4.3. ELECTROSTATICS 57
associated with acceleration that drop off as 1=R.
As elements of a Clifford algebra, the field strengths admit geometrical interpretation. e
factor z ^ ˇ in f
ret
represents the plane spanned by the velocity ˇ and the line of observation z.
Similarly, we recognize
z ˇR D ˇ
2
z C ˇ
.
z ˇ
/
D
.
z ^ ˇ
/
ˇ
representing the projection of ˇ onto the z ˇ plane, and so we have
f
ret
D
e
4
'
.
R
/
z ^ ˇ
R
3
ret
D
c
5
c
f
ret
ˇ
for the retarded fields. Similarly, using
a
.
b ^ c ^ d
/
D
.
a b
/
c ^ d
.
a c
/
b ^ d C
.
a d
/
b ^ c
and z
2
D 0, we see that
z ^ W D
z ^ ˇ ^
P
ˇ
z
in f
rad
represent the projection of z onto the volume spanned by z, ˇ, and
P
ˇ. Similarly,
ret
is
proportional to zQ D .
P
ˇ z/z=c, the projection of z onto the acceleration
P
ˇ.
4.3 ELECTROSTATICS
e covariant equivalent of a spatially static charge is a uniformly evolving event
X
.
/
D u D
u
0
; u
with constant timelike velocity
P
X D u D ˇc, which in its rest frame simply advances along the
time axis as t D ˇ
0
. As a result, and given the geometrical interpretation of the Clifford forms,
the field strengths are essentially kinematical in structure.
Writing the timelike velocity ˇ in terms of the unit vector
O
ˇ
ˇ
2
< 0 ˇ D
j
ˇ
j
O
ˇ
O
ˇ
2
D 1 ˇ
2
D
j
ˇ
j
2
the observation line z can be separated into components
z
k
D
O
ˇ
O
ˇ z
z
?
D z C
O
ˇ
O
ˇ z
which satisfy
z
2
k
D
O
ˇ
2
O
ˇ z
2
D
O
ˇ z
2
z
2
?
D z
2
C 2
O
ˇ z
2
O
ˇ z
2
D
O
ˇ z
2
D z
2
k
.
ˇ z
/
2
D
j
ˇ
j
2
O
ˇ z
2
D
j
ˇ
j
2
z
2
k
:
(4.16)
58 4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS
e condition of retarded causality
z
2
D c
2
2
R
ˇ
2
2c
R
ˇ x C x
2
D 0
relates the field to the location of the event along the backward lightcone of the observation
point. is implicit choice of
R
and its gradient
0 D d.z
2
/ D 2
c
2
R
d
R
ˇ
2
c
R
ˇ cd
R
ˇ x C x
D 2
Œ
cRd
R
C z
lead to the following expressions:
d
R
D
z
cR
.
ˇ d
/
R
D
ˇ z
ˇ z
D 1
.
z d
/
R
D
z
2
cR
D 0
d
.
ˇ z
/
D d
ˇ x ˇ
2
R
D
.
ˇ z
/
ˇ ˇ
2
z
ˇ z
D
ˇ
ˇ
ˇ
2
ˇ
ˇ
z
?
cR
d
1
R
n
D .1/
n
n
.
ˇ z
/
ˇ ˇ
2
z
.
ˇ z
/
nC2
D
n
ˇ
ˇ
ˇ
2
ˇ
ˇ
z
?
R
nC2
d z D d
.
x cˇ
R
/
D d x cˇ d
R
D 3
d ^ z D d ^
.
x cˇ
R
/
D cd
R
^ ˇ D
ˇ ^ z
R
d ^ Oz D d ^
z
j
z
j
D
1
j
z
j
ˇ ^ z
R
Oz ^
z
j
z
j
2
D
ˇ ^ Oz
R
:
Using these expressions, the pre-Maxwell equations (4.14) can be easily verified for the case of
a uniform velocity event [5]. For example, recalling '
0
D '"=, the exterior derivative of f
is
d ^ f D
e
4
d ^
'
.
R
/
z ^ ˇ
R
3
ˇ
2
C '
0
.
R
/
z ^ ˇ
cR
2
which produces terms of the type:
d'
.n/
^
.
z ^ ˇ
/
D '
.nC1/
z
cR
^
.
z ^ u
/
D 0
d ^
.
z ^ ˇ
/
D
.
d ^ z
/
^ ˇ D
ˇ ^ z
R
^ ˇ D 0
d
1
R
n
^
.
z ^ u
/
D
"
n
ˇ
ˇ
ˇ
2
ˇ
ˇ
z
?
R
nC2
#
^
.
z
?
^ u
/
D 0
and thus we recover
d ^ f D 0
4.3. ELECTROSTATICS 59
from kinematics.
It is convenient to write the field strengths in 3-vector and scalar form
.e/
i
D f
0i
.b/
i
D "
ij k
f
jk
./
i
D f
5i
0
D f
50
for which the field equations split into four generalizations of the 3-vector Maxwell equations
r e
1
c
5
@
@
0
D
e
c
j
0
'
D e
0
'
r b D 0
r b
1
c
@
@t
e
1
c
5
@
@
D
e
c
j
'
r e C
1
c
@
@t
b D 0
(4.17)
and three new equations for the fields and
0
r C
1
c
@
@t
0
D
e
c
j
5
'
D
ec
5
c
'
r
55
1
c
5
@
@
b D 0
r
0
C
1
c
@
@t
C
55
1
c
5
@
@
e D 0:
(4.18)
Writing d D e
0
@
0
C r and f D e
0
^ e C
1
2
f
jk
e
j
^ e
k
we find that
d ^ f D 0 !
