Maxwells equation for the static electric field with charge density

If we define the scalar potential in any of the usual ways

 

When this is LaPlaceís equation and for we get Poissonís equation.

 

Expressing the LaPlace equation in spherical coordinates

 

 

This equation can be solved by separation of variables assuming a product of the

 

 

The solution for the angular part leads to the spherical harmonics

 

If you assume no dependence we get solutions in terms of where . These functions are Legendre polynomials. For general dependence we find the associated Legendre polynomials as shown above.

 

 

 

 

LegendreP

Wolfram http://mathworld.wolfram.com/LegendrePolynomial.html

 

where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).

The first few Legendre polynomials are

P_0(x)

=

1

(2)

P_1(x)

=

x

(3)

P_2(x)

=

1/2(3x^2-1)

(4)

P_3(x)

=

1/2(5x^3-3x)

(5)

P_4(x)

=

1/8(35x^4-30x^2+3)

(6)

P_5(x)

=

1/8(63x^5-70x^3+15x)

(7)

P_6(x)

=

1/(16)(231x^6-315x^4+105x^2-5).

(8)

 

 x^n=sum_(l=n,n-2,...)((2l+1)n!)/(2^((n-l)/2)(1/2(n-l))!(l+n+1)!!)P_l(x)

 

Wolfram http://mathworld.wolfram.com/LegendrePolynomial.html

 

 

http://en.wikipedia.org/wiki/Legendre_polynomials

In mathematics, Legendre functions are solutions to Legendre's differential equation:

{d \over dx} \left[ (1-x^2) {d \over dx} P_n(x) \right] + n(n+1)P_n(x) = 0.

 

Legendre polynomials in multipole expansions

Figure 2

Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):

\frac{1}{\sqrt{1 + \eta^{2} - 2\eta x}} = \sum_{k=0}^{\infty} \eta^{k} P_{k}(x)

which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials.

As an example, the electric potential Φ(r) (in spherical coordinates) due to a point charge located on the z-axis at z = a (Figure 2) varies like

\Phi (r, \theta ) \propto \frac{1}{R} = \frac{1}{\sqrt{r^{2} + a^{2} - 2ar \cos\theta}}.

If the radius r of the observation point P is greater than a, the potential may be expanded in the Legendre polynomials

\Phi(r, \theta) \propto
\frac{1}{r} \sum_{k=0}^{\infty} \left( \frac{a}{r} \right)^{k} 
P_{k}(\cos \theta)

where we have defined η = a/r < 1 and x = cos θ. This expansion is used to develop the normal multipole expansion.

Conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion.