Suppose there are more than one point source
charges providing forces on a field charge. A fairly simple example
with three source charges (shown in green and indexed by
subscripts) and one field charge (in red, designated
q
q), is diagramed
below. We assume that the source charges are fixed in space, and
the field charge
q
q is subject to
forces from the source charges.
Note the coordinate system that has been
chosen. All of the charges lie on the corners of a square, and the
origin is chosen to collocate with the lower right source charge,
and aligned with the square. Since we can have only one origin of
coordinates, no more than one of the source points can lie at the
origin, and the displacements from different source points to the
field point differ.
The total force on the field charge
q
q is due to
applications of the force described in Equation (1):
F
⇀
E
=
k
q
1
q
2
r
2
r
^
F
⇀
E
=
k
q
1
q
2
r
2
r
^
, from each of
the source charges. So the total force is the sum of these
individual forces. (In physics, when quantities add linearly in
this way, we say that the quantities superpose. To say that the
total force is the sum of the individual forces, we say that the
forces are subject to superposition.)
The displacements of the field charge from
each source charge are shown as light blue arrows in the diagram
below.
Using the principle of superposition for this
example, and therefore applying Equation (1) three times and
summing the results, we find that
F
⇀
E
q
=
k
q
(
q
1
r
1
2
r
^
1
+
q
2
r
2
2
r
^
2
+
q
3
r
3
2
r
^
3
)
F
⇀
E
q
=
k
q
(
q
1
r
1
2
r
^
1
+
q
2
r
2
2
r
^
2
+
q
3
r
3
2
r
^
3
)
(1)To actually use this relationship, we need to
determine the values of the
r
n
r
n
and the
r
^
n
r
^
n
, where
n
=
{
1
,
2
,
3
}
n
=
{
1
,
2
,
3
}
. To see how this can be done,
it is useful to add to the figure the vectors that locate each of
the source charges.
This figure repays some study. First note that
the simple symbol
r
⇀
r
⇀
has been reserved to mean the
location of the field point. (This is a conventional usage.) Other
locations are designated with subscripts and primes. The vectors
from source points to the field point (needed in the use of
Equation (1)) carry subscripts and are shown in light blue. The
vectors locating source points (shown in brown in the figure) carry
subscripts and a prime sign to indicate they are source locations.
By virtue of the geometry of the diagram and choice of coordinate
system,
r
⇀
1
′
=
0
r
⇀
1
′
=
0
and
r
⇀
1
=
r
⇀
r
⇀
1
=
r
⇀
. For each source point
r
⇀
n
′
r
⇀
n
′
, there is a triangle formed by
vectors:
r
⇀
n
′
+
r
⇀
n
=
r
⇀
r
⇀
n
′
+
r
⇀
n
=
r
⇀
. This fact (which is always
true for any three points in space) is the key for doing
calculations for the force on the field charge, because the vectors
needed in the use of Equation (1) are given for each source charge
by
r
⇀
n
=
r
⇀
-
r
⇀
n
′
r
⇀
n
=
r
⇀
-
r
⇀
n
′
.
Assume that the side of the square in this
example is
a
a. Then for the first
source charge
r
⇀
1
=
r
⇀
=
a
y
^
r
⇀
1
=
r
⇀
=
a
y
^
. For the second source charge
r
⇀
2
′
=
a
(
-
x
^
)
r
⇀
2
′
=
a
(
-
x
^
)
and
r
⇀
2
=
a
(
x
^
+
y
^
)
r
⇀
2
=
a
(
x
^
+
y
^
)
. For the third source charge
r
⇀
3
′
=
a
(
(
-
x
^
)
+
y
^
)
r
⇀
3
′
=
a
(
(
-
x
^
)
+
y
^
)
and
r
⇀
3
=
a
x
^
r
⇀
3
=
a
x
^
. Substituting these values
into Equation (2) provides the solution for this example
F
⇀
E
q
=
k
q
2
a
2
(
(
2
q
3
+
q
2
)
x
^
+
(
2
q
1
+
q
2
)
y
^
)
F
⇀
E
q
=
k
q
2
a
2
(
(
2
q
3
+
q
2
)
x
^
+
(
2
q
1
+
q
2
)
y
^
)
(2).
We can generalize Equation (2) to give a basic
result for the Coulomb force on a field point charge
q
q due to an arbitrary
number
N
N of fixed source
point charges
q
n
q
n
in a compact form:
F
⇀
E
q
[
r
⇀
]
=
k
q
∑
n
=
1
N
q
n
r
n
2
r
^
n
F
⇀
E
q
[
r
⇀
]
=
k
q
∑
n
=
1
N
q
n
r
n
2
r
^
n
(3)Equation (3), properly applied, provides the
solution for the force on any point field charge affected by the
electric force from an arbitrary number of fixed source charges
with known values and locations.
Coulomb Law Forces
