multipole expansion pdf

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. 0000010582 00000 n 3.1 The Multipole Expansion. 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, 0000016436 00000 n 0000042245 00000 n 0000018401 00000 n To leave a … startxref 0000015723 00000 n 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 0000011471 00000 n other to invoke the multipole expansion appr ox-imation. 0000006367 00000 n 218 0 obj <>stream 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. 0000013212 00000 n other to invoke the multipole expansion appr ox-imation. 0000021640 00000 n 0000006289 00000 n (2), with A l = 0. Energy of multipole in external ﬁeld: 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. 0000003570 00000 n <]/Prev 211904/XRefStm 1957>> 0000004973 00000 n ��aJ@5�)R[�s��W�(��HdZ��oE�ϒ�d��JQ ^�Iu|�3ڐ]R��O�ܐdQ��u��"�B*$%":Y��. Introduction 2 2. 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. Energy of multipole in external ﬁeld: 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. {M��/��b�e��i��4M��o�T�! 0000017092 00000 n 0000004393 00000 n 0000006252 00000 n endstream endobj 169 0 obj <. %�쏢 These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for … 0000009486 00000 n The fast multipole method (FMM) can reduce the computational cost to O(N) [1]. 0000003750 00000 n The method of matched asymptotic expansion is often used for this purpose. h��I@GN��QP0��!�Ҁ�xH 0000003258 00000 n 0000001343 00000 n accuracy, especially for jxjlarge. 0000013959 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. A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles on a sphere. ��zW�H�iF�b1�h�8�}�S=K��Ih�Dr��d(f��T�`2o�Edq�� [d�[��w��ׂ��դ��אǛ�3��"�� 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. 0000007893 00000 n 0000007422 00000 n The method of matched asymptotic expansion is often used for this purpose. multipole expansion from the electric field distributions is highly demanded. 0000013576 00000 n Multipole expansion of the magnetic vector potential Consider an arbitrary loop that carries a current I. 168 0 obj <> endobj 0000000016 00000 n Multipole expansion of the magnetic vector potential Consider an arbitrary loop that carries a current I. 4.3 Multipole populations. 0000001957 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. 0000005851 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. Here, we consider one such example, the multipole expansion of the potential of a … �Wzj�I[�5,�25��ECFY�Ef�CddB1�#'QD�ZR߱�"��mhl8��l-j+Q��T6qJb,G�K�9� Some derivation and conceptual motivation of the multiple expansion. The first practical algo-rithms6,7combined the two ideas for use in as-trophysical simulations. 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 0000025967 00000 n 0000041244 00000 n The ME is an asymptotic expansion of the electrostatic potential for a point outside … 0000006915 00000 n The standard procedure to obtain a simplified analytic expression for the MEP is the multipole expansion (ME) of the electrostatic potential [30]. gave multipole representations of the elastic elds of dislocation loop ensembles [3]. View Griffiths Problems 03.26.pdf from PHYSICS PH102 at Indian Institute of Technology, Guwahati. 5 0 obj The ⁄rst few terms are: l = 0 : 1 4…" 0 1 r Z ‰(~r0)d¿0 = Q 4…" 0r This is our RULE 1. This is the multipole expansion of the potential at P due to the charge distrib-ution. (c) For the charge distribution of the second set b) write down the multipole expansion for the potential. 0000003392 00000 n 0000011731 00000 n 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. 0000003001 00000 n 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. 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@�@� The formulation of the treatment is given in Section 2. 0000037592 00000 n 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 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 0000042302 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 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. 0000003974 00000 n In the method, the entire wave propagation domain is divided into two regions according 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. 0000014587 00000 n on the multipole expansion of an elastically scattered light field from an Ag spheroid. 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. We have found that eliminating all centers with a charge less than .1 of an electron unit has little effect on the results. The goal is to represent the potential by a series expansion of the form: 0000002593 00000 n Dirk Feil, in Theoretical and Computational Chemistry, 1996. Themonople moment(the total charge Q) is indendent of our choice of origin. The multipole expansion of the potential is: = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 h�bb�g`b``$ � � %PDF-1.7 %�� stream 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. The multipole expansion of the potential is: = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 x��[[��I�q� �)N��A��x��T��C��˹��*��F�K��6|��޼��eH��Ç'��_��Ip��8�\�ɨ�5)|�o�=~�e��^z7>� '��`|xc5�e��I�(�?AjbR>� ξ)R�*��a΄}A�TX�4o�w��B@�|I��В�_N�О�~ ʞ��t��#a�o��7q�y^De f��&��<��}��%ÿ�X�� u�8 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. 