Winter 2008
Lecture 3
Page 1 of 5
Lecture 3
Electric Forces and Fields due to surface charge distributions
Surface Charges
: In most practical circuits and systems, there will be metallic surfaces
(for example, microstrip lines for IC and high frequency circuits, wires for low frequency
systems, and metallic tubes for waveguides).
To model a surface charge we need to define the distribution (or density function) of the
charge on the line. The notation is typically
(
29
2
m
C
S
r
for surface charge density. This
may be a constant or a function of position along the line.
In this case we can replace the point charge with a piece of the line charge (the piece will
have an area of
dS
). This leads to
dS
dQ
S
r
=
.
Since the charge is a continuous distribution, it will be more effective to integrate the
contributions (summing up each contribution for a continuous function is the job of an
integral). This allows us to write the electric field as
∫
=
3
4
1
R
dS
R
E
S
r
pe
.
As in the previous case of a line charge, the electric field due to a surface charge will
depend upon the geometry of the problem as well as the distribution of the charge.
The following example will illustrate the use of the above relationships.
EXAMPLE
A circular (radius “a”) of charged surface with uniform charge density
S
r
is centered on
the origin. Compute the electric field at a point (
0, 0 ,z
o
).
Figure 1 The geometry for the surface charge
SOLUTION:
The first step is to set up the relationship. The charge distribution is a patch of surface so
that the elemental charge is given by
f
r
r
r
r
d
d
dS
dQ
S
S
=
=
.
The vector linking the charge (0 0 z) to the observer at (x
o
y
o
z
o
) is
(
29
(
29
(
29
z
o
y
x
a
z
a
y
a
x
R
ˆ
ˆ
ˆ
+

+

=
=
z
o
a
z
a
ˆ
ˆ
+

r
r
.
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