In Figure 2’s oct-tree decomposition, ever-larger regions of space that represent in-creasing numbers of particles can interact through individual multipole expansions at in-creasing distances. Multipole expansion of the magnetic vector potential Consider an arbitrary loop that carries a current I. h���I@GN���QP0�����!�Ҁ�xH Note that … Let’s start by calculating the exact potential at the field point r= … Energy of multipole in external field: Formal Derivation of the Multipole Expansion of the Potential in Cartesian Coordinates Consider a charge density ρ(x) confined to a finite region of space (say within a sphere of radius R). The formulation of the treatment is given in Section 2. Tensors are useful in all physical situations that involve complicated dependence on directions. ?9��7���R�߅G.�����$����VL�Ia��zrV��>+�F�x�J��nw��I[=~R6���s:O�ӃQ���%må���5����b�x1Oy�e�����-�$���Uo�kz�;fn��%�$lY���vx$��S5���Ë�*�OATiC�D�&���ߠ3����k-Hi3
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W.}V^�8l�1>�� I���2K[a'����J�������[)'F2~���5s��Kb�AH�D��{I�`����D�''���^�A'��aJ-ͤ��Ž\���>��jk%�]]8�F�:���Ѩ��{���v{�m$��� xref
x��[[����I�q� �)N����A��x�����T����C���˹��*���F�K��6|������eH��Ç'��_���Ip�����8�\�ɨ�5)|�o�=~�e��^z7>� The goal is to represent the potential by a series expansion of the form: In addition to the well-known formulation of multipole expansion found in textbooks of electrodynamics,[38] some expressions have been developed for easier implementation in designing (2), with A l = 0. Multipole Expansion e171 Multipole Series and the Multipole Operators of a Particle With such a coordinate system, the Coulomb interaction of particles 1 and 2 (with charges q1 and q2) can be expanded using the following approximation2: q1q2 r12 mnk k=0 snl l=0 m=−s Akl|m|R −(k+ l+1)M ˆ(k,m) a (1) ∗M( ,m) b (2), (X.2) where the coefficient ������aJ@5�)R[�s��W�(����HdZ��oE�ϒ�d��JQ ^�Iu|�3ڐ]R��O�ܐdQ��u�����"�B*$%":Y��. 0000011471 00000 n
2 Multipole expansion of time dependent electromagnetic fields 2.1 The fields in terms of the potentials Consider a localized, oscillating source, located in otherwise empty space. 0000013959 00000 n
We have found that eliminating all centers with a charge less than .1 of an electron unit has little effect on the results. Themonople moment(the total charge Q) is indendent of our choice of origin. trailer
The relevant physics can best be made obvious by expanding a source distribution in a sum of specific contributions. The various results of individual mul-tipole contributions and their dependence on the multipole-order number and the size of spheroid are given in Section 3. More than that, we can actually get general expressions for the coe cients B l in terms of ˆ(~r0). 0000017487 00000 n
In the method, the entire wave propagation domain is divided into two regions according In the next section, we will con rm the existence of a potential (4), divergence-free property of the eld (5), and the Poisson equation (7). 0000011731 00000 n
Let’s start by calculating the exact potential at the field point r= … 4.3 Multipole populations. 0000005851 00000 n
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The fast multipole method (FMM) can reduce the computational cost to O(N) [1]. accuracy, especially for jxjlarge. The multipole expansion of the electric current density 6 4. 3.1 The Multipole Expansion. MULTIPOLE EXPANSION IN ELECTROSTATICS 3 As an example, consider a solid sphere with a charge density ˆ(r0)=k R r02 (R 2r0)sin 0 (13) We can use the integrals above to find the first non-zero term in the series, and thus get an approximation for the potential. This expansion was the rst instance of what came to be known as multipole expansions. The ⁄rst few terms are: l = 0 : 1 4…" 0 1 r Z ‰(~r0)d¿0 = Q 4…" 0r This is our RULE 1.