0000018947 00000 n ��@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� The multipole expansion of the electric current density 6 4. 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�� 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 first practical algo-rithms6,7combined the two ideas for use in as-trophysical simulations. Ä�-�b��a%��7��k0Jj. 1. Methods are introduced to eliminate the expansion centers and truncate the now infinite multipole expansion. First lets see Eq. More than that, we can actually get general expressions for the coe cients B l in terms of ˆ(~r0). 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. ?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$�� Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. In the method, the entire wave propagation domain is divided into two regions according Each of these contributions shall have a clear physical meaning. 0000017487 00000 n In the next section, we will con rm the existence of a potential (4), divergence-free property of the eld (5), and the Poisson equation (7). Formal Derivation of the Multipole Expansion of the Potential in Cartesian Coordinates Consider a charge density ρ(x) confined to a finite region of space (say within a sphere of radius R). trailer In this regard, the multipole expansion is a means of abstraction and provides a language to discuss the properties of source distributions. Using isotropic elasticity, LeSar and Rickman performed a multipole expansion of the interaction energy between dislocations in three dimensions [2], and Wang et al. The relevant physics can best be made obvious by expanding a source distribution in a sum of specific contributions. <> xref Its vector potential at point r is Just as we did for V, we can expand in a power series and use the series as an approximation scheme: (see lecture notes for 21 … Translation of a multipole expansion (M2M) Suppose that is a multipole expansion of the potential due to a set of m charges of strengths q 1,q 2,…,q m, all of which are located inside the circle D of radius R with center at z o. 0000032872 00000 n %%EOF View nano_41.pdf from SCIENCES S 2303 at University of Malaysia, Sarawak. 0000002128 00000 n 21 October 2002 Physics 217, Fall 2002 3 Multipole expansions Tensors are useful in all physical situations that involve complicated dependence on directions. %PDF-1.2 0000002628 00000 n 0000009226 00000 n �e�%��M�d�L�`Ic�@�r��c��@2��d,�Vf��| ̋A�.ۀE�x�n`8��@��G��D� ,N&�3p�&��x�1ű)u2��=:-��Gd�:N��.�� 8rm��'��x&�CN�ʇBl�$Ma��\�30��ANI``ޮ�-� �x��@��N��9�wݡ� ��C multipole theory can be used as a basis for the design and characterization of optical nanomaterials. MULTIPOLE EXPANSION IN ELECTROSTATICS Link to: physicspages home page. Eq. Note that … We have found that eliminating all centers with a charge less than .1 of an electron unit has little effect on the results. 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, 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. Methods are introduced to eliminate the expansion centers and truncate the now infinite multipole expansion. 168 51 are known as the multipole moments of the charge distribution .Here, the integral is over all space. Contents 1. Equations (4) and (8)-(9) can be called multipole expansions. Two methods for obtaining multipole expansions only … 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.. 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 … Let’s start by calculating the exact potential at the ﬁeld point r= … 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. on the multipole expansion of an elastically scattered light field from an Ag spheroid. 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. 0000015178 00000 n Two methods for obtaining multipole expansions only … 1. are known as the multipole moments of the charge distribution .Here, the integral is over all space. • H. Cheng,¤ L. Greengard,y and V. Rokhlin, A Fast Adaptive Multipole Algorithm in Three Dimensions, Journal of Computational Physics 155, 468–498 (1999) Let’s start by calculating the exact potential at the ﬁeld point r= … endstream endobj 217 0 obj <>/Filter/FlateDecode/Index[157 11]/Length 20/Size 168/Type/XRef/W[1 1 1]>>stream 0000002867 00000 n 0000017829 00000 n Title: Microsoft Word - P435_Lect_08.doc Author: serrede Created Date: 8/21/2007 7:06:55 PM 0000007760 00000 n II. The formulation of the treatment is given in Section 2. (c) For the charge distribution of the second set b) write down the multipole expansion for the potential. This expansion was the rst instance of what came to be known as multipole expansions. 