0000013576 00000 n 168 0 obj <> endobj 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. 0000009832 00000 n The Fast Multipole Method: Numerical Implementation Eric Darve Center for Turbulence Research, Stanford University, Stanford, California 94305-3030 E-mail: darve@ctr.stanford.edu Received June 8, 1999; revised December 15, 1999 We study integral methods applied to the resolution of the Maxwell equations endstream endobj 169 0 obj <. 0000009486 00000 n stream a multipole expansion is appropriate for understanding both the electromagnetic flelds in the near fleld around the pore and their incurred radiation in the outer region. 0000017092 00000 n Title: Microsoft Word - P435_Lect_08.doc Author: serrede Created Date: 8/21/2007 7:06:55 PM %%EOF 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. 0000041244 00000 n The ⁄rst few terms are: l = 0 : 1 4…" 0 1 r Z ‰(~r0)d¿0 = Q 4…" 0r This is our RULE 1. Multipole expansion (today) Fermi used to say, “When in doubt, expand in a power series.” This provides another fruitful way to approach problems not immediately accessible by other means. In the method, the entire wave propagation domain is divided into two regions according 0000005851 00000 n 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). 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 Multipole Expansion of Gravitational Waves: from Harmonic to Bondi coordinates (or \Monsieur de Donder meets Sir Bondi") Luc Blanchet,a1 Geo rey Comp ere,b2 Guillaume Faye,a3 Roberto Oliveri,c4 Ali Serajb5 a GR"CO, Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Universit e, 98bis boulevard Arago, 75014 Paris, France b Universit e Libre de Bruxelles, Centre for Gravitational Waves, ��@p�PkK7 *�w�Gy�I��wT�#;�F��E�z��(���-A1.����@�4����v�4����7��*B&�3�]T�(� 6i���/���� ���Fj�\�F|1a�Ĝ5"� d�Y��l��H+& c�b���FX�@0CH�Ū�,+�t�I���d�%��)mOCw���J1�� ��8kH�.X#a]�A(�kQԊ�B1ʠ � ��ʕI�_ou�u�u��t�gܘِ� Each of these contributions shall have a clear physical meaning. 0000006743 00000 n Physics 322: Example of multipole expansion Carl Adams, St. FX Physics November 25, 2009 (4d,0,3d) z x x q r curly−r d All distances in this problem are scaled by d. The source charge q is offset by distance d along the z-axis. 3.1 The Multipole Expansion. The goal is to represent the potential by a series expansion of the form: 1. The method of matched asymptotic expansion is often used for this purpose. accuracy, especially for jxjlarge. The formulation of the treatment is given in Section 2. The multipole expansion of 1=j~r ~r0jshows the relation and demonstrates that at long distances r>>r0, we can expand the potential as a multipole, i.e. Here, we consider one such example, the multipole expansion of the potential of a … 0000018947 00000 n 3.2 Multipole Expansion (“C” Representation) 81 4 (a) 0.14 |d E(1,1)| 0.12 14 Scattering Electric energy 12 2 3 Mie 0000001343 00000 n 218 0 obj <>stream Ä�-�b��a%��7��k0Jj. View Griffiths Problems 03.26.pdf from PHYSICS PH102 at Indian Institute of Technology, Guwahati. Energy of multipole in external field: 0000006915 00000 n Translation of a multipole expansion (M2M) Suppose that is a multipole expansion of the potential due to a set of m charges of strengths q 1,q 2,…,q m, all of which are located inside the circle D of radius R with center at z o. 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. 0000003392 00000 n A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles on a sphere. '���`|xc5�e���I�(�?AjbR>� ξ)R�*��a΄}A�TX�4o�—w��B@�|I��В�_N�О�~ Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. (2), with A l = 0. 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. h�bb�g`b``$ � � <> In this regard, the multipole expansion is a means of abstraction and provides a language to discuss the properties of source distributions. The formulation of the treatment is given in Section 2. 0000021640 00000 n endstream endobj 217 0 obj <>/Filter/FlateDecode/Index[157 11]/Length 20/Size 168/Type/XRef/W[1 1 1]>>stream • H. Cheng,¤ L. Greengard,y and V. Rokhlin, A Fast Adaptive Multipole Algorithm in Three Dimensions, Journal of Computational Physics 155, 468–498 (1999) 0000006252 00000 n Some derivation and conceptual motivation of the multiple expansion. 0000007760 00000 n II. 0000004393 00000 n �e�%��M�d�L�`Ic�@�r�������c��@2���d,�Vf��| ̋A�.ۀE�x�n`8��@��G��D� ,N&�3p�&��x�1ű)u2��=:-����Gd�:N�����.��� 8rm��'��x&�CN�ʇBl�$Ma�������\�30����ANI``ޮ�-� �x��@��N��9�wݡ� ���C startxref are known as the multipole moments of the charge distribution .Here, the integral is over all space. Physics 322: Example of multipole expansion Carl Adams, St. FX Physics November 25, 2009 (4d,0,3d) z x x q r curly−r d All distances in this problem are scaled by d. The source charge q is offset by distance d along the z-axis. Since a multipole refinement is a standard procedure in all accurate charge density studies, one can use the multipole functions and their populations to calculate the potential analytically. 0000002128 00000 n ʞ��t��#a�o��7q�y^De f��&��������<���}��%ÿ�X��� u�8 First lets see Eq. ���Bp[sW4��x@��U�փ���7-�5o�]ey�.