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Electric Multipoles Appendix A Electric Multipoles In Sects.7.1.4 and 8.3, the general multipole expansion was discussed. In Exercise (7.1), a multipole expansion of the electrostatic potential is performed. In this and the next appendix, the expansion will be obtained from a general interaction picture in the electric and the magnetic case respectively. The starting point is the basic energy formulas (8.89) and (8.88). The electric energy is given by (8.89), Fig.A.1, 1 ρ(r¯)ρ (r¯ ) Ue = dV dV (A.1) 4πε0 R V V where R¯ =¯r −¯r . If the systems interact at a large distance, the energy may be expressed through the multipole expansion. The energy is then series-expanded (also known as Taylor or Maclaurin expansion) with respect to the distance between the systems. In the first step, let the system V interact with a pointlike system V with the position vector r¯, where r r , and expand the energy arisen about the centre of V . This means that the system V is expressed in multipole terms. The second step involves a series expansion about the centre of the system V , for each multipole order of V , thereby expressing the system V in terms of multipoles. The calculation will be made up to the order 1/d5 corresponding to a quadrupole– quadrupole interaction, where d is the distance between the centres of the systems. The following definitions will be used: For a system with volume τ, the total charge is q = ρ(x, y, z)dτ (A.2) τ the dipole moment pi = ri ρ(x, y, z)dτ (A.3) τ © Springer International Publishing Switzerland 2015 239 K. Prytz, Electrodynamics: The Field-Free Approach, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-13171-9 240 Appendix A: Electric Multipoles V d x ρdV r V’ R r’ ρdV ρ’dV’ ρ’dV’ Char ge elements R Distance between elements r, r’ P osition of elements d Distance between system's centers x Position of unprimed element relative the center of system V Fig. A.1 Two systems interact electrically the quadrupole moment Qij = ri r j ρ(x, y, z)dτ (A.4) τ and so on for higher order moments, where r1 = x, r2 = y, r3 = z. Note that for- mula (A.4) differs from the definition of quadrupole moment in Exercise (7.1). The latter is known as reduced quadrupole moment and is the more common form. Here, formula (A.4) is chosen as this definition is simpler and results in more transparent formulas. A.1 Multipole Expansion of System V In analogy with Exercise (7.1), the series expansion up to quadrupole order of the primed system gives ⎡ 1 ρ(r¯)ρ(r¯) ρρr¯ ·¯r U ≈ ⎣ dVdV + dVdV e 3 4πε0 r r V V V V ⎤ 2 2 2 1 3(r¯ ·¯r ) r r + ρρ − dV dV⎦ (A.5) 2 r 5 r 5 V V Appendix A: Electric Multipoles 241 Fig. A.2 Interaction point Interaction point between two charge elements r R V’ r’ ρ’dV’ These terms correspond to monopole, dipole and quadrupole structures of the primed system (Fig. A.2). A.2 Multipole Expansion of System V Now let the multipoles of the primed system interact individually with the different multipoles of the system V . A.2.1 V Monopole The first term in (A.5) corresponds to a monopole of system V : ρ(¯)ρ(¯) m− = 1 r r Ue dV dV (A.6) 4πε0 r V V A series expansion is performed about the unprimed system’s centre in order to include its structure, Fig. A.1. Let the vector x¯ be the distance between the centre of the system V and an element of charge, i.e. r¯ =¯x + d¯, where d x. Then 2 + ¯ ·¯ 2 + ¯ ·¯ 2 1 = 1 ≈ 1 − 1 x 2d x + 3 x 2d x / 1 (A.7) r 2+ ¯·¯ 1 2 d 2 d2 8 d2 d + x 2d x 1 d2 Terms which are linear and quadratic in x correspond to dipole and quadrupole of the system V . The first term gives the monopole structure, i.e. the point charge. Thus, Monopole–monopole mm = 1 1 ρ(¯)ρ(¯) = qq Ue r r dV dV (A.8) 4πε0 d 4πε0d V V 242 Appendix A: Electric Multipoles Monopole-dipole 1 1 1 2d¯ ·¯x qdˆ ·¯p U md = ρ(r¯)ρ(r¯) − dVdV =− e 2 2 (A.9) 