Axial multipole moments

Axial multipole moments are a series expansionof the electric potential of acharge distribution localized close to the origin along one Cartesian axis, denoted here as the z-axis. However, the axial multipole expansion can also be applied to any potential or field that varies inverselywith the distance to the source, i.e., as 1 R {displaystyle {frac {1}{R}}} .For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density λ ( z ) {displaystyle lambda (z)} localized to the z-axis. Axial multipole moments are a series expansionof the electric potential of acharge distribution localized close to the origin along one Cartesian axis, denoted here as the z-axis. However, the axial multipole expansion can also be applied to any potential or field that varies inverselywith the distance to the source, i.e., as 1 R {displaystyle {frac {1}{R}}} .For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density λ ( z ) {displaystyle lambda (z)} localized to the z-axis. The electric potential of a point charge q located onthe z-axis at z = a {displaystyle z=a} (Fig. 1) equals If the radius r of the observation point is greater than a, we may factor out 1 r {displaystyle {frac {1}{r}}} and expand the square rootin powers of ( a / r ) < 1 {displaystyle (a/r)<1} using Legendre polynomials where the axial multipole moments M k ≡ q a k {displaystyle M_{k}equiv qa^{k}} contain everything specific to a given charge distribution; the other parts of the electric potential depend only on the coordinates of the observation point P. Special cases include the axialmonopole moment M 0 = q {displaystyle M_{0}=q} , the axial dipolemoment M 1 = q a {displaystyle M_{1}=qa} and the axial quadrupolemoment M 2 ≡ q a 2 {displaystyle M_{2}equiv qa^{2}} . This illustrates the general theorem that the lowestnon-zero multipole moment is independent of the origin of the coordinate system, but higher multipole moments are not (in general). Conversely, if the radius r is less than a, we may factor out 1 a {displaystyle {frac {1}{a}}} and expandin powers of ( r / a ) < 1 {displaystyle (r/a)<1} , once again using Legendre polynomials where the interior axial multipole moments I k ≡ q a k + 1 {displaystyle I_{k}equiv {frac {q}{a^{k+1}}}} containeverything specific to a given charge distribution;the other parts depend only on the coordinates of the observation point P. To get the general axial multipole moments, we replace the point charge of the previous section with an infinitesimalcharge element λ ( ζ )   d ζ {displaystyle lambda (zeta ) dzeta } , where λ ( ζ ) {displaystyle lambda (zeta )} represents the charge density atposition z = ζ {displaystyle z=zeta } on the z-axis. If the radius rof the observation point P is greater than the largest | ζ | {displaystyle left|zeta ight|} for which λ ( ζ ) {displaystyle lambda (zeta )} is significant (denoted ζ max {displaystyle zeta _{ ext{max}}} ), the electric potential may be written where the axial multipole moments M k {displaystyle M_{k}} are defined Special cases include the axial monopole moment (=total charge)