backslash mathbf\_(\backslash mathbf,t)=\backslash frac(\backslash mathbf\backslash times\backslash mathbf)\backslash frac$$$$\backslash mathbf\_(\backslash mathbf,t)=\backslash frac(\backslash mathbf\_\backslash times\backslash mathbf)$$

Pure magnetic dipole power

The average power radiated per unit solid angle by a magnetic dipole is

$$\backslash frac=\backslash frack^4\backslash |\backslash mathbf\backslash |\_2^2\backslash sin^2\backslash theta$$

where θ is measured with respect to the magnetic dipole

m . The total power radiated is:

$$P\_=\backslash frack^4\backslash |\backslash mathbf\backslash |\_2^2$$

Electric quadrupole radiation

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.

$$\backslash frac\; \backslash int\; d^3\backslash mathbf\backslash left((\backslash mathbf\backslash cdot\backslash mathbf)\backslash mathbf(\backslash mathbf)+(\backslash mathbf\backslash cdot\backslash mathbf(\backslash mathbf))\backslash mathbf\backslash right)=\backslash frac\; \backslash int\; d^3\backslash mathbf\; \backslash mathbf\; (\backslash mathbf\backslash cdot\backslash mathbf)\backslash rho(\backslash mathbf)$$

$$\backslash mathbf\_(\backslash mathbf,t)\; =\; \backslash frac\; \backslash frac\backslash int\; d^3\backslash mathbf\; \backslash mathbf\; (\backslash mathbf\backslash cdot\backslash mathbf)\backslash rho(\backslash mathbf)$$

This corresponds to the traceless electric quadrupole moment tensor $Q\_\; =\; \backslash int\; d^3\backslash mathbf(3\; x\text{'}\_\backslash alpha\; x\text{'}\_\backslash beta\; -\; \backslash |\backslash mathbf\backslash |^2\; \backslash delta\_)\backslash rho(\backslash mathbf)$. Contracting the second index with the normal vector $[Q(\backslash mathbf\{n\})]\_\backslash alpha\; =\; \backslash sum\_\backslash beta\; Q\_\; n\_\backslash beta$ allows the vector potential to be expressed as

$$\backslash mathbf\_(\backslash mathbf,t)\; =\; \backslash frac\; \backslash frac\backslash frac\backslash mathbf$$

Electric quadrupole fields

The resulting magnetic and electric fields are:

$$\backslash mathbf\_(\backslash mathbf,t)\; =\; \backslash frac\; \backslash frac\backslash mathbf\backslash times\backslash mathbf$$$$\backslash mathbf\_(\backslash mathbf,t)=Z\_0(\backslash mathbf\_\backslash times\backslash mathbf)$$

Pure electric quadrupole power

The average power radiated per unit solid angle by an electric quadrupole is

$$\backslash frac=\backslash frack^6\backslash |(\backslash mathbf\backslash times\backslash mathbf)\backslash times\backslash mathbf\backslash |^2$$

where θ is measured with respect to the magnetic dipole

m . The total power radiated is:

$$P\_=\backslash frack^6\backslash sum\_\; Q\_^2$$

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

\omega is considered. Hence the charge, current, and intrinsic magnetization densities are given by

$$\backslash rho(\backslash mathbf,t)=\backslash rho(\backslash mathbf)e^$$$$\backslash mathbf(\backslash mathbf,t)=\backslash mathbf(\backslash mathbf)e^$$$$\backslash mathbf(\backslash mathbf,t)=\backslash mathbf(\backslash mathbf)e^$$

respectively. The resulting electric and magnetic fields share the same time-dependence as the sources.

$$\backslash mathbf(\backslash mathbf,t)=\backslash mathbf(\backslash mathbf)e^$$$$\backslash mathbf(\backslash mathbf,t)=\backslash mathbf(\backslash mathbf)e^$$

