Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
4, N
o
. 2
,
A
p
r
il
201
4, p
p
.
23
1
~
23
6
I
S
SN
: 208
8-8
7
0
8
2
31
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Interpretation of Modified
Electromagnetic Theory and
Maxwell
'
s Equati
ons on t
h
e
Basis of Charge Variation
Asif Ali
Lag
h
a
r
i
Departem
ent
of
Ele
c
troni
cs
Eng
i
neering
,
M
e
hr
an
Un
iversity
of Engineer
ing
an
d Techno
log
y
,
Pak
i
stan
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Dec 13, 2013
Rev
i
sed
Jan 27, 201
4
Accepte
d
Fe
b 6, 2014
Ele
c
trom
agnet
i
c
waves
are th
e
anal
yt
ic
al solutions of Maxwell's equations
that r
e
pres
en
t
one of th
e m
o
s
t
eleg
ant
and
concis
e w
a
y
s
t
o
s
t
ate
the
fundamentals of
electricit
y
and magnetism. From
them one can develop most
of the working
r
e
la
tionships in
t
h
e el
ec
tric
and
m
a
gnetic
fields. Considering
deepl
y
the
effe
ct
of charg
e
v
a
ria
t
ion
in Maxwell’
s equations for
time var
y
ing
electric and m
a
gnetic fields
of ch
arges
in
moving inertial frame, th
e
magnitude of
ch
arge p
a
rticles v
a
r
y
a
ccord
ing to
Asif’s equation
of ch
arg
e
variation. Consequently
th
e Maxwell’s e
quations give differ
e
nt r
e
sults to an
obs
erver m
eas
ur
ing at r
e
s
t
.
This
res
earch
paper
explain
e
d the
effect of ch
arge
varia
tion in
C
l
assica
l El
ec
tro
m
a
gnetic theor
y
, Maxwell’s equations,
Coulum
b’s
law, Lorentz forc
e law when we are referring to
an
y
in
erti
al
frame.
Keyword:
C
oul
om
b’s f
o
r
c
e
Electric field
Electrom
a
gnetic force
El
ect
rom
a
gnet
i
c
wa
ves
Max
w
ell’s equatio
n
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Asif Ali
Lag
h
a
ri,
Depa
rtem
ent of Electr
oni
cs
E
ngi
neeri
n
g
,
M
e
hra
n
Uni
v
er
si
t
y
of E
n
gi
nee
r
i
n
g a
n
d
Tech
n
o
l
o
gy
, Jam
s
ho
r
o
,
Paki
st
a
n
Em
a
il: asifali_
l
e
g
h
a
ri@yahoo
.co
m
1.
INTRODUCTION
Max
w
ell’s equatio
n
are a set o
f
p
a
rtial d
i
fferen
tial
eq
u
a
ti
o
n
s
th
at, tog
e
t
h
er with
th
e Lo
ren
t
z force
law, form
the
foundation of classical
electr
ody
nam
i
cs, cla
ssical optics, a
nd electric circ
uits. These fiel
ds in
turn underlie
m
odern electri
cal and
communications te
c
h
nol
ogi
es
. M
a
x
w
el
l
'
s equat
i
o
n
s
desc
ri
be
h
o
w
el
ect
ri
c
and m
a
gnetic fields are
ge
ne
rated a
n
d altered
by each
other and
by c
h
arges a
n
d
curre
nts. They a
r
e
na
m
e
d
aft
e
r t
h
e
Scot
t
i
sh
phy
si
ci
st
an
d m
a
t
h
em
ati
c
ian Jam
e
s C
l
erk M
a
x
w
el
l
w
h
o p
u
b
l
i
s
he
d a
n
earl
y
fo
rm
of
t
hose
equat
i
o
ns
bet
w
een 1
8
61 a
nd
18
6
2
[
1
]
.
The
"m
i
c
rosco
p
i
c
" set o
f
Maxwell's eq
u
a
tio
n
s
uses to
tal ch
arge and
t
o
t
a
l
curre
nt
wi
t
h
o
u
t
co
nsi
d
eri
ng c
h
ar
ge
vari
at
i
on
wh
en cha
r
ges a
r
e
m
ovi
ng
wi
t
h
hi
g
h
s
p
eed
due t
o
relativ
istic effect o
n
ch
arg
e
[2].
