Page 178 - Engineering Electromagnetics, 8th Edition

160 ENGINEERING ELECTROMAGNETICS

6.6 POISSON’S AND LAPLACE’S EQUATIONS
In preceding sections, we have found capacitance by first assuming a known charge
distribution on the conductors and then finding the potential difference in terms of
the assumed charge. An alternate approach would be to start with known potentials
on each conductor, and then work backward to find the charge in terms of the known
potential difference. The capacitance in either case is found by the ratio Q/V .
The first objective in the latter approach is thus to find the potential function
between conductors, given values of potential on the boundaries, along with possible
volume charge densities in the region of interest. The mathematical tools that enable
this to happen are Poisson’s and Laplace’s equations, to be explored in the remainder
of this chapter. Problems involving one to three dimensions can be solved either ana-
lytically or numerically. Laplace’s and Poisson’s equations, when compared to other
methods, are probably the most widely useful because many problems in engineering
practice involve devices in which applied potential differences are known, and in
which constant potentials occur at the boundaries.
Obtaining Poisson’s equation is exceedingly simple, for from the point form of
Gauss’s law,
(21)
∇ · D = ρ ν
the definition of D,

D = E (22)

and the gradient relationship,

E =−∇V (23)
by substitution we have

∇ · D =∇ · ( E) =−∇ · ( ∇V ) = ρ ν
or

ρ ν
∇ · ∇V =− (24)

for a homogeneous region in which is constant.
Equation (24) is Poisson’s equation, but the “double ∇” operation must be inter-
preted and expanded, at least in rectangular coordinates, before the equation can be
useful. In rectangular coordinates,

∂A x ∂A y ∂A z
∇ · A = + +
∂x ∂y ∂z
∂V ∂V ∂V
∇V = a x + a y + a z
∂x ∂y ∂z