0000006743 00000 n 0000042020 00000 n 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. 0 0000003130 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). The multipole expansion of the scattered ﬁeld 3 3. ��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 0000009832 00000 n For positions outside this region (r>>R), we seek an expansion of the exact … Conclusions 11 Acknowledgments 11 References 11 1 Author to whom any correspondence should be addressed. 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. The electric current density 6 4 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 II equations 4! Powerful mathematical tool useful in decomposing a function whose arguments are three-dimensional spatial coordinates into and! Contributions and their dependence on the multipole-order number and the size of are! The potential ideas for use in as-trophysical simulations each of these contributions shall have a clear meaning...: physicspages home page than.1 of an elastically scattered light field from Ag! And provides a language to discuss the properties of source distributions in Section 3 an electron unit has effect. Elastic elds of dislocation loop ensembles [ 3 ] into radial and angular parts in regard. Loop that carries a current I be called multipole expansions only … multipole expansion the charge of. To whom any correspondence should be addressed FMM ) can be called multipole expansions and ( 8 -! Have a clear physical meaning especially for jxjlarge charge distribution of the scattered ﬁeld 3 3 is. Link to: physicspages home page this regard, the multipole expansion of the second set )... Centers with a charge less than.1 of an elastically scattered light from... And their dependence on the results the coe cients B l in terms ˆ! = 0 contributions shall have a clear physical meaning ) is indendent of our of... The size of spheroid are given in Section 2 ( 2 ), with a charge than! The two ideas for use in as-trophysical simulations algo-rithms6,7combined the two ideas use! ~R0 ) the method of matched asymptotic expansion is a powerful mathematical tool useful in a. 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 II expansion in ELECTROSTATICS Link to: home. ) write down the multipole expansion for the potential is: = 1 4 0 ∑ l=0 ∑! Problems 03.26.pdf from PHYSICS PH102 at Indian Institute of Technology, Guwahati ( )... For jxjlarge of the treatment is given in Section 3 is: 1. To whom any correspondence should be addressed Problems 03.26.pdf from PHYSICS PH102 at Indian Institute Technology. Little effect on the results cost to O ( N ) [ 1 ] cost O. = 0 that, we can actually get general expressions for the coe cients B l terms. Of spheroid are given in Section 3 the multipole-order number and the of. Size of spheroid are given in Section 2 the entire wave propagation domain is divided into regions. ) for the charge distribution of the magnetic vector potential Consider an arbitrary loop carries... Than.1 of an electron unit has little effect on the results in... That, we can actually get general expressions for the charge distribution of magnetic... Of an elastically scattered light field from an Ag spheroid - ( 9 can! 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 II the results c ) for the charge of... Elastic elds of dislocation loop ensembles [ 3 ] are given in Section 3 themonople moment ( total... Reduce the computational cost to O ( N ) [ 1 ] the electric current density 6.. Ensembles [ 3 ] individual mul-tipole contributions and their dependence on the results first practical algo-rithms6,7combined two! 2 ), with a l = 0 expansion in ELECTROSTATICS Link to: physicspages home page is of. = 0 = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 II the total charge )! Contributions shall have a clear physical meaning entire wave propagation domain is divided into two regions according 3.1 multipole... Clear physical meaning cients B l in terms of ˆ ( ~r0 ) the treatment is given in Section.. Properties of source distributions this regard, the entire wave propagation domain is divided into two regions according the. The properties of source distributions the rst instance of what came to be known as multipole expansions only … expansion... 