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Chapter 1
Introduction and Survey
1.1
Maxwell’s equations in a vacuum
1.1.1
Electrostatics
The results of the numerous investigations of electromagnetic phenomena carried out during
the 18th and 19th centuries led to the development of a set of equations which govern these phenomena. Coulomb’s law
(an action at a distance law) provided the description of the force,
F
12
, on a stationary point particle at
r
1
with an electric
charge
q
1
due to a stationary point particle with charge
q
2
located at
r
2
Note that the
r
1
and
r
2
are vectors. In the following
equations of electrostatics the parameter
k
1
is a constant determined by the system of units. In the modern system of units,
SI,
k
1
=(4
πε
o
)
−
1
with
ε
o
the permittivity of free space (
=
8.854 x 10
−
12
farad per meter (F/m) ). In Gaussian units ,
k
1
=1
See Table 1 on page 779.
F
12
=
k
1
q
1
q
2

r
1
−
r
2

3
(
r
1
−
r
2
)
....
(1.01)
where in SI units
k
1
=
1
4
π±
o
=10
−
7
c
2
and
q
is in Coulombs (C)
The concept of an electric
f
eld,
E
(
r
1
−
r
2
)
,
generated by electrically charged
point
particles
E
(
r
1
−
r
2
)=
k
1
q
2

r
1
−
r
2

3
(
r
1
−
r
2
)
(1.02)
allowed a local force law on point charges to be developed.
F
12
=
q
1
E
(
r
1
−
r
2
)
(1.03)
Superposition principle
The superposition principle holds for the electric
f
eld and if
{
q
i
,i
, ..., N
}
are the charges of N point particles
located at the points
{
r
i
, ..., N
}
the electric
f
eld at a point
r
6
=
r
i
is
2
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(
r
)=
k
1
N
X
i
=1
q
i

r
−
r
i

3
(
r
−
r
i
1
4
π±
o
N
X
i
=1
q
i

r
−
r
i

3
(
r
−
r
i
)
.
SI units
(1.04)
The relationship between the charged particles and the
f
elds which they generate is generalized by Gauss’ law. This law
relates electric
f
elds to electric charge densities,
ρ
(
r
)
,
in the area of electrostatics.
∇
·
E
(
r
)=4
πk
1
ρ
(
r
ρ
(
r
)
/±
o
in SI units
(1.05)
Point charges and the Dirac Delta function
In the case that
ρ
(
r
qδ
(3)
(
r
−
r
2
)
,where
δ
(3)
(
r
−
r
2
)
is a three dimensional ‘Dirac delta function’ (see next two
pages)
1
, a solution to this partial differential equation is the electric
f
eld due to a charged point particle located at
r
2
.The
(one dimensional) Dirac delta function is de
f
ned by
Z
b
a
f
(
x
0
)
δ
(
x
−
x
0
)
dx
0
=
f
(
x
)
if
a<x<b
and
0
otherwise.
(1.06)
If
x
=
a
or
x
=
b
the integral is not well de
f
ned, but a common de
f
nition is
Z
b
a
f
(
x
0
)
δ
(
x
−
x
0
)
dx
0
=
1
2
f
(
x
)
x
=
a
or
x
=
b
The three dimensional Dirac delta function (recognized henceforth with vectors as variables) is equal to the product of
three one dimensional Dirac delta functions
δ
(
r
−
r
0
)
≡
δ
(
x
−
x
0
)
δ
(
y
−
y
0
)
δ
(
z
−
z
0
)
(1.07)
so that
ZZZ
V
f
(
r
0
)
δ
(
r
−
r
0
)
dx
0
dy
0
dz
0
=
f
(
r
)
if
r
is interior to the volume,
V
and
0
otherwise
(1.08)
If the argument of a Diract delta function is a function of x,
δ
(
g
(
x
)) =
X
i
δ
(
x
−
x
i
)

dg
dx

x
=
x
i
where
g
(
x
i
)=0
denote simple zeros of
g
(
x
)
(1.09)
1
The Dirac delta function is de
f
ned by the condition that
]
b
a
f
±
x
0
²
δ
±
x
−
x
0
²
dx
0
=
f
(
x
)
(1.1)
if
and
]
b
a
f
±
x
0
²
δ
±
x
−
x
0
²
dx
0
=0
(1.2)
if
x<a
or
x>b
.
If
x
=
a
or
x
=
b
the integral is not well de
f
ned.
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 Spring '10
 G
 Electrostatics

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