Digital Signal Processing Reference
In-Depth Information
E
=
a
r
E
r
Q
Q
E
r
=
4
pe
0
r
2
r
Figure 2-14
Electric field generated by a point charge
Q
.
where is the volume density of the charge within volume
V
in C/m
3
and
Q
enc
is the total charge enclosed by the surface
S
and contained in the volume
V
in
units of coulombs.
As shown in Figure 2-14, a point charge has an electric field that radiates
out in all directions, necessitating the use of spherical coordinates. In spherical
coordinates, the
φ
and
θ
omponents of the electric field are zero, leaving only
an
r
component directed outward from the point charge. Therefore, since the
surface area of a sphere is 4
πr
2
, the electric field around a point charge in free
space is derived from (2-59):
Q
4
πε
0
r
2
E
=
a
r
E
r
=
V
/
m
(2-60)
Substituting (2-60) into (2-58) allows the calculation of the work done per unit
charge when a charge is moved along
r
from point
a
to point
b
in an electrostatic
field. Note that if the test charge was moved along
φ
or
θ
, there would be no work
done because
E
·
dl
=
E
·
dl
=
E
r
dr
.
0; however, along the radial component,
W
q
b
b
Q
4
πε
0
r
2
dr
E
·
l
=−
b
=−
d
a
a
a
→
(2-61)
1
r
b
−
Q
4
πε
0
1
r
a
=
=
(a)
−
(b)
V
Therefore, we can define the electrostatic potential as work done to move a charge
in an electrostatic field from point
a
to point
b
:
Q
4
πε
0
r
ab
ab
=
V
(2-62)
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