4πε0 d 2 d 4πε0d V V which is formula (7.7). Monopole-quadrupole 2 1 1 1 x2 3 2d¯ ·¯x U mq = ρ(r¯)ρ(r¯) − + dVdV e 2 2 4πε0 d 2 d 8 d V V ⎛ ⎞ q ⎝ 3 ⎠ =− Qii − di d j Qij (A.10) 8πε d3 d2 0 i i, j Compare with Exercise (7.1). A.2.2 V Dipole Now consider the next term in the V -structure, formula (A.5), corresponding to a dipole. Similarly to the procedure above, a series expansion is performed about the centre of the unprimed system. The integrand of the second term in formula (A.5) becomes ρρ ¯ ·¯ ( ¯ + ¯) ·¯ ¯ ·¯ ¯ ·¯ r r = ρρ x d r = ρρ x r + ρρ d r r 3 r 3 r 3 r 3 2 ¯ 2 ¯ 2 x¯ ·¯r 3 x + 2d ·¯x 15 x + 2d ·¯x ≈ ρρ 1 − + d3 2 d2 8 d2 ¯ 2 ¯ 2 ¯ 2 d ·¯r 3 x + 2d ·¯x 15 x + 2d ·¯x + ρρ 1 − + (A.11) d3 2 d2 8 d2 The dipole-monopole term is equivalent to the monopole-dipole term, which was calculated above. The next order is therefore dipole–dipole. Dipole–dipole This order corresponds to terms which are linear in r and x: 1 ρρr¯ ·¯r U dd = dVdV e 3 4πε0 r 1 x¯ ·¯r d¯ ·¯r 3 2d¯ ·¯x ≈ ρρ + − dVdV 3 3 2 4πε0 d d 2 d Appendix A: Electric Multipoles 243 1 = ((p¯ ·¯p) − (dˆ ·¯p)(dˆ ·¯p)) 3 3 (A.12) 4πε0d which is formula (7.16). Dipole-quadrupole In this order, terms which are linear in r and quadratic in x contribute: ¯ ¯ 2 ¯ 2 1 x¯ ·¯r 3 2d ·¯x d ·¯r 3 x 15 2d ·¯x U dq ≈ ρρ − + − + dV dV e 3 2 3 2 2 4πε0 d 2 d d 2 d 8 d ⎡ ⎤ 3 ⎣ 1 ¯ 5 ¯ ⎦ = − Qij p d j − d ·¯p Qii + d ·¯p di d j Qij (A.13) 4πε d5 i 2 2d2 0 i, j i i, j A.2.3 V Quadrupole Quadrupole-monopole and quadrupole-dipole were calculated above but with reversed systems. The energy is of course independent of which system is primed or unprimed. Therefore, there is only one remaining interaction at this level: Quadrupole–quadrupole The quadrupole–quadrupole interaction corresponds to the quadratic terms with respect to both r and x. The quadrupole term in (A.5)is − 1 1 3(r¯ ·¯r )2 r 2r 2 U q = ρρ − dVdV e 5 5 (A.14) 4πε0 2 r r ρρ(3(r¯ ·¯r )2 − r 2r 2) r 5 2 ¯ 2 ¯ 2 1 5 x + 2d ·¯x 35 x + 2d ·¯x ≈ ρρ (3(x¯ + d¯) ·¯r )2 − r 2(x¯ + d¯)2) 1 − + d5 2 d2 8 d2 1 = ρρ[ ((x¯ ·¯r )2 + (x¯ ·¯r )(d¯ ·¯r ) + (d¯ ·¯r )2) − r 2x2 − r 2 (x¯ · d¯) − r 2d2] 3 2 2 5 d 2 5 x + ρρ [3((x¯ ·¯r )2 + 2(x¯ ·¯r )(d¯ ·¯r ) + (d¯ ·¯r )2) − r 2x2 − r 22(x¯ · d¯) − r 2d2] − 2 d7 ¯ d ·¯x + ρρ [3((x¯ ·¯r )2 + 2(x¯ ·¯r )(d¯ ·¯r ) + (d¯ ·¯r )2) − r 2x2 − r 22(x¯ · d¯) − r 2d2] −5 d7 ¯ 2 35 (d ·¯x) + ρρ [3((x¯ ·¯r )2 + 2(x¯ ·¯r )(d¯ ·¯r ) + (d¯ ·¯r )2) − r 2x2 − r 22(x¯ · d¯) − r 2d2] 2 d9 (A.15) where terms which are up to quadratic in r and x have been kept. 244 Appendix A: Electric Multipoles The quadratic terms result in 2 1 5 x ρρ [3(x¯ ·¯r )2 − r 2x2] + ρρ [3(d¯ ·¯r )2 − r 2d2] − d5 2 d7 ¯ d ·¯x + ρρ [3(2(x¯ ·¯r )(d¯ ·¯r )) − r 22(x¯ · d¯)] −5 d7 ¯ 2 35 (d ·¯x) + ρρ [3(d¯ ·¯r )2 − r 2d2] 2 d9 3(x¯ ·¯r )2 + 3 r 2x2 15 x2 30(x¯ ·¯r )(d¯ ·¯r )(d¯ ·¯x) = ρρ 2 − (d¯ ·¯r )2 − d5 2 d7 d7 15r 2(d¯ ·¯x)2 ( ¯ ·¯)2( ¯ ·¯)2 − 2 + 105 d r d x d7 2 d9 3 ˜ ˜ 1 ˜ ˜ 15 ˜ ˜ = QijQ + Qii Q − di d j Q Qii d5 ij 2 jj 2d7 ij i, j i, j i 30 ˜ ˜ 15 ˜ ˜ − QikQ d j dk − Q di d j Qij d7 ij 2d7 ii i, j,k i i, j 105 ˜ ˜ + di d j Q di d j Qij (A.16) 2d9 ij i, j i, j where Q˜ denotes the density of quadrupole moment. The quadrupole–quadrupole energy is ⎡ qq 1 3 ⎣ 1 5 U = (2QijQ + Qii Q ) − di d j Q Qii e 4πε 4 d5 ij jj d7 ij 0 i, j i, j i 20 5 − Q Q d d − Q d d Q 7 ik ij j k 7 ii i j ij d , , d , i j k ⎤i i j 35 + d d Q d d Q ⎦ (A.17) d9 i j ij i j ij i, j i, j The total energy in quadrupole order becomes = mm + md + mq + dm + dd + dq + qm + qd + qq Ue Ue Ue Ue Ue Ue Ue Ue Ue Ue = mm + dd + qq + md + mq + dq Ue Ue Ue 2Ue 2Ue 2Ue Appendix A: Electric Multipoles 245 Exercises A.1 Use formula (A.10) to determine the electric potential from a pure quadrupole.
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  • 1 Chapter 3 Dipole and Quadrupole Moments 3.1
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