8
ˆ
<
ˆ
:
r b D 0
r e C
1
c
@
@t
b D 0
expressing the absence of electromagnetic monopoles.
In the rest frame of a charged event, we may set
P
t D 1 ! ˇ D e
0
, so for an observation
point
x
D
.
ct;
x
/
z
2
D 0 !
8
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
:
R
D t
jxj
c
R D e
0
.
x c
R
e
0
/
D jxj
z D
.
c
.
t
R
/
; x
/
D R
.
e
0
C Ox
/
and the field strengths reduce to
f .x; / D
e
4
'
.
R
/
e
0
^ Ox
R
2
1 C
"
.
R
/
c
R
D e
0
^ e.x; /
.x; / D
c
5
c
e
4
'
.
R
/
Ox
R
2
C
"
.
R
/
cR
e
0
1 C
c
2
c
2
5
C Ox
:
60 4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS
We thus find that the magnetic field b is zero, while
e D
e
4
'
.
R
/
R
2
'
0
.
R
/
R
Ox D
c
5
c
e (4.19)
and
0
.x; / D
e
4
'
0
.
R
/
R
c
5
c
C
c
c
5
: (4.20)
Because we obtained f .x; / using only the leading term G
Maxwell
in the Green’s function, we
expect errors on the order of the neglected term G
Correlation
. In particular, we notice that
@
@
C
55
c
2
5
@
2
G
Maxwell
D ı
4
.
x
/
ı
.
/
1
2
55
c
2
5
ı.x
2
/ ı
00
./;
where the second term on the right is canceled when G
Correlation
is included in the wave equation.
As a result, calculating
r D
c
5
c
e
4
'
.
R
/
ı
3
.
x
/
'
00
.
R
/
cR
;
where we use r .Ox=R
2
/ D 4ı
3
.x/, and
1
c
@
@t
0
D
e
4
'
00
.
R
/
cR
c
5
c
C
c
c
5
leads to the Gauss law as
1
c
@
@t
0
C r D
c
5
c
e
4
'
.
R
/
ı
3
.
x
/
C
c
c
5
e
4
'
00
.
R
/
cR
exposing an error at the order of ı
00
.
R
/.
We now consider a long straight charged line oriented along the z-axis, with charge per
unit length
e
. In cylindrical coordinates
x D
.
; z
/
D
.
x; y
/
D
O
D
p
x
2
C y
2
the fields and e are found by replacing R D
p
2
C z
2
in (4.19) and (4.20) and integrating
along the z-axis to find
e D
e
4
Z
dz
0
B
B
@
'
t C
.
2
Cz
2
/
1=2
c
.
2
C z
2
/
3=2
'
0
t C
.
2
Cz
2
/
1=2
c
c
.
2
C z
2
/
1
C
C
A
.
O
; z
/
0
D
e
4
c
5
c
Z
dz
'
0
t C
.
2
Cz
2
/
1=2
c
c
.
2
C z
2
/
1=2
:
4.3. ELECTROSTATICS 61
To get a sense of these expressions, we may use (3.15) to approximate '.x/ D ı.x/ which
permits us to easily carry out the z-integration to obtain
e D
e
2
0
B
@
.
t =c
/
c
.
t
/
2
2
=c
2
3=2
ı
.
t =c
/
q
.
t
/
2
2
=c
2
1
C
A
O
which vanishes >
R
D t =c as required for retarded causality. Since
Z
d
'
.
/
D 1
Z
d
'
0
.
/
D 0
the concatenated electric field is found as
E.x/ D
Z
d
e.x; / D
e
4
Z
dz
1
.
2
C
z
2
/
3=2
.
O
; z
/
D
e
2
.
O
; 0
/
in agreement with the standard expression.
To obtain the field of a charged sheet in the x y plane with charge per unit area , it is
convenient to start from the potential from a charged event, and integrating over x and y with
R D
p
x
2
C y
2
C z
2
. us,
a
0
.x; / D
c
4
Z
dx
0
dy
0
'
t C
1
c
q
.
x x
0
/
2
C
.
y y
0
/
2
C z
2
c
q
.
x x
0
/
2
C
.
y y
0
/
2
C z
2
and a
5
.x; / D .c
5
=c/a
0
.x; /. Changing to radial coordinates .x; y/ ! .; / we obtain
a
0
.x; / D
c
4
Z
dd
'
t C
1
c
p
2
C z
2
c
p
2
C z
2
which by change of variable D
1
c
p
2
C z
2
becomes
a
0
.x; / D
c
2
Z
1
j
z
j
=c
'
.
t C
/
d :
We calculate the fields from
e.x; / D ra
0
D
2
'
t C
j
z
j
c
r
j
z
j
D
2
".z/'
t C
j
z
j
c
Oz;
where .x; / D .c
5
=c/e.x; / and
0
D
55
c
5
@
a
0
C
1
c
@
t
a
5
D
1
c
55
c
c
5
c
5
c
@
a
0
D
c
55
c
c
5
c
5
c
'
t C
j
z
j
c
:
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