MULTIPOLE FIELDS. Multipoles, 2 l poles. Monopoles, dipoles, quadrupoles, octupoles... Electric Dipole R 1 R 2. P(r,θ,φ) e r
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1 MULTIPOLE FIELDS Mutpoes poes. Monopoes dpoes quadupoes octupoes Eectc Dpoe +q O θ e R R P(θφ) -q e The potenta at the fed pont P(θφ) s ( θϕ )= q R R
2 Bo E. Seneus : Now R = ( e) = + cosθ R = ( + e) = + + cosθ and = ( + ) = + cosθ R cosθ a a aa ( ) aa ( ) ( a ) = ( + ) = = + + cosθ ( cos θ ) + ( 5cos θ cosθ)+ = + cosθ cos θ R 5 ( cos θ cosθ )+ The potenta becomes appomatey cosθ ( θϕ ) q We have assumed that <<. The eectc dpoe moment s defned as p = qe The owest ode (n /) contbuton to the potenta fom the pa of chages s the dpoe contbuton. An dea dpoe a pont dpoe s obtaned f we et the sepaaton of the chages go to zeo and at the same tme et the chages go
3 Bo E. Seneus : to nfnty so that q s kept constant (q=p). If we do that ony the owest ode contbuton to the potenta suvves. To be noted s that f we keep the sepaaton of the chages fnte even f the chages ae pont chages thee ae hghe ode mutpoe contbutons. The dpoe potenta can be wtten as = p e = p The eectc fed s E = and n spheca coodnates we have E = p = cosθ snθ Eθ = = p θ Eϕ = = 0 snθ ϕ
4 Bo E. Seneus :4 MULTIPOLE EXPANSION R =- P( ) q ( ) O e Unpmed -vectos epesent fed ponts. Pmed epesent souce ponts. Uppe case R vectos epesent vectos fom souce ponts to fed ponts. Geek subscpts efe to ndvdua chages o patces. Roman subscpts efe to coodnate aes. The potenta fom chage q s = q R We want to epand /R aound the souce ogn keepng the fed pont fed.
5 Bo E. Seneus :5 The genea Tayo epanson s f f f f!! = = = Ths gves fo the potenta fom one of the pont chages ()= = = q q R q R R R Now snce R = we have R R R R = = = =
6 Bo E. Seneus :6 Ths means that q q + q + ()= The tota potenta fom a chages may then be wtten as () 4 ()= ()= ()+ ()+ ()+ ()+ whee () q ()= q = ()= q 4 ()= q ()= ( ) q! whee the fst tem s the monopoe potenta the second the dpoe potenta the thd the quadupoe potenta and so on. The ast genea tems s caed the th mutpoe potenta. To be noted s that each tem dops off faste wth dstance wth one powe n than the pevous tem. () + Ths means that fo age dstances the owest ode non-vanshng mutpoe potenta domnates.
7 Bo E. Seneus :7 THE DIPOLE POTENTIAL ( ) ()= q = q gad = p gad = p Thus ( ) p e ()= whee the dpoe moment of the system s defned as: p= q In contnuous case: p= d ρ ( )
8 Bo E. Seneus :8 The eectc dpoe fed vecto s E = gad gad p = = gad( p ) ( p ) gad = p+ ( p ) 5 = ( p ) p 5 Ths may atenatvey be wtten on the fom E= φp ; E = φp whee φ s the so-caed dpoe-dpoe tenso wth eements: φ δ = 5
9 Bo E. Seneus :9 THE QUADRUPOLE POTENTIAL 4 ()= q Ths epesson may be used as t stands but we w ntoduce the quadupoe tenso. Fst we note that snce / s a souton to Lapaces equaton we have 0 0 = > Ths may be wtten as δ = 0 > 0 Snce ths s a nu quantty any constant tmes ths may be added to (4) wthout ateng ts vaue. We choose the constant to be 6 q and obtan 4 ()= 6 q δ Ths may be wtten as 4 ()= 6 = 6 Q δ Q 5 = 6 Q φ
10 Bo E. Seneus :0 whee the quadupoe tenso Q has the eements. Q = q δ Ths tenso s symmetc whch means at most 6 ndependent eements. Actuay thee ae at most 5 ndependent eements snce one can show that the tace sum of the dagona eements s zeo. Aso the dpoe-dpoe tenso s symmetc whch means that ( 4 ) ()= Qφ ()= Qφ () 6 6 = [ Q φ() ] = T Q φ() 6 6 [ ] If the quadupoe tenso s efeed to pncpa aes t becomes dagona. Ths togethe wth the vanshng tace means that thee ae ony ndependent eements. In many appcatons the chage dstbuton has an as of symmety. If we choose ths to be as then Q = Q. Then thee s ony one ndependent eement. Q 0 0 Q = 0 Q Q Fo contnuous chage dstbuton: Q = d d d V ρ
11 Bo E. Seneus : Wth ths smpe speca case the quadupoe potenta becomes 4 cos θ ( θ) = Q 4 Dscusson: The mutpoe moments n genea depend on the poston of the ogn. Howeve the owest non-vanshng moment s ndependent of the poston of the ogn.
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