M. Fabbrichesi , SISSA

Laplace eq. in spherical coordinates

In this lesson, the Laplace equation is solved by using the *separation of variables *approach and cartesian coordinates for the case of a rectangular box with five sides at zero potential, while the sixth (z=c) has a specified potential Φ = V(x,y) [Jackson, sect. 2.9]. Next, the Laplace equation is solved in *spherical coordinates *which represent a more natural choice of coordinates for a large class of practical problems [Jackson, sec. 3.1]. Legendre equation and Legendre polynomials are then introduced in order to derive the general solution of the Laplace equation. Solution for the azimuthal symmetry and definition of spherical harmonics are also given [Jackson, sec 3.3 and 3.5]. In the last part of the lesson, the expansion of the Green function 1/|**r**-**r**'| in terms of spherical harmonics is used to derive a multipole expansion for the potential generated by a localized charge [Jackson, sec. 4.1]. Finally, the multipole expansion of the energy of a charge distribution is derived by expanding Φ in a Taylor series [Jackson, 4.2].