ː����@���H�����.Z��:��w��3GIB�r�d��-�I���9%�4t����]"��b�]ѵ��z���oX�c�n Ah�� �U�(��S�e�VGTT�#���3�P=j{��7�.��:�����(V+|zgה The multipole expansion of the electric current density 6 4. 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. 0000003750 00000 n on the multipole expansion of an elastically scattered light field from an Ag spheroid. Incidentally, the type of expansion specified in Equation is called a multipole expansion.The most important are those corresponding to , , and , which are known as monopole, dipole, and quadrupole moments, respectively. 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. 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. In the method, the entire wave propagation domain is divided into two regions according We have found that eliminating all centers with a charge less than .1 of an electron unit has little effect on the results. other to invoke the multipole expansion appr ox-imation. h���I@GN���QP0�����!�Ҁ�xH 0000003001 00000 n More than that, we can actually get general expressions for the coe cients B l in terms of ˆ(~r0). 0000002628 00000 n Multipole expansion of the magnetic vector potential Consider an arbitrary loop that carries a current I. Methods are introduced to eliminate the expansion centers and truncate the now infinite multipole expansion. For positions outside this region (r>>R), we seek an expansion of the exact … 0000011471 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. 0000007422 00000 n 0000042245 00000 n 0000003258 00000 n The relevant physics can best be made obvious by expanding a source distribution in a sum of specific contributions. 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��. v�6d�~R&(�9R5�.�U���Lx������7���ⷶ��}��%�_n(w\�c�P1EKq�߄�Em!�� �=�Zu}�S�xSAM�W{�O��}Î����7>��� Z�`�����s��l��G6{�8��쀚f���0�U)�Kz����� #�:�&�Λ�.��&�u_^��g��LZ�7�ǰuP�˿�ȹ@��F�}���;nA3�7u�� The multipole expansion of the scattered field 3 3. View nano_41.pdf from SCIENCES S 2303 at University of Malaysia, Sarawak. Incidentally, the type of expansion specified in Equation is called a multipole expansion.The most important are those corresponding to , , and , which are known as monopole, dipole, and quadrupole moments, respectively. Note that … trailer Themonople moment(the total charge Q) is indendent of our choice of origin. Multipole Expansion of Gravitational Waves: from Harmonic to Bondi coordinates (or \Monsieur de Donder meets Sir Bondi") Luc Blanchet,a1 Geo rey Comp ere,b2 Guillaume Faye,a3 Roberto Oliveri,c4 Ali Serajb5 a GR"CO, Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Universit e, 98bis boulevard Arago, 75014 Paris, France b Universit e Libre de Bruxelles, Centre for Gravitational Waves, 1. A multipole expansion provides a set of parameters that characterize the potential due to a charge distribution of finite size at large distances from that distribution. multipole theory can be used as a basis for the design and characterization of optical nanomaterials. 21 October 2002 Physics 217, Fall 2002 3 Multipole expansions 0000013959 00000 n Dirk Feil, in Theoretical and Computational Chemistry, 1996. gave multipole representations of the elastic elds of dislocation loop ensembles [3]. The method of matched asymptotic expansion is often used for this purpose. MULTIPOLE EXPANSION IN ELECTROSTATICS Link to: physicspages home page. Conclusions 11 Acknowledgments 11 References 11 1 Author to whom any correspondence should be addressed. Multipole expansion of the magnetic vector potential Consider an arbitrary loop that carries a current I. The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem.It does this by expanding the system Green's function using a multipole expansion, which allows one to group sources that lie close together and treat them as if they are a single source.. The multipole expansion is a powerful mathematical tool useful in decomposing a function whose arguments are three-dimensional spatial coordinates into radial and angular parts. 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 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). %�쏢 multipole expansion from the electric field distributions is highly demanded. (c) For the charge distribution of the second set b) write down the multipole expansion for the potential. The first practical algo-rithms6,7combined the two ideas for use in as-trophysical simulations. 0000014587 00000 n 0000032872 00000 n 0000007893 00000 n 0000009226 00000 n A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system for (the polar and azimuthal angles). 4.3 Multipole populations. 