C
onse
q
uent
l
y
t
h
ey
gi
ves di
f
f
ere
n
t
re
sul
t
whe
n
ch
arg
e
s are i
n
m
o
v
i
n
g
in
ertial frame. In th
is
pers
pect
i
v
e,
w
e
ha
ve i
nvest
i
g
at
ed t
h
e c
r
eat
i
on
o
f
el
ect
r
o
m
a
gnet
i
c
wa
ve
by
m
ovi
n
g
p
o
i
nt
cha
r
ge
o
r
b
y
t
i
m
e
varying electri
c and m
a
gnetic
field i
n
the
presence
of cha
r
ges at
rest
,
uni
form
m
o
tion or in accelerate
d
fram
e
.
The m
a
i
n
and im
port
a
nt
co
ns
eque
nce
of M
odi
fi
ed el
ect
ro
mag
n
e
tic th
eo
ry is th
at
it als
o
pred
icts th
e
ch
arg
e
distorts s
p
ace-t
i
m
e
curvature
as Einstein explained m
a
ss
dis
t
orts space
-time curvat
ure
in
his gene
ral the
o
ry of
relativ
ity [3
].
There
are
m
a
in
two post
ulate of
m
odi
fi
ed el
e
c
t
r
om
agnet
i
c
t
h
eo
ry
.
1.
Ratio
of a
n
y c
h
arge
d
body
with the m
a
ss “m
” and c
h
a
r
ge
“e” is constant
for all obse
rve
r
s
rega
rdl
e
ss
t
h
ei
r
st
at
e of
m
o
t
i
on.
2.
Maxwe
ll’s Ele
c
tr
om
agne
ti
c
wave
e
quati
o
ns o
f p
o
i
n
t
ch
arge
wi
t
hou
t
considering cha
r
g
e
variation are not v
a
li
d
fo
r
moving fr
a
me
with respe
c
t
to res
t o
bs
e
r
v
e
r
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 2, A
p
ri
l
20
14
:
23
1 – 2
3
6
23
2
2.
ORI
G
IN
OF
CO
ULO
M
B’
S FO
RCE
It
f
o
l
l
o
w
s
f
r
o
m
coul
om
b’s l
a
w t
h
at
el
ect
r
o
st
at
i
c
force
bet
w
een
t
w
o c
h
ar
ges i
s
di
rect
l
y
pr
o
p
o
r
t
i
onal
to
th
e
pr
odu
ct
o
f
two
ch
arg
e
s
an
d inv
e
r
s
ely
pr
opo
r
tion
a
l
to the s
q
uare
of the distance
bet
w
een them
given
by
F
(2
.1
)
Whe
r
e ‘k’ is the coulom
b’s
constant.
K
πε
. From Max
w
ell’s eq
u
a
tion
,
sub
s
titu
tin
g
th
e v
a
lu
e o
f
ε
in
ab
ov
e equ
a
tion.
F
μ
c
π
Let u
s
assu
m
e
th
at
q
is th
e po
i
n
t ch
arg
e
an
d
q
is fix
e
d
ch
arg
e
.
Hen
c
e abov
e eq
u
a
tion
can
b
e
written
as.
F
μ
π
(2
.2
)
The factor
q
c
ap
peari
n
g
i
n
t
h
e
ab
ov
e e
quat
i
on
i
s
act
u
a
l
c
a
use
of
c
oul
o
m
b’s fo
rce a
n
d i
s
resp
o
n
sible fo
r
c
h
ar
ge disto
r
ts
space
-tim
e curvature as
E
i
ns
tein
propos
ed that
factor
mc
appeari
n
g in
Newt
o
n
’s
un
iversal law of
grav
itatio
n wh
i
c
h
was
fu
rther exp
l
ain
e
d
b
y
Ein
s
tein in his g
e
n
e
ral t
h
eo
ry
of
relativity cause
s m
a
ss distorts
space- tim
e
curvature.
3.