Using these definitions and the continuity equation allows Maxwell's equations to be written as

$$\backslash boldsymbol\backslash cdot\backslash mathbf(\backslash mathbf)=-\backslash frac\backslash boldsymbol\backslash cdot\backslash mathbf(\backslash mathbf)$$$$\backslash boldsymbol\backslash cdot\backslash mathbf(\backslash mathbf)=-\backslash boldsymbol\backslash cdot\backslash mathbf(\backslash mathbf)$$$$\backslash boldsymbol\backslash times\backslash mathbf(\backslash mathbf)=ikZ\_0\backslash left(\backslash mathbf(\backslash mathbf)+\backslash mathbf(\backslash mathbf)\backslash right)$$$$\backslash boldsymbol\backslash times\backslash mathbf(\backslash mathbf)=-\backslash frac\backslash mathbf(\backslash mathbf)+\backslash mathbf(\backslash mathbf)$$

These equations can be combined by taking the curl of the last equations and applying the identity

\boldsymbol{\nabla} x (\boldsymbol{\nabla} x V ) = \boldsymbol{\nabla}(\boldsymbol{\nabla} ⋅ V ) - \boldsymbol{\nabla} ^{2V} . This gives the vector forms of the non-homogeneous Helmholtz equation.

$$(\backslash nabla^2+k^2)\backslash mathbf(\backslash mathbf)=-\backslash left[ikZ\_0\backslash mathbf\{J\}(\backslash mathbf\{x\})+ikZ\_0\backslash boldsymbol\{\backslash nabla\}\backslash times\backslash mathbf\{M\}(\backslash mathbf\{x\})+\backslash frac\{iZ\_0\}\{k\}\backslash boldsymbol\{\backslash nabla\}(\backslash boldsymbol\{\backslash nabla\}\backslash cdot\backslash mathbf\{J\}(\backslash mathbf\{x\}))\backslash right]$$$$(\backslash nabla^2+k^2)\backslash mathbf(\backslash mathbf)=-\backslash left[k^2\backslash mathbf\{M\}(\backslash mathbf\{x\})+\backslash boldsymbol\{\backslash nabla\}\backslash times\backslash mathbf\{J\}(\backslash mathbf\{x\})+\backslash boldsymbol\{\backslash nabla\}(\backslash boldsymbol\{\backslash nabla\}\backslash cdot\backslash mathbf\{M\}(\backslash mathbf\{x\}))\backslash right]$$

Solutions of the wave equation

The homogeneous wave equations that describes electromagnetic radiation with frequency

\omega in a source-free region have the form.

$$(\backslash boldsymbol^2+k^2)\backslash boldsymbol(\backslash mathbf)=0$$

The wave function

\boldsymbol{\Psi}( x ) can be expressed as a sum of vector spherical harmonics

$$\backslash boldsymbol(\backslash mathbf)=\backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; f\_(k\; r)\backslash mathbf\_(\backslash theta,\; \backslash phi)$$$$f\_(k\; r)=A\_^\; h\_\backslash ell^(k\; r)\; +\; A\_^\; h\_\backslash ell^(k\; r)$$

Where $\backslash mathbf\_(\backslash theta,\backslash phi)=\backslash mathbfY\_(\backslash theta,\backslash phi)/\backslash sqrt$ are the normalized vector spherical harmonics and

h ^{( 1 )} _{\ell} and h ^{( 2 )} _{\ell} are spherical Hankel functions. See spherical Bessel functions. The differential operator L =- i x x \boldsymbol{\nabla} is the angular momentum operator with the property L ^{2} Y _{\ell} = \ell(\ell +1 ) Y _{\ell} . The coefficients A ^{( 1 )} _{\ell m} and A ^{( 2 )} _{\ell m} correspond to expanding and contracting waves respectively. So A ^{( 2 )} _{\ell m} =0 for radiation. To determine the other coefficients, the Green's function for the wave equation is applied. If the source equation is

$$(\backslash boldsymbol^2+k^2)\backslash boldsymbol(\backslash mathbf)=-\backslash mathbf(\backslash mathbf)$$

then the solution is:

$$\backslash Psi\_\backslash alpha(\backslash mathbf)=\backslash sum\_\backslash beta\; \backslash int\; d^3\backslash mathbf\; G\_(\backslash mathbf,\backslash mathbf)\; V\_\backslash beta(\backslash mathbf)$$

The Green function can be expressed in vector spherical harmonics.

