Page 179 - Engineering Electromagnetics, 8th Edition

CHAPTER 6 Capacitance 161

and therefore

∂ ∂V ∂ ∂V ∂ ∂V
∇ · ∇V = + +
∂x ∂x ∂y ∂y ∂z ∂z
2
2
2
∂ V ∂ V ∂ V
= + + (25)
∂x 2 ∂y 2 ∂z 2
2
Usually the operation ∇ · ∇ is abbreviated ∇ (and pronounced “del squared”), a good
reminder of the second-order partial derivatives appearing in Eq. (5), and we have
2
2
2
∂ V ∂ V ∂ V
2 ρ ν
∇ V = + + =− (26)
∂x 2 ∂y 2 ∂z 2
in rectangular coordinates.
If ρ ν = 0, indicating zero volume charge density, but allowing point charges,
line charge, and surface charge density to exist at singular locations as sources of the
field, then
2
∇ V = 0 (27)
2
which is Laplace’s equation. The ∇ operation is called the Laplacian of V.
In rectangular coordinates Laplace’s equation is

2
2
2
∂ V ∂ V ∂ V
2
∇ V = + + = 0 (rectangular) (28)
∂x 2 ∂y 2 ∂z 2
2
and the form of ∇ V in cylindrical and spherical coordinates may be obtained by using
the expressions for the divergence and gradient already obtained in those coordinate
systems. For reference, the Laplacian in cylindrical coordinates is
2
2
1 ∂ ∂V 1 ∂ V ∂ V
2 (cylindrical) (29)
∇ V = ρ + 2 2 + 2
ρ ∂ρ ∂ρ ρ ∂φ ∂z
and in spherical coordinates is
2
2 1 ∂ 2 ∂V 1 ∂ ∂V 1 ∂ V
∇ V = r + sin θ + (spherical)
2
r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ 2
2
2
2
(30)
These equations may be expanded by taking the indicated partial derivatives, but it is
usuallymorehelpfultohavethemintheformsjustgiven;furthermore,itismucheasier
to expand them later if necessary than it is to put the broken pieces back together again.
Laplace’s equation is all-embracing, for, applying as it does wherever volume
charge density is zero, it states that every conceivable configuration of electrodes