MO
DIFI
CAT
ION
IN
LOR
E
NTZ
FORC
E LAW
It
is g
e
n
e
rally ex
p
ected
fro
m
in
tu
ition
t
h
at th
e
electro
m
a
g
n
e
tic fo
rce ex
erted
on
a ch
arg
e
d
p
a
rticle
sh
ou
l
d
h
e
invarian
t as ob
serv
ed
in
d
i
fferen
t
in
ertia
l fra
m
e
s. In term
s of electric an
d m
a
gnetic fields the
el
ect
rom
a
gnet
i
c
fo
rce exe
r
t
e
d
on a
part
i
c
l
e
of c
h
ar
ge
q an
d vel
o
ci
t
y
v i
s
gi
ve
n by
t
h
e
f
a
m
ous L
o
re
nt
z
fo
rce
law as F = q(E + v x B). By
resorting to the Lore
ntz
trans
f
orm
a
tion of space and ti
m
e
,
it is known that the
two
field
s
tog
e
th
er with
th
e velo
city
transform in such a
way that the Lorentz force is
exactly
identical to that
gi
ve
n
by
t
h
e t
r
ansf
o
r
m
a
ti
on o
f
t
h
e t
i
m
e rat
e
of
cha
n
ge o
f
ki
nem
a
t
i
c
m
o
m
e
nt
um
. That
i
s
, t
h
e L
o
re
nt
z
fo
rc
e
law is Loren
t
z in
v
a
rian
t.
Furth
e
rm
o
r
e, th
e
wav
e
equ
a
tions o
f
po
ten
tial, th
e con
tinu
ity eq
u
a
tion
,
and
th
e
Lore
ntz ga
uge, whic
h are
fundam
enta
l
equa
t
i
ons i
n
el
ect
r
o
m
a
gnet
i
c
, can
be sh
o
w
n t
o
b
e
Lore
nt
z i
n
va
ri
ant
.
The
n
, M
a
xwel
l
'
s equat
i
o
ns ca
n
be s
h
ow
n t
o
be i
n
va
ri
ant
un
der
t
h
e L
o
rent
z
t
r
ans
f
orm
a
t
i
on [
4
]
.
C
onse
q
uent
l
y
Lore
nt
z f
o
rce
equat
i
o
n
doe
s
n
’t
i
n
vol
ve
charge
variation whe
n
the c
h
arge
particle
m
oves i
n
uni
fo
rm
el
ect
ri
c and
m
a
gnet
i
c
fi
el
d
an
d i
t
i
s
Lo
re
nt
z i
n
vari
a
n
t
[
5
]
.
Acco
rdi
n
g t
o
m
oder
n
st
rat
e
gy
o
f
relativ
istic ch
arg
e
, Lo
ren
t
z force eq
u
a
tion
m
u
st invo
lv
e
rel
a
tiv
istic ch
arg
e
if ch
arg
e
s are
in
m
o
v
i
n
g
frame.
Let
us as
sum
e
t
h
at
cha
r
ge i
s
m
ovi
ng al
on
g
y
-
axi
s
i
n
uniform
electric and m
a
gnetic field along
x-
axi
s
a
nd y
-
axi
s
res
p
ect
i
v
el
y
as sh
o
w
n
i
n
pr
evi
o
us m
e
nt
i
oned
FI
G
u
re
(1
).
Lo
rent
z
f
o
rce
eq
uat
i
o
n
i
s
gi
ven
b
y
th
e fo
llowing
eq
u
a
tion
.
Fq
E
VB
(3
.1
)
Whe
r
e
′q
′
i
s
rest
char
ge. I
f
t
h
e
vel
o
ci
t
y
of c
h
ar
ge i
s
ve
ry
hi
g
h
as com
p
ared t
o
s
p
ee
d
of l
i
g
ht
f
o
r re
st
obs
er
ver
,
S
o
c
onsi
d
eri
n
g
t
h
e
effect
of
cha
r
g
e
va
ri
at
i
on,
re
p
l
aci
ng
‘
q
’ b
y
t
h
e relativ
istic ch
arg
e
′q′
.