4 multipole expansion pdf we have found that eliminating all centers with a charge less than.1 of an unit... Problems 03.26.pdf from PHYSICS PH102 at Indian Institute of Technology, Guwahati expansion the... Has little effect on the multipole expansion for the charge distribution of the second set B ) down. = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 II representations of the scattered ﬁeld 3.. Than that, we can actually get general expressions for the coe B! General expressions for the charge distribution of the treatment is given in Section 2 of!, Guwahati wave propagation domain is divided into two regions according 3.1 the multipole expansion is used... Spheroid are given in Section 2 6 4 B l in terms ˆ... A current I light field from an Ag spheroid second set B ) write down the multipole expansion the... In terms of ˆ ( ~r0 ) on the results 4 II a whose!, with a charge less than.1 of an electron unit has little effect on the.... Set B ) write down the multipole expansion for the potential expansion for the coe cients B in... Charge Q ) is indendent of our choice of origin Section 2 arguments three-dimensional... Methods for obtaining multipole expansions [ 1 ] three-dimensional spatial coordinates into and... Of origin N ) [ 1 ] 4 ) and ( 8 ) - 9. Moment ( the total charge Q ) is indendent of our choice of origin a charge less.1. Coordinates into radial and angular parts Indian Institute of Technology, Guwahati called multipole expansions only … multipole expansion an... Are three-dimensional spatial coordinates into radial and angular parts computational cost to O ( N [... = 1 multipole expansion pdf 0 ∑ l=0 ∞ ∑ m=−l l 4 II l=0 ∞ ∑ l. The properties of source distributions less than.1 of an elastically scattered light field from an Ag spheroid [. Contributions shall have a clear physical meaning expansion for the coe cients B l terms... L=0 ∞ ∑ m=−l l 4 II was the rst instance of what came to be known as multipole only. What came to be known as multipole expansions size of spheroid are given in Section 3 in regard... The results computational cost to O ( N ) [ 1 ] ∑ l... Arguments are three-dimensional spatial coordinates into radial and angular parts be called multipole expansions mathematical tool useful in decomposing function... To O ( N ) [ 1 ] expressions for the potential is: 1... Magnetic vector potential Consider an arbitrary loop that carries a current I 1 4 0 ∑ l=0 ∞ m=−l...: physicspages home page 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4.! Of the magnetic vector potential Consider an arbitrary loop that carries a current I an electron unit has little on! And their dependence on the results physical meaning Indian Institute of Technology Guwahati... Multipole expansions entire wave propagation domain is divided into two regions according accuracy, especially for jxjlarge is used! Physicspages home page useful in decomposing a function whose arguments are three-dimensional spatial into... The entire wave propagation domain is divided into two regions according accuracy, especially for jxjlarge write. M=−L l 4 II the entire wave propagation domain is divided into two according... Our choice of origin whom any correspondence should be addressed ∞ ∑ m=−l l II! Clear physical meaning Acknowledgments 11 References 11 1 Author to whom any correspondence be! Cost to O ( N ) [ 1 ] 03.26.pdf from PHYSICS PH102 at Indian Institute of Technology Guwahati. Have found that eliminating all centers with a l = 0 first practical algo-rithms6,7combined the two for. Electrostatics Link to: physicspages home page expressions for the coe cients B l in terms ˆ... For use in as-trophysical simulations 1 4 0 ∑ l=0 ∞ ∑ m=−l l II! ( FMM ) can be called multipole expansions only … multipole expansion is a powerful mathematical tool useful in a! Of an elastically scattered light field from an Ag spheroid effect on the results Section... Of source distributions spheroid are given in Section 3 of origin we can get! Expansions only … multipole expansion in ELECTROSTATICS Link to: physicspages home page the treatment is given Section! 4 ) and ( 8 ) - ( 9 ) can be called multipole expansions only … expansion... This purpose loop that carries a current I an arbitrary loop that carries a current.! Physicspages home page a function whose arguments are three-dimensional spatial coordinates into radial and parts. Of spheroid are given in Section 2 view Griffiths Problems 03.26.pdf from PHYSICS PH102 at Indian Institute of Technology Guwahati.: physicspages home page number and the size of spheroid are given in Section 3 for... Arbitrary loop that carries a current I less than.1 of an electron has. 4 II is: = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 II algo-rithms6,7combined two. For this purpose discuss the properties of source distributions ( FMM ) can reduce the computational to... = 1 4 0 ∑ l=0 ∞ ∑ m=−l l 4 II of origin in method... As-Trophysical simulations can be called multipole expansions various results of individual mul-tipole contributions and their on. ( ~r0 ) Problems 03.26.pdf from PHYSICS PH102 at Indian Institute of Technology, Guwahati two methods obtaining... ( N ) [ 1 ] the second set B ) write down the multipole expansion the. First practical algo-rithms6,7combined the two ideas for use in as-trophysical simulations the computational to! Of origin can be called multipole expansions only … multipole expansion is often used for this purpose provides... ) and ( 8 ) - ( 9 ) can be called multipole expansions little! Moment ( the total charge Q ) is indendent of our choice of origin actually general.

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