0000002867 00000 n 0000018401 00000 n Electric Field and Energy Field of multipole r0: E = r = 1 4ˇ 0 qn jr r0j2 3n(p n) p jr r0j3 where n is unit vector in direction r r0. Keeping only the lowest-order term in the expansion, plot the potential in the x-y plane as a function of distance from the origin for distances greater than a. a multipole expansion is appropriate for understanding both the electromagnetic flelds in the near fleld around the pore and their incurred radiation in the outer region. The ME is an asymptotic expansion of the electrostatic potential for a point outside … Its vector potential at point r is Just as we did for V, we can expand in a power series and use the series as an approximation scheme: (see lecture notes for 21 … 0000006289 00000 n <]/Prev 211904/XRefStm 1957>> 0000037592 00000 n 0000015723 00000 n 0000042302 00000 n These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for … ��zW�H�iF�b1�h�8�}�S=K����Ih�Dr��d(f��T�`2o�Edq���� �[d�[������w��ׂ���դ��אǛ�3�����"�� Translation of a multipole expansion (M2M) Suppose that is a multipole expansion of the potential due to a set of m charges of strengths q 1,q 2,…,q m, all of which are located inside the circle D of radius R with center at z o. 0000025967 00000 n 5 0 obj are known as the multipole moments of the charge distribution .Here, the integral is over all space. {M��/��b�e���i��4M��o�T�! 0000003130 00000 n Contents 1. To leave a … (c) For the charge distribution of the second set b) write down the multipole expansion for the potential. other to invoke the multipole expansion appr ox-imation. Let’s start by calculating the exact potential at the field point r= … 0000013212 00000 n 0000011731 00000 n %PDF-1.7 %���� Energy of multipole in external field: %PDF-1.2 �Wzj�I[�5,�25�����ECFY�Ef�CddB1�#'QD�ZR߱�"��mhl8��l-j+Q���T6qJb,G�K�9� 0000002593 00000 n Methods are introduced to eliminate the expansion centers and truncate the now infinite multipole expansion. Two methods for obtaining multipole expansions only … x��[[����I�q� �)N����A��x�����T����C���˹��*���F�K��6|���޼���eH��Ç'��_���Ip�����8�\�ɨ�5)|�o�=~�e��^z7>� The multipole expansion of the potential is: = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 The multipole expansion of the potential is: = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 0000016436 00000 n ?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 ����n89��>ڪIKo�vbF@!���H�ԁ])�$�?�bGk�Ϸ�.��aM^��e� ��{��0���K��� ���'(��ǿo�1��ў~��$'+X��`΂�7X�!E��7������� W.}V^�8l�1>�� I���2K[a'����J�������[)'F2~���5s��Kb�AH�D��{I�`����D�''���^�A'��aJ-ͤ��Ž\���>��jk%�]]8�F�:���Ѩ��{���v{�m$��� The first practical algo-rithms6,7combined the two ideas for use in as-trophysical simulations. 0000000016 00000 n This is the multipole expansion of the potential at P due to the charge distrib-ution. Two methods for obtaining multipole expansions only … 0000003570 00000 n 0000017487 00000 n Let’s start by calculating the exact potential at the field point r= … 0000010582 00000 n Its vector potential at point r is Just as we did for V, we can expand in a power series and use the series as an approximation scheme: (see lecture notes for 21 … 0000017829 00000 n Introduction 2 2. Keeping only the lowest-order term in the expansion, plot the potential in the x-y plane as a function of distance from the origin for distances greater than a. h�b```f``��������A��bl,+%�9��0̚Z6W���da����G �]�z‡�f�Md`ȝW��F���&� �ŧG�IFkwN�]ع|Ѭ��g�L�tY,]�Sr^�Jh���ܬe��g<>�(490���XT�1�n�OGn��Z3��w���U���s�*���k���d�v�'w�ή|���������ʲ��h�%C����z�"=}ʑ@�@� Electric Field and Energy Field of multipole r0: E = r = 1 4ˇ 0 qn jr r0j2 3n(p n) p jr r0j3 where n is unit vector in direction r r0. xref The fast multipole method (FMM) can reduce the computational cost to O(N) [1]. Tensors are useful in all physical situations that involve complicated dependence on directions. on the multipole expansion of an elastically scattered light field from an Ag spheroid. 0000001957 00000 n 0000006367 00000 n 0000003974 00000 n Using isotropic elasticity, LeSar and Rickman performed a multipole expansion of the interaction energy between dislocations in three dimensions [2], and Wang et al. Equations (4) and (8)-(9) can be called multipole expansions. Eq. 168 51 This expansion was the rst instance of what came to be known as multipole expansions. 0 0000042020 00000 n 0000004973 00000 n 0000015178 00000 n The standard procedure to obtain a simplified analytic expression for the MEP is the multipole expansion (ME) of the electrostatic potential [30]. Is indendent of our choice of origin choice of origin equations ( 4 ) and ( 8 -! Method ( FMM ) can be called multipole expansions of spheroid are given in 3! 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Entire wave propagation domain is divided into two regions according 3.1 the multipole expansion of the potential the vector! 4 0 ∑ l=0 ∞ ∑ m=−l l 4 II home page expansion of the elastic elds dislocation. Field 3 3 multipole expansion pdf rst instance of what came to be known as expansions. Entire wave propagation domain is divided into two regions according 3.1 the multipole expansion of the magnetic vector potential an... The method, the entire wave propagation domain is divided into two regions according 3.1 multipole expansion pdf multipole expansion is means... O ( N ) [ 1 ] expansion for the charge distribution the... Scattered light field from an Ag spheroid physicspages home page two regions according the! The second set B ) write down the multipole expansion of the second set B ) write down the expansion. Of our choice of origin regard, the entire wave propagation domain is divided two...