$$G\_(\backslash mathbf,\backslash mathbf)=\backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; i\; k\; h\_\backslash ell^(kr)\; j\_\backslash ell(kr\text{'})\; X\_(\backslash theta,\backslash phi)\; X\_^*(\backslash theta\text{'},\backslash phi\text{'})$$

Note that $\backslash mathbf\_^*=Y\_^*\backslash mathbf/\backslash sqrt$ is a differential operator that acts on the source function

V . Thus, the solution to the wave equation is:

$$\backslash boldsymbol(\backslash mathbf)=\; \backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; \backslash frac\; h\_\backslash ell^(kr)\; \backslash mathbf\_(\backslash theta,\backslash phi)\; \backslash int\; d^3\backslash mathbf\; j\_\backslash ell(kr\text{'})\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\; \backslash mathbf\backslash cdot\backslash mathbf(\backslash mathbf)$$

Electric multipole fields

Applying the above solution to the electric multipole wave equation

$$(\backslash nabla^2+k^2)\backslash mathbf(\backslash mathbf)=-\backslash left[k^2\backslash mathbf\{M\}(\backslash mathbf\{x\})+\backslash boldsymbol\{\backslash nabla\}\backslash times\backslash mathbf\{J\}(\backslash mathbf\{x\})\; +\; \backslash boldsymbol\{\backslash nabla\}(\backslash boldsymbol\{\backslash nabla\}\backslash cdot\backslash mathbf\{M\}(\backslash mathbf\{x\}))\backslash right]$$

gives the solution for the magnetic field:

$$\backslash mathbf^(\backslash mathbf)=\backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; a\_^\; h\_\backslash ell^(kr)\; \backslash mathbf\_(\backslash theta,\; \backslash phi)$$$$a\_^=\backslash frac\; \backslash int\; d^3\backslash mathbf\; j\_\backslash ell(kr\text{'})\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\; \backslash mathbf\backslash cdot\backslash left[k^2\backslash mathbf\{M\}(\backslash mathbf\{x\text{'}\})+\backslash boldsymbol\{\backslash nabla\text{'}\}\backslash times\backslash mathbf\{J\}(\backslash mathbf\{x\text{'}\})+\backslash boldsymbol\{\backslash nabla\text{'}\}(\backslash boldsymbol\{\backslash nabla\text{'}\}\backslash cdot\backslash mathbf\{M\}(\backslash mathbf\{x\text{'}\}))\backslash right]$$

The electric field is:

$$\backslash mathbf^(\backslash mathbf)=\backslash frac\backslash boldsymbol\backslash times\backslash mathbf^(\backslash mathbf)$$

The formula can be simplified by applying the identities

$$\backslash mathbf\backslash cdot\backslash mathbf(\backslash mathbf)=i\backslash boldsymbol\backslash cdot(\backslash mathbf\backslash times\backslash mathbf(\backslash mathbf))$$$$\backslash mathbf\backslash cdot(\backslash boldsymbol\backslash times\backslash mathbf(\backslash mathbf))=i\backslash nabla^2(\backslash mathbf\backslash cdot\backslash mathbf(\backslash mathbf))-\backslash frac(r^2\backslash boldsymbol\backslash cdot\backslash mathbf(\backslash mathbf))$$$$\backslash mathbf\backslash cdot\backslash boldsymbols(\backslash mathbf)=0$$

to the integrand, which results in

$$a\_^=\backslash frac\; \backslash int\; d^3\backslash mathbf\; j\_\backslash ell(kr\text{'})\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\; \backslash left[-ik\backslash boldsymbol\{\backslash nabla\}\; \backslash cdot\; (\backslash mathbf\{x\text{'}\}\backslash times\backslash mathbf\{M\}(\backslash mathbf\{x\text{'}\}))-\backslash frac\{i\}\{k\}\backslash nabla^2(\backslash mathbf\{x\text{'}\}\; \backslash cdot\; \backslash mathbf\{J\}(\backslash mathbf\{x\text{'}\}))-\backslash frac\{c\backslash partial\}\{r\text{'}\backslash partial\; r\text{'}\}(r\text{'}^2\backslash rho(\backslash mathbf\{x\text{'}\}))\backslash right]$$

