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Background
x_2
x_1
x_2  x_1
q_2
q_1
O
Figure 1: Forces exerted by static charges.
We develop Maxwell’s equations from their experimental foun
dations in differential and integral form for both static and dynamic
applications. We consider first the microscopic and then the macro
scopic forms of the equations. Maxwell’s equations are numbered
in what follows.
Coulomb’s law
Stationary point charges exert a mutual central force on one an
other. The force exerted on charge
q
2
under the influence of
q
1
in
Figure 1 is governed by the following empirical law:
F
21
=
kq
1
q
2
x
2

x
1

x
2

x
1

3
≡
q
2
E
21
and is attributed to the electric field established by
q
1
:
E
(
x
)
=
kq
1
x

x
1

x

x
1

3
In MKS units,
k
=
1
4
π²
◦
and
²
◦
=
8
.
854
×
10

12
F/m. The electric
(Coulomb) force is a central force with a
1
/r
2
dependence on radial
distance. The contributions from
n
point charges add linearly and
exert the total force on the
i
th particle:
F
i
=
q
i
4
π²
◦
n
X
j
6
=
i
q
j
x
i

x
j

x
i

x
j

3
E
(
x
i
)
=
1
4
π²
◦
n
X
j
6
=
i
q
j
x
i

x
j

x

x
j

3
≡
lim
q
i
→
0
F
i
q
i
where we note that a point charge exerts no self force. For a con
tinuous charge distribution, the electric field becomes:
E
(
x
)
=
1
4
π²
◦
Z
ρ
(
x
0
)
x

x
0

x

x
0

3
d
3
x
0
S
V
q
E
n
r
da
Figure 2: Gaussian surface
S
enclosing a volume
V
with a point
charge
q
within.
where the primed and unprimed coordinates denote the source and
observation points, respectively, and where
ρ
(
x
)
is the volume charge
density, with MKS units of Coulombs per cubic meter. Substitut
ing
ρ
(
x
) =
∑
n
j
=1
q
j
δ
(
x

x
j
)
recovers the electric field due to
n
discrete, static point charges.
Gauss’ law
Evaluating Coulomb’s law for any but the simplest charge distribu
tions can be cumbersome and requires specification of the charge
distribution a priori. An alternative and often more useful relation
ship between electric fields and charge distributions is provided by
Gauss’ law. Gauss’ law will become part of a system of differential
equations that permit the evaluation of the electric field even before
the charge distribution is known under some circumstances.
Consider the electric flux through the closed surface
S
sur
rounding the volume
V
containing the point charge
q
shown in
Figure 2. According to Coulomb’s law, the flux through the differ
ential surface area element
da
is
E
·
ˆ
nda
=
q
4
π²
◦
cos
α
r
2
da
where
r
is the distance from the point charge to the surface area
element,
ˆ
n
is the unit normal of
da
, and
α
is the angle between
ˆ
n
and
E
. Note next that
cos
αda
is the projection of the area element
da
on the surface of a sphere of radius
r
concentric with the point
charge. This projection could be expressed as
ds
=
r
2
sin
θdθdφ
in spherical coordinates or, more generally, as
ds
=
r
2
d
Ω
, where
d
Ω
is the differential solid angle spanned by
da
as viewed from the
point charge. Consequently,
E
·
ˆ
nda
=
q
4
π²
◦
d
Ω
1
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View Full Document Integrating over the surface of
S
then yields (noting that the total
solid angle subtended by any exterior closed surface is
4
π
Sr.)
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This note was uploaded on 09/26/2007 for the course ECE 486 taught by Professor Hysell during the Spring '06 term at Cornell University (Engineering School).
 Spring '06
 HYSELL
 Electromagnet, Electromotive Force, Magnetic Field, Faraday, current density

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