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

Famous quotes containing the words moments, point and/or charge:

“To the artist is sometimes granted a sudden, transient insight which serves in this matter for experience. A flash, and where previously the brain held a dead fact, the soul grasps a living truth! At moments we are all artists.”
—Arnold Bennett (1867–1931)

“Competition has been shown to be useful up to a certain point and no further, but cooperation, which is the thing we must strive for today, begins where competition leaves off.”
—Franklin D. Roosevelt (1882–1945)

“Carlyle must undoubtedly plead guilty to the charge of mannerism. He not only has his vein, but his peculiar manner of working it. He has a style which can be imitated, and sometimes is an imitator of himself.”
—Henry David Thoreau (1817–1862)