Green's theorem and integration by parts manipulates the formula into

$$a\_^=\backslash frac\; \backslash int\; d^3\backslash mathbf\; j\_\backslash ell(kr\text{'})\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\; \backslash left[-ik\backslash boldsymbol\{\backslash nabla\}\backslash cdot(\backslash mathbf\{x\text{'}\}\backslash times\backslash mathbf\{M\}(\backslash mathbf\{x\text{'}\}))+ik\backslash mathbf\{x\text{'}\}\; \backslash cdot\; \backslash mathbf\{J\}(\backslash mathbf\{x\text{'}\})\backslash right]\; +\; c\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\backslash rho(\backslash mathbf)\backslash frac(r\text{'}\; j\_\backslash ell(kr\text{'}))$$

j _{\ell(kr')} can also be simplified by assuming that the radiation length scale is much larger than the source length scale, which is true for most antennas.

$$j\_\backslash ell(kr\text{'})=\backslash frac\; +\; \backslash mathcal\; O((kr\text{'})^)$$

Retaining only the lowest order terms results in the simplified form for the electric multipole coefficients:

$$a\_^=\backslash frac\backslash left(\backslash frac\backslash right)^[Q\_\{\backslash ell\; m\}+Q\_\{\backslash ell\; m\}\text{'}]$$$$Q\_=\backslash int\; d^3\backslash mathbf\; r\text{'}^\backslash ell\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\backslash rho(\backslash mathbf)$$$$Q\_\text{'}=-\backslash frac\backslash int\; d^3\backslash mathbf\; r\text{'}^\backslash ell\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\; \backslash boldsymbol\backslash cdot(\backslash mathbf\backslash times\backslash mathbf(\backslash mathbf))$$

Q _{\ell} is the same as the electric multipole moment in the static case if it were applied to the static charge distribution \rho( x ) whereas Q _{\ell} ' corresponds to an induced electric multipole moment from the intrinsic magnetization of the source material.

Magnetic multipole fields

Applying the above solution to the magnetic multipole wave equation

$$(\backslash nabla^2+k^2)\backslash mathbf(\backslash mathbf)=-\backslash left[ikZ\_0\backslash mathbf\{J\}(\backslash mathbf\{x\})+ikZ\_0\backslash boldsymbol\{\backslash nabla\}\backslash times\backslash mathbf\{M\}(\backslash mathbf\{x\}))+\backslash frac\{iZ\_0\}\{k\}\backslash boldsymbol\{\backslash nabla\}(\backslash boldsymbol\{\backslash nabla\}\backslash cdot\backslash mathbf\{J\}(\backslash mathbf\{x\}))\backslash right]$$

gives the solution for the electric field:

$$\backslash mathbf^(\backslash mathbf)=\backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; a\_^\; h\_\backslash ell^(kr)\; \backslash mathbf\_(\backslash theta,\; \backslash phi)$$$$a\_^=\backslash frac\; \backslash int\; d^3\backslash mathbf\; j\_\backslash ell(kr\text{'})\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\; \backslash mathbf\backslash cdot\backslash left[ikZ\_0\backslash mathbf\{J\}(\backslash mathbf\{x\})+ikZ\_0\backslash boldsymbol\{\backslash nabla\}\backslash times\backslash mathbf\{M\}(\backslash mathbf\{x\}))+\backslash frac\{iZ\_0\}\{k\}\backslash boldsymbol\{\backslash nabla\}(\backslash boldsymbol\{\backslash nabla\}\backslash cdot\backslash mathbf\{J\}(\backslash mathbf\{x\}))\backslash right]$$

The magnetic field is:

$$\backslash mathbf^(\backslash mathbf)=-\backslash frac\backslash boldsymbol\backslash times\backslash mathbf^(\backslash mathbf)$$

As before, the forumula simplifies to:

$$a\_^=\backslash frac\; \backslash int\; d^3\backslash mathbf\; j\_\backslash ell(kr\text{'})\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\; \backslash left[\backslash boldsymbol\{\backslash nabla\}\backslash cdot(\backslash mathbf\{x\text{'}\}\backslash times\backslash mathbf\{J\}(\backslash mathbf\{x\text{'}\}))-k^2\backslash mathbf\{x\text{'}\}\backslash cdot\backslash mathbf\{M\}(\backslash mathbf\{x\text{'}\})\backslash right]\; +\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\backslash boldsymbol\backslash cdot\backslash mathbf(\backslash mathbf)\backslash frac(r\text{'}\; j\_\backslash ell(kr\text{'}))$$

