Week 4 Lex 3: Multipole Expansion II

The multipole expansion arises from the idea that the potential due to a point charge q at a distance r goes as 1/r. If we calculate the approximate potential at a large distance from a physical dipole which is a pair of equal and opposite charges +q and -q at a distance d  where the vector d is from the -ve to the +ve charge is given by a binomial expansion to first order ( higher orders neglected) as 
V=(1/4πε₀)[ q d cos θ/r²] and goes as 1/r² to a first order.


This tells us that it may be possible to obtain the approximate potential at a large distance from an arbitrary charge configuration in terms of a series of contributions in powers of (1/r). This is somewhat like expressing a complex waveform in terms of some simple waves of different frequencies ( fundamentals and harmonics), called a Fourier expansion.. What allows this is the Superposition Principle.


When we expand the denominator of the the expression for the potential at a faraway point r due to an arbitrary volume charge distribution ρ(r') given as
V=(1/4πε₀)∫_τ ρ(r')dτ'/r
( note that r is the Griffiths script/curly r. )
in a binomial expansion, we obtain a series of contributions in powers of (1/r) which is the distance between the source point r' and the field point r. The coefficient of each power of (1/r) is an integral that provides the respective moments like the monopole moment ( total charge Q=∫ ρ(r')dτ') then a dipole moment p= ∫_τ r'ρ(r')dτ' and
the contribution to the potential is given as V_dip=(1/4πε₀)p. r cap /r² please note that r cap is a unit vector along r. Note that θ or θ' is simply the angle between the position vectors r and r'. It is not the polar angle unless z axis is chosen along the r' or r direction. Note that this is the dipole moment of a PURE DIPOLE


*For a point charge at origin the monopole term is the only non zero term and the potential due to a point charge is EXACT all higher multipoles vanishing. ( Check it out)


*The dipole moment expression works for a line charge λ, surface charge σ distribution replacing the volume distribution ρ and integrating over
a surface or a line.


For a discrete point charge distribution of q_i at r'_i the pure dipole moment is p=∑_i=1ⁿ q_i r'_i. A physical dipole moment is a special case of this for just two charges +q and -q and is given as p=qd where the vector d is the vector from - to the + charge. This is consistent as it gives the correct potential where
for a physical dipole. But note that this potential is approximate because for a physical dipole of two unlike but equal charges the monopole term is zero as total charge Q=0 but higher multipole terms are not zero. They are smaller than the dipole contribution and hence neglected in the approximation that the field point r is far away from the source.


It should be obvious now that for a single point charge away from the origin ( that is r' not equal to 0) will have all possible moments starting from the monopole moment ( total charge Q=q in this case) and all possible such terms in the expression for the potential V at a field point r.


* The physical dipole reduces to a pure dipole in the limit q
goes to infinity and d goes to 0 with qd=p held fixed. So a physical dipole are finitely separated charges and a pure dipole is more like a point. The pure dipole is the second moment in the multipole expansion. For large distances the field
of a pure dipole and a physical dipole are identical but are very different close to the source. ( See the two figures given in Griffiths).


The multipole expansion is dependent obviously on the coordinate system. The example of a point charge q away from the origin is an example of this where all higher multipoles also contribute although the dominant term is the monopole moment.


For a shift in origin the monopole moment Q does not change. The (pure) dipole moment changes due to the shift except when the total charge Q=0 ( algebraic sum). In that case ( for total charge Q=0) the (pure) dipole moment is unchanged. See the example for a shift of origin by a vector a  in Griffiths at the end of the section which was discussed in class.

For quick reference

http://cr4.globalspec.com/blogentry/2842/Multipole-Expansion