Dipole Source Modeling 41 Multipole Expansion of

Equations (3.15) and (3.18) are the general solutions for $ and B given an arbitrary current source density Js in the absence of boundaries. The integrand of each function involves derivatives of Js(r'), and the integration kernel 1/|r — r'| called the Green's function (Jackson 1975). The basis of the multipole expansion is to assume that Js(r') is confined to some finite region of space, and that the point r at which the field is being computed or measured is far away compared to the size of the source distribution, i.e., |r| ^ |r'|. Computing the Taylor series of 1/|r — r'| through the first two terms gives

1-77 = 77 + TIT + 7/ 7 \ 5 rirj + ..., |r'| < |r| (4.1)

The first term is called the monopole term, and falls off as 1/|r|. The second term is called the dipole term, and falls off as 1/1 r |2. The third term is called the quadrupole term, and falls off as 1/|r|3, and so on. Inserting (4.1) into (3.15) gives

and so on.

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