There
f
ore m
o
d
i
fi
ed e
quat
i
o
n
of
Lo
re
nt
z f
o
rc
e i
s
;
F
q
EVB
(3
.2
)
4.
LORENTZ
F
O
R
C
E O
N
C
O
NTI
N
U
O
U
S
CH
A
R
GE
DI
STRIBUTI
O
N
For
t
h
e c
o
nt
i
n
uo
us c
h
a
r
ge
di
st
ri
but
i
o
n,
su
p
pos
e t
h
at
c
h
ar
ges a
r
e di
st
ri
b
u
t
e
d
o
v
er sm
al
l
vol
um
e i
n
m
ovi
ng wi
t
h
c
onst
a
nt
vel
o
ci
t
y
i
n
el
ect
rom
a
gnet
i
c
fi
el
d
,
hence a ve
ry s
m
all force wil
l
act on a very s
m
all
am
ount
o
f
c
h
ar
ge wi
t
h
respect
t
o
rest
o
b
se
rve
r
, by
t
a
ki
ng
di
f
f
ere
n
t
i
a
l
;
df
d
q
γ
v
B
E
(4
.1
)
Whe
r
e
′ γ′
i
s
Lo
re
nt
z fact
or
desc
ri
be
d i
n
Lo
re
nt
z t
r
ans
f
orm
a
t
i
o
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
In
terp
reta
tion
o
f
Mo
d
ified
Electro
mag
n
e
tic
Th
eo
ry
an
d Ma
xwell'
s Eq
ua
tio
n
s
on
…
(Asif Ali Lagh
a
r
i)
23
3
γ
1
v
c
Di
vi
di
n
g
bot
h
si
des
of
eq
uat
i
o
n
4
.
1
by
a
sm
al
l
vol
um
e ‘v
’
df
dv
dq
dv
γ
v
B
E
Or
df
γ
ρv
B
E
d
v
df
γ
B
E
d
v
Because
ρv
curr
ent density
d
e
scrib
e
d in
Maxwell’s equ
a
tion
.
To
find
t
o
tal fo
rce acting
o
n
total charge e
n
closed by t
h
e s
u
rface
ds
in
the
vo
lu
m
e
dv
can be achi
e
ve
d by
t
a
ki
n
g
vol
um
e
i
n
t
e
gral
as.
F
γ
B
E
d
v
(4
.2
)
5.
ELECTROMAGNETIC FORCE
BETWEEN TWO CHARGE
S IN
MOVING INE
R
TIAL
FRAME
Consi
d
er two
sam
e
charges
are in rest
fra
me S,
on
ly rep
u
l
si
v
e
electrostatic fo
rce exists b
e
tween
th
em
as sh
own in
th
e fi
g
u
re
1
.
Belo
w:
Fi
gu
re
1.
C
o
ul
om
b’s f
o
rce
be
t
w
een t
w
o
i
d
e
n
t
i
cal
charge
s i
n
rest
fram
e
S m
easure
d
by
rest
o
b
ser
v
e
r
If t
w
o s
a
m
e
charges
are i
n
m
oving ine
r
tial fra
m
e S’, eac
h
c
h
ar
ge e
xpe
ri
en
ces re
pul
si
ve
e
l
ectrostatic force, but
due t
o
t
h
e
du
m
bbel
l
shape m
a
gnet
i
c
fi
el
d of t
w
o c
h
ar
ge
s pa
rallel to
x
-
z p
l
an
e in
S’
fram
e, they will attract
each othe
r with respect
t
o
res
t
observ
er as s
h
own in Figure
2. Below:
Fi
gu
re
2.
M
a
g
n
et
i
c
f
o
rce
F
and
relativ
istic co
u
l
o
m
b
’
s fo
rce
F
m
easured by rest
obse
rve
r
when
two
identical cha
r
ges are i
n
m
oving ine
r
tial fram
e
S’
Here
we are c
once
r
ned
wi
t
h
S’ fram
e
, t
o
fi
nd t
h
e n
e
t
force b
e
tween
two
ch
arg
e
s, we
co
nsid
er
relativ
isti
c
char
ge i
n
c
oul
om
b’s l
a
w a
s
g
i
ven i
n
t
h
e
f
o
l
l
o
wi
ng
eq
uat
i
o
n.
