Axial Multipole Moments - Axial Multipole Moments of A Point Charge

Axial Multipole Moments of A Point Charge

The electric potential of a point charge q located on the z-axis at (Fig. 1) equals

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon} \frac{1}{R} = \frac{q}{4\pi\varepsilon} \frac{1}{\sqrt{r^{2} + a^{2} - 2 a r \cos \theta}}.$

If the radius r of the observation point is greater than a, we may factor out and expand the square root in powers of using Legendre polynomials

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon r} \sum_{k=0}^{\infty} \left( \frac{a}{r} \right)^{k} P_{k}(\cos \theta ) \equiv \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} M_{k} \left( \frac{1}{r^{k+1}} \right) P_{k}(\cos \theta )$

where the axial multipole moments 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, the axial dipole moment and the axial quadrupole moment . This illustrates the general theorem that the lowest non-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 and expand in powers of using Legendre polynomials

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon a} \sum_{k=0}^{\infty} \left( \frac{r}{a} \right)^{k} P_{k}(\cos \theta ) \equiv \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k} r^{k} P_{k}(\cos \theta )$

where the interior axial multipole moments contain everything specific to a given charge distribution; the other parts depend only on the coordinates of the observation point P.

Read more about this topic: Axial Multipole Moments

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