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# electric potential

Amount of work needed to move a unit electric charge from a reference point to a specific point against an electric field. The potential energy of a positive charge increases when it moves against an electric field, and decreases when it moves with the field. Electric potential can be thought of as potential energy per unit charge. The work done in moving a unit charge from one point to another, as in an electric circuit, is equal to the difference in potential energies at each point. Electric potential is expressed in units of joules per coulomb, or volts.

In physics, the Laplace expansion of a 1/r - type potential is applied to expand Newton's gravitational potential or Coulomb's electrostatic potential. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the interelectronic repulsion.

The Laplace expansion is in fact the expansion of the inverse distance between two points. Let the points have position vectors r and r', then the Laplace expansion is


frac{1}
= sum_{ell=0}^infty frac{4pi}{2ell+1} > sum_{m=-ell}^{ell} (-1)^m frac{r_^ell }{r_^{ell+1} } Y^{-m}_ell(theta, varphi) Y^{m}_ell(theta', varphi'). Here r has the spherical polar coordinates (r, θ, φ) and r' has (r', θ', φ'). Further r< is min(r, r') and r> is max(r, r'). The function $Y^m_\left\{ell\right\}$ is a normalized spherical harmonic function. The expansion takes a simpler form when written in terms of solid harmonics,

frac{1}

## Derivation

The derivation of this expansion is simple. One writes

frac{1}
> = frac{1}{sqrt{r^2 + (r')^2 - 2 r r' cosgamma}} = frac{1}{r_ sqrt{1 + h^2 - 2 h cosgamma}} quadhbox{with}quad h equiv frac{r_}{r_} . We find here the generating function of the Legendre polynomials $P_ell\left(cosgamma\right)$ :

frac{1}{sqrt{1 + h^2 - 2 h cosgamma}} = sum_{ell=0}^infty h^ell P_ell(cosgamma). Use of the spherical harmonic addition theorem

P_{ell}(cos gamma) = frac{4pi}{2ell + 1} sum_{m=-ell}^{ell} Y^{-m}_{ell}(theta, varphi) Y^m_{ell}(theta', varphi') gives the desired result.

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