A. F. Peterson:
Notes on Electromagnetic Fields & Waves
10/04
Fields & Waves Note #9
Electrostatic Equations and Boundary Conditions
Objectives:
Review the equations assembled up to now relating the various
fields with the associated charge distribution.
Introduce the scalar Laplacian
operator.
Develop conditions on the electric field and electric flux density at
material interfaces.
Electrostatic Equations
At this point it is appropriate to summarize the various fields that have been introduced and
the equations that relate them.
Our study of electrostatics began with the force field
F
(N)
associated with a distribution of electric charge.
This force field was renormalized to yield
the electric field
E
(V/m).
An alternative normalization gave us the electric flux density
D
(C/m
2
). Finally, in Note #8 we introduced the voltage field
V
(V), which provides yet
another way of looking at the same force field, from the perspective of a scalar rather than a
vector quantity.
In integral form, we have two equations that describe the static electric field and flux density:
E
d
∑
=
Ú
l
any closed path
0
(9.1)
D
dS
Q
∑
=
ÚÚ
any closed surface
enclosed
(9.2)
These integral relations involve the average values of the field over the path or surface in
question.
We also have two equations in differential form that involve these fields at points
within a region:
E
V
= —
(9.3)
—
∑
=
D
v
r
(9.4)
We previously observed that (9.3) implies (9.1).
In Note #7, (9.4) was obtained from (9.2)
by a limiting procedure.
Thus, these two sets of equations are somewhat equivalent.
In conducting materials, the field and flux density satisfy
E
D
=
=
0
0
,
(9.5)
whereas, in dielectric materials,
D
E
r
=
e e
0
(9.6)
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A. F. Peterson:
Notes on Electromagnetic Fields & Waves
10/04
We observe that equations (9.4) and (9.6) could be combined together to yield
—
∑
=
(
)
e
r
E
v
(9.7)
and, furthermore, that (9.3) could be introduced to obtain
—
∑
—
=
(
)
e
r
V
v
(9.8)
Equation (9.8) is a generalized form of
Poisson’s equation
. If the permittivity
e
is a
constant within the region of interest, (9.8) can be rewritten as
—
∑
—
= 
(
)
V
v
r
e
(9.9)
The combination of the two derivative operators on the lefthand side of (9.9) is equivalent
to
—
∑
—
=
∂
∂
∂
∂
Ê
Ë
Á
ˆ
¯
˜
+
∂
∂
∂
∂
Ê
Ë
Á
ˆ
¯
˜
+
∂
∂
∂
∂
Ê
Ë
Á
ˆ
¯
˜
=
∂
∂
+
∂
∂
+
∂
∂
= —
(
)
V
x
V
x
y
V
y
z
V
z
V
x
V
y
V
z
V
2
2
2
2
2
2
2
(9.10)
which is known as the
scalar Laplacian
.
Equation (9.9) can be written as
—
= 
2
V
v
r
e
(9.11)
Equation (9.11) is the ordinary form of Poisson’s equation.
In a region containing nonzero
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 Spring '08
 CITRIN
 Electrostatics, Electromagnet, Electric charge, Fundamental physics concepts, A. F. Peterson

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