Chapter
6
ELECTROSTATIC BOUNDARY
VALUE PROBLEMS
Our schools had better get on with what is their overwhelmingly most important
task: teaching their charges to express themselves clearly and with precision in
both speech and writing; in other words, leading them toward mastery of their
own language. Failing that, all their instruction in mathematics and science is a
waste of time.
—JOSEPH WEIZENBAUM, M.I.T.
b.1 INTRODUCTION
The procedure for determining the electric field E in the preceding chapters has generally
been using either Coulomb's law or Gauss's law when the charge distribution is known, or
using E = —
W
when the potential
V
is known throughout the region. In most practical
situations, however, neither the charge distribution nor the potential distribution is known.
In this chapter, we shall consider practical electrostatic problems where only electro
static conditions (charge and potential) at some boundaries are known and it is desired to
find E and
V
throughout the region. Such problems are usually tackled using Poisson's
1
or
Laplace's
2
equation or the method of images, and they are usually referred to as
boundary
value
problems. The concepts of resistance and capacitance will be covered. We shall use
Laplace's equation in deriving the resistance of an object and the capacitance of a capaci
tor. Example 6.5 should be given special attention because we will refer to it often in the
remaining part of the text.
.2 POISSON'S AND LAPLACE'S EQUATIONS
Poisson's and Laplace's equations are easily derived from Gauss's law (for a linear mater
ial medium)
V • D = V • eE =
p
v
(6.1)
'After Simeon Denis Poisson (17811840), a French mathematical physicist.
2
After Pierre Simon de Laplace (17491829), a French astronomer and mathematician.
199
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Electrostatic BoundaryValue Problems
and
E = VV
Substituting eq. (6.2) into eq. (6.1) gives
V(eVV) = p
v
for an inhomogeneous medium. For a homogeneous medium, eq. (6.3) becomes
V
2
y =
 ^
(6.2)
(6.3)
(6.4)
This is known as
Poisson's equation. A
special case of this equation occurs when p
v
= 0
(i.e., for a chargefree region). Equation (6.4) then becomes
V
2
V= 0
(6.5)
which is known as
Laplace's equation.
Note that in taking s out of the lefthand side of
eq. (6.3) to obtain eq. (6.4), we have assumed that
e
is constant throughout the region in
which
V
is defined; for an inhomogeneous region, e is not constant and eq. (6.4) does not
follow eq. (6.3). Equation (6.3) is Poisson's equation for an inhomogeneous medium; it
becomes Laplace's equation for an inhomogeneous medium when
p
v
=
0.
Recall that the Laplacian operator V
2
was derived in Section 3.8. Thus Laplace's equa
tion in Cartesian, cylindrical, or spherical coordinates respectively is given by
(6.6)
(6.7)
(6.8)
depending on whether the potential is
V(x,
y,
z), V(p, 4>, z),
or
V(r, 6, 4>).
Poisson's equation
in those coordinate systems may be obtained by simply replacing zero on the righthand
side of eqs. (6.6), (6.7), and (6.8) with
—p
v
/e.
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 Fall '10
 DinavahiandIyer
 Electrostatics, Laplace, Electric charge, Eqs, Electrostatic BoundaryValue Problems

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