Chapter.1.Appendix.Multipole.Expansion

# Chapter.1.Appendix.Multipole.Expansion - 146 Chapter 4...

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Unformatted text preview: 146 Chapter 4 Multipoles, Electrostatics of Macroscopic Media, Dielectrics—SI Consequently the coefﬁcients in (4.1) are: qgm : J Y’fmw', (ﬁv')r’[p(x') d3x' (4.3) These coefﬁcients are called multipole moments. To see the physical interpreta- tion of them we exhibit the ﬁrst few explicitly in terms of Cartesian coordinates: 1 r 3:_; Quuzmjp(x)dx ’Wq (4‘4) _ _ r _ v r 1 3 F f _ i _ - qn * lg” (x W )p(x ) d x — lg” (PX my) l3 J’ #3 i 4 r v r 2 7 6110 417 Z P“) x 47sz 1 15 _ , r l 1 15 _ (122 :3 157i (X' - ty )2p(x ) 11% = E [ET (Q11 * 21Q12 - Q22) 15 1 15 Q2I = —\/;TJ 2196' " iyr)P(X') dsx' = "g '87T(Q|3 _ iQn) (4‘6) _1 12_ :2 I 31713 (12‘0’2 l4,” (3z r~)p(x)dx —2 47Tst Only the moments with m 2 0 have been given, since (3.54) shows that for a real charge density the moments with m < 0 are related through (4.5) ghim : (—1)mqgi<m In equations (4.4)—(4.6), q is the total charge, or monopole moment, p is the electric dipole moment: p : Jx'p(x’) d3x’ (4.8) and QU— is the traceless quadrupole moment tensor: 9.. = l <3xzx3- e r'26;,-)p(x’) oer (4.9) We see that the 1th multipole coefﬁcients [(2] + 1) in number] are linear com- binations of the corresponding multipoles expressed in rectangular coordinates. The expansion of @(x) in rectangular coordinates (MK): 1 [g+p-3x+%2ij’ﬂ+m] (4,10) 471'60 r r r5 by direct Taylor series expansion of 1i ix * x’l will be left as an exercise for the reader. It becomes increasingly cumbersome to continue the expansion in (4.10) beyond the quadrupole terms. The electric ﬁeld components for a given multipole can be expressed most 150 Chapter 4 Multipoles, Electrostatics of Macroscopic Media, Dielectrics—SI The added delta function does not contribute to the ﬁeld away from the site of the dipole. Its purpose is to yield the required volume integral (4.18), with the convention that the spherically symmetric (around x0) volume integral of the ﬁrst term is zero (from angular integration), the singularity at x : x0 causing an otherwise ambiguous result. Equation (4.20) and its magnetic dipole counterpart (5.64), when handled carefully, can be employed as if the dipoles were idealized point dipoles, the delta function terms carrying the essential information about the actually ﬁnite distributions of charge and current. 4.2 Multipole Expansion of the Energy of 4:: Charge Distribution in an External Field If a localized charge distribution described by p(x) is placed in an external potential @(x), the electrostatic energy of the system is: W = f p(x)¢(x.) ex (421) If the potential (I) is slowly varying over the region where p(x) is nonnegligible, then it can be expanded in a Taylor series around a suitably chosen origin: 32(1) ax,- ax} ' 1 @(x) : @(9)_+ x . V<I>(0) + E E Z xli diff—J'tﬂ'i ﬂy'rfh I Utilizing the deﬁnition of the electric ﬁeld E : ~VIIJ, the last two terms can be rewritten. Then (4.22) becomes: (0) + - - - (4.22) i 6E- C1)(x) =(1)(0) —x-E(O) Exixj%(0)+ 2 z I 6):; Since V - E = 0 for the external ﬁeld, we can subtract érzv - E(0) from the last term to obtain ﬁnally the expansion: 3E. @(x) = (12(0) — x . 13(0) — é 2‘, (3x,x,. — ﬂag) (0) + (4.23) When this is inserted into (4.21) and the deﬁnitions of total charge, dipole mo— ment (4.8), and quadrupole moment (4.9) are employed, the energy takes the form: . 6E- W=qct>(0)—p-E(0)—%22Q,j5§(0)+m (4.24) 1 j J' This expansion shows the characteristic way in which the various multipoles in- teract with an external ﬁeld—the charge with the potential, the dipole with the electric ﬁeld, the quadrupole with the ﬁeld gradient, and so on. In nuclear physics the quadrupole interaction is of particular interest. Atomic nuclei can possess electric quadrupole moments, and their magnitudes and signs reﬂect the nature of the forces between neutrons and protons, as well as the ...
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## This note was uploaded on 12/19/2010 for the course PHYS 423 taught by Professor G. during the Spring '10 term at Missouri S&T.

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Chapter.1.Appendix.Multipole.Expansion - 146 Chapter 4...

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