Retaining only the lowest order terms results in the simplified form for the magnetic multipole coefficients:

$$a\_^=\backslash frac\backslash left(\backslash frac\backslash right)^[M\_\{\backslash ell\; m\}+M\_\{\backslash ell\; m\}\text{'}]$$$$M\_=\backslash frac\backslash int\; d^3\backslash mathbf\; r\text{'}^\backslash ell\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\; \backslash boldsymbol\backslash cdot(\backslash mathbf\backslash times\backslash mathbf(\backslash mathbf))$$$$M\_\text{'}=\backslash int\; d^3\backslash mathbf\; r\text{'}^\backslash ell\; Y\_^*(\backslash theta\text{'},\; \backslash phi\text{'})\backslash boldsymbol\backslash cdot\backslash mathbf(\backslash mathbf)$$

M _{\ell} is the magnetic multipole moment from the effective magnetization x x J ( x )/ 2 while M _{\ell} ' corresponds to the intrinsic magnetization M ( x ) .

General solution

The electric and magnetic multipole fields combine to give the total fields: $$\backslash mathbf(\backslash mathbf,t)=\backslash Re\backslash left(\backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; \backslash left[a\_\{\backslash ell\; m\}^\{(M)\}\; h\_\backslash ell^\{(1)\}(kr)\; \backslash mathbf\{X\}\_\{\backslash ell\; m\}(\backslash theta,\; \backslash phi)+\backslash frac\{iZ\_0\}\{k\}a\_\{\backslash ell\; m\}^\{(E)\}\backslash boldsymbol\{\backslash nabla\}\backslash times(h\_\backslash ell^\{(1)\}(kr)\backslash mathbf\{X\}\_\{\backslash ell\; m\}(\backslash theta,\; \backslash phi))\backslash right]e^\backslash right)$$$$\backslash mathbf(\backslash mathbf,t)=\backslash Re\backslash left(\backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; \backslash left[a\_\{\backslash ell\; m\}^\{(E)\}\; h\_\backslash ell^\{(1)\}(kr)\; \backslash mathbf\{X\}\_\{\backslash ell\; m\}(\backslash theta,\; \backslash phi)-\backslash frac\{i\}\{kZ\_0\}a\_\{\backslash ell\; m\}^\{(M)\}\backslash boldsymbol\{\backslash nabla\}\backslash times(h\_\backslash ell^\{(1)\}(kr)\backslash mathbf\{X\}\_\{\backslash ell\; m\}(\backslash theta,\; \backslash phi))\backslash right]e^\backslash right)$$

Note that the radial function

h ^{( 1 )} _{\ell} (kr) can be simplified in the far field limit 1 /r \ll 1 .

$$h\_\backslash ell^(kr)=(-i)^\backslash frac\; +\; \backslash mathcal\; O(1/r^2)$$

Thus the radial dependence of radiation is recovered.

See also

• Multipole expansion
• Spherical harmonics
• Vector spherical harmonics
• Near and far field
• Quadrupole formula

References

^{[2] [3] [4] [5] [6] [7]}

Notes and References

1. Web site: Robert G. Brown . Vector Calculus: Integration by Parts . 2007-12-28 . Classical Electrodynamics: Part II.
2. Book: Hartle, James B. . 2003 . Gravity: An Introduction to Einstein's General Relativity . . 0-8053-8662-9.
3. Book: Rose, M. E. . 1955 . Multipole Fields . John Wiley & Sons.
4. Book: Blatt . John M. . Weisskopf . Victor F. . 1963 . Theoretical Nuclear Physics - Seventh Printing . . 0-471-30932-X.
5. Book: Raab . Roger E. . de Lange . Owen L. . 2004 . Multipole Theory in Electromagnetism . . 978-0-19-856727-1.
6. Book: Jackson, John David . 1999 . Classical Electrodynamics - Third Edition . . 0-471-30932-X.
7. Book: Hafner, Christian . 1990 . The Generalized Multipole Technique for Computational Electromagnetics . . 0-89006-429-6.