ABSTRACT
Equation 6.1 is known as the Laplace-Poisson equation. When rTRUEis equal to zero, it is called the Laplace equation, and when it is not equal to zero, it is called the Poisson equation. In vacuum, Eq. 6.1 is modified as Df f f f r
e = ∂ ∂
+ ∂ ∂
+ ∂ ∂
= - 2
0x y z TOTAL (6.2)
For convenience, we rewrite Eq. 6.1 in polar coordinate (r, q, j) and cylindrical coordinate (r, j, z) as follows (Fig. 6.1):
X
Y
Z
j
q r
P(r, q,j)
X
Y
Z
r
P(r, j, Z)
j
Df f q q
q f q q
f f
= ∂ ∂
∂ ∂
Ê ËÁ
ˆ ¯˜ + ∂
∂ ∂ ∂
Ê ËÁ
ˆ ¯˜ + ∂
∂
Ï1 1 1 2
2r r r
r sin sin
sin Ì Ô
ÓÔ
¸ ˝ Ô Ô˛ = -
r e TRUE
(6.3) In cylindrical coordinates, the equation isDf f f j
f r e
= ∂ ∂
∂ ∂
Ê ËÁ
ˆ ¯˜ + ∂
∂ + ∂ ∂
= -1 1 2
2r r r
r r z TRUE (6.4)
When a distribution of charges is given, the electric potential can be found everywhere by using this equation. At boundaries between dielectrics, the following boundary conditions hold for scalar potential: f1 = f2 (6.5) For metals, E ^ surfaces e f
x e
f x1
2 ∂ ∂
= ∂ ∂
(6.6) This means that the electric potential f is determined by Eqs. 6.1 and 6.6 uniquely. Namely, the specification of a potential under the closed surface determines the outstanding potential distribution in a space surrounded by the surface (Dirichlet boundary conditions). The specification of an electric field everywhere under the closed surface also determines a unique potential distribution (Neumann boundary conditions). There are four ways of solving the Laplace-Poisson equation under given electromagnetic boundary conditions: method of images, a direct solution, conformal mapping, and Green’s function. All solutions will be explained in the following sections.