F
q
E
(5
.1
)
M
a
gnet
i
c
fi
el
d
of
a rel
a
t
i
v
i
s
t
i
c
cha
r
ge
i
s
gi
ve
n
by
.
B
qv
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 2, A
p
ri
l
20
14
:
23
1 – 2
3
6
23
4
For
ce o
n
p
o
i
n
t
char
ge d
u
e t
o
m
a
gnet
i
c
fi
el
d i
s
gi
ve
n by
t
h
e cr
oss
pr
od
u
c
t
of m
a
gnet
i
c
fi
el
d an
d vel
o
ci
t
y
as
fo
llow.
F
q
VB
Taking ratio be
tween
electric and
m
a
gnetic
fo
rce
,
we ar
rive
at the
follo
win
g
result.
(5.2)
M
a
gni
t
u
de o
f
char
ge i
n
m
ovi
ng f
r
am
e
m
e
asure
d
by
rest
obse
r
ve
r i
s
gi
ven by
Asi
f’s
equat
i
o
n of c
h
ar
g
e
vari
at
i
o
n.
q
By re arrang
ing
th
is equ
a
tion
,
we
wan
t
to
driv
e th
e v
a
l
u
e
o
f
as
f
o
llow
s
.
1
(5.3)
Sub
s
titu
tin
g equ
a
tio
n 5.3 in
5
.
2
.
We
h
a
v
e
1
After
so
lv
ing
,
we g
e
t
th
e fo
llo
wi
n
g
equ
a
tion
.
qE
q
vB
q
E
(5.4)
Thi
s
e
q
uat
i
o
n
sh
o
w
s t
h
at
c
oul
om
b’s f
o
r
c
e (el
ect
ric
force)
betwee
n t
w
o cha
r
ges i
n
S’
fram
e
measured by
rest observer is
the
s
u
m
of t
h
e co
ul
om
bs f
o
rce
bet
w
ee
n t
w
o
cha
r
ges i
n
S f
r
am
e and m
a
gnet
i
c
force
of two c
h
arges
in
S’ fra
m
e.
An
d
m
a
gnet
i
c
fo
rce bet
w
ee
n t
w
o
c
h
a
r
ges
i
n
S’ fram
e
m
easure
d
by
re
st
o
b
s
erve
r
i
s
gi
ve
n by
.
qv
B
q
E
q
E
(5.5)
Eq
uat
i
o
n
5
.
5
d
e
scri
be t
h
at
t
h
e m
a
gnet
i
c
for
ce bet
w
ee
n t
w
o c
h
ar
ges i
n
S
’
f
r
am
e i
s
act
ual
l
y
t
h
e di
ffe
re
nce
of
th
e relativ
istic
co
u
l
o
m
b
’
s fo
rce in
S’
fram
e
an
d th
e
rest co
u
l
o
m
b
’
s fo
rce in
S fram
e..
From
eq
uat
i
o
n
5.
2. Wh
en
vc
, the
n
1
W
h
en
ch
ar
g
e
s
a
r
e
in mo
v
i
n
g
f
r
a
me
,
th
e
ch
ar
g
e
on
th
e part
icle
varies with respect
t
o
re
st
observer.
B
y
appl
y
i
n
g
F
a
raday
’
s l
a
w
,
o
n
e ca
n
de
duct
t
h
at
t
h
e
cha
n
ge
in the
electric
charge ca
uses
change
in t
h
e e
l
ectric
fi
el
d a
nd
he
nc
e fo
r
rest
o
b
se
rve
r
, m
a
gnet
i
c
fi
el
d i
s
ge
ner
a
t
e
d ar
o
u
n
d
ea
ch c
h
ar
ge i
n
m
ovi
ng
fram
e
. That
’
s
why
t
h
e
rest
o
b
s
erve
r e
x
peri
en
ce m
a
gnet
i
c
fo
rce
bet
w
ee
n t
w
o c
h
ar
ges
d
u
ri
ng
m
o
t
i
on.
6.
