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 $frac\{1\}\{R\}$.For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density $lambda(z)$localized to the "z"-axis.

Axial multipole moments of a point charge

The electric potential of a point charge "q" located onthe "z"-axis at $z=a$ (Fig. 1) equals

:$Phi(mathbf\{r\})\; =\; frac\{q\}\{4pivarepsilon\}\; frac\{1\}\{R\}\; =frac\{q\}\{4pivarepsilon\}\; frac\{1\}\{sqrt\{r^\{2\}\; +\; a^\{2\}\; -\; 2\; a\; r\; cos\; heta.$

If the radius "r" of the observation point is greater than "a", we may factor out $frac\{1\}\{r\}$ and expand the square rootin powers of using Legendre polynomials

:$Phi(mathbf\{r\})\; =\; frac\{q\}\{4pivarepsilon\; r\}\; sum\_\{k=0\}^\{infty\}left(\; frac\{a\}\{r\}\; ight)^\{k\}\; P\_\{k\}(cos\; heta\; )\; equivfrac\{1\}\{4pivarepsilon\}\; sum\_\{k=0\}^\{infty\}\; M\_\{k\}left(\; frac\{1\}\{r^\{k+1\; ight)\; P\_\{k\}(cos\; heta\; )$

where the axial multipole moments $M\_\{k\}\; equiv\; q\; a^\{k\}$ 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 axial
monopole moment $M\_\{0\}=q$, the axial dipolemoment $M\_\{1\}=q\; a$ and the axial quadrupolemoment $M\_\{2\}\; equiv\; q\; a^\{2\}$. This illustrates the general theorem that the lowestnon-zero multipole moment is independent of the
origin of the coordinate system, but higher multipole multipole moments are not (in general).

Conversely, if the radius "r" is less than "a", we may factor out $frac\{1\}\{a\}$ and expandin powers of using Legendre polynomials

:$Phi(mathbf\{r\})\; =\; frac\{q\}\{4pivarepsilon\; a\}\; sum\_\{k=0\}^\{infty\}left(\; frac\{r\}\{a\}\; ight)^\{k\}\; P\_\{k\}(cos\; heta\; )\; equiv\; frac\{q\}\{4pivarepsilon\}\; sum\_\{k=0\}^\{infty\}\; I\_\{k\}r^\{k\}\; P\_\{k\}(cos\; heta\; )$where the interior axial multipole moments $I\_\{k\}\; equiv\; frac\{q\}\{a^\{k+1$ containeverything specific to a given charge distribution;the other parts depend only on the coordinates of the observation point P.

General axial multipole moments

To get the general axial multipole moments, we replace the point charge of the previous section with an infinitesimalcharge element $lambda(zeta)\; dzeta$, where $lambda(zeta)$ represents the charge density atposition $z=zeta$ on the "z"-axis. If the radius "r"of the observation point P is greater than the largest $left|\; zeta\; ight|$ for which $lambda(zeta)$is significant (denoted $zeta\_\{max\}$), the electric potential may be written

:$Phi(mathbf\{r\})\; =\; frac\{1\}\{4pivarepsilon\}\; sum\_\{k=0\}^\{infty\}\; M\_\{k\}left(\; frac\{1\}\{r^\{k+1\; ight)\; P\_\{k\}(cos\; heta\; )$

where the axial multipole moments $M\_\{k\}$ are defined

:$M\_\{k\}\; equiv\; int\; dzeta\; lambda(zeta)\; zeta^\{k\}$

Special cases include the axial monopole moment (=total charge)

:$M\_\{0\}\; equiv\; int\; dzeta\; lambda(zeta)$,

the axial dipole moment $M\_\{1\}\; equiv\; int\; dzeta\; lambda(zeta)\; zeta$, andthe axial quadrupole moment $M\_\{2\}\; equiv\; int\; dzeta\; lambda(zeta)\; zeta^\{2\}$.Each successive term in the expansion varies inversely with a greater power of $r$, e.g., the monopole potential varies as $frac\{1\}\{r\}$, the dipole potential varies as $frac\{1\}\{r^\{2$, the quadrupole potential varies as $frac\{1\}\{r^\{3$, etc. Thus, at large distances($frac\{zeta\_\{max\{r\}\; ll\; 1$), the potential is well-approximatedby the leading nonzero multipole term.

The lowest non-zero axial multipole moment is invariant under a shift "b" in
origin, but higher moments generallydepend on the choice of origin. The shifted multipole moments$M\_\{k\}^\{prime\}$ would be

:$M\_\{k\}^\{prime\}\; equiv\; int\; dzeta\; lambda(zeta)\; left(zeta\; +\; b\; ight)^\{k\}$

Expanding the polynomial under the integral:$left(\; zeta\; +\; b\; ight)^\{l\}\; =\; zeta^\{l\}\; +\; l\; b\; zeta^\{l-1\}\; +\; ldots\; +\; l\; zeta\; b^\{l-1\}\; +\; b^\{l\}$leads to the equation:$M\_\{k\}^\{prime\}\; =\; M\_\{k\}\; +\; l\; b\; M\_\{k-1\}\; +\; ldots\; +\; l\; b^\{l-1\}\; M\_\{1\}\; +\; b^\{l\}\; M\_\{0\}$If the lower moments $M\_\{k-1\},\; M\_\{k-2\},ldots\; ,\; M\_\{1\},\; M\_\{0\}$are zero, then $M\_\{k\}^\{prime\}\; =\; M\_\{k\}$. The same equation shows that multipole moments higher than the first non-zero moment do depend on the choice of origin (in general).

Interior axial multipole moments

Conversely, if the radius "r" is smaller than the smallest$left|\; zeta\; ight|$ for which $lambda(zeta)$is significant (denoted $zeta\_\{min\}$), the electric potential may be written

:$Phi(mathbf\{r\})\; =\; frac\{1\}\{4pivarepsilon\}\; sum\_\{k=0\}^\{infty\}\; I\_\{k\}r^\{k\}\; P\_\{k\}(cos\; heta\; )$

where the interior axial multipole moments $I\_\{k\}$ are defined

:$I\_\{k\}\; equiv\; int\; dzeta\; frac\{lambda(zeta)\}\{zeta^\{k+1$

Special cases include the interior axial monopole moment ($eq$ the total charge)

:$M\_\{0\}\; equiv\; int\; dzeta\; frac\{lambda(zeta)\}\{zeta\}$,

the interior axial dipole moment $M\_\{1\}\; equiv\; int\; dzeta\; frac\{lambda(zeta)\}\{zeta^\{2$,etc. Each successive term in the expansion varies with a greater power of $r$, e.g., the interior monopole potential varies as $r$, the dipole potential varies as $r^\{2\}$, etc. At short distances ($frac\{r\}\{zeta\_\{min\; ll\; 1$), the potential is well-approximated by the leading nonzero interior multipole term.

ee also

*Potential theory
*Multipole moments
*Multipole expansion
*Legendre polynomials
*Spherical multipole moments
*Cylindrical multipole moments

References

External links