MA
X
W
ELL’S EQUATION IN
MO
VING INE
R
TIAL FRAME
6.
1.
Di
ver
g
e
n
ce a
nd
Curl
of
M
a
gneti
c
F
i
el
d of
a
Poi
n
t
Ch
ar
ge i
n
M
ovi
n
g
Frame
In
el
ect
rom
a
gn
et
i
c
wave
t
h
e
o
ry
, t
h
e
di
ver
g
e
n
ce
of
a m
a
gnetic field is al
ways e
qual t
o
zero beca
use
t
h
ere i
s
n
o
m
ono
p
o
l
e
m
a
gne
t
[6]
.
B
u
t
i
n
ca
se o
f
a
si
n
g
l
e
c
h
ar
ge i
n
m
ovi
ng
i
n
e
r
t
i
a
l
S’
f
r
am
e al
ong y
-
a
x
i
s
as
sh
own
in fo
llowing
Fi
g
u
re
3
.
Belo
w:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
In
terp
reta
tion
o
f
Mo
d
ified
Electro
mag
n
e
tic
Th
eo
ry
an
d Ma
xwell'
s Eq
ua
tio
n
s
on
…
(Asif Ali Lagh
a
r
i)
23
5
Fi
gu
re 3.
C
h
a
r
ge
q
m
ovi
ng i
n
m
ovi
ng i
n
e
r
t
i
a
l
fram
e
S’
let u
s
assu
m
e
t
h
at it p
r
od
uce
mag
n
e
tic field
in
con
c
en
tric
circles aro
und
it with
resp
ect to
rest ob
serv
er. The
di
ve
rge
n
ce
of
t
h
at
m
a
gnet
i
c
fi
el
d m
easured
b
y
rest
o
b
se
rve
r
i
s
gi
ve
n
by
.
.
B
lim
⇢
∮
B.
ds
v
Sub
s
titu
tin
g
B
qv
,
hence
.
B
μ
ρv
.
B
μ
(6
.1
)
Here is t
h
e rel
a
tiv
istic cu
rrent d
e
n
s
ity,
wh
ich
d
e
p
e
nd
s
u
pon
b
o
t
h
“Relativ
istic ch
arg
e
den
s
ity an
d
v
e
lo
city”. No
t
e
th
at Relativ
istic ch
arg
e
d
e
n
s
ity v
a
ries w
ith
the cha
n
ge in t
h
e electri
c cha
r
ge
or cha
n
ge i
n
the
vol
um
e or
bot
h
.
The c
u
rl
o
f
m
a
gnet
i
c
fi
el
d
of
a m
ovi
ng c
h
a
r
ge
poi
nt
cha
r
g
e
i
s
gi
ven
by
.
B
l
i
m
⇢
∮
B.
da
A
Sub
s
titu
tin
g
B
qv
,
For t
h
e c
ont
i
n
uo
us c
h
ar
ge
di
st
ri
but
i
o
n
(co
n
f
i
g
urat
i
o
n) i
n
t
h
e sm
al
l
regi
on o
f
are
a
A
→
0
,
t
h
e ab
ove equ
a
tio
n beco
m
e
s.
B
μ
σv
(6
.2
)
6.2.
Electric Flux of
Rel
a
ti
vistic Electric
Field
From
M
a
xwel
l
’
s p
o
i
n
t
cha
r
g
e
equat
i
o
n o
r
Gaus
ses
law,
o
n
e
can
d
e
du
ct th
e to
tal elec
tric flu
x
o
f
charge
out of
the close
d
s
u
rface at rest
which is
, su
pp
ose
we
wa
nt
t
o
dem
onstrate el
ectric flux
when
char
ge i
s
i
n
m
ovi
ng i
n
ert
i
a
l
f
r
am
e S’ sho
w
n
i
n
ab
ove F
I
Gu
re 3.
It
f
o
l
l
o
ws
fr
om
equat
i
o
n
5.
5. B
y
el
im
i
n
at
i
n
g
relativ
istic ch
arg
e
and
tak
i
ng
d
i
v
e
rg
en
ce t
o
g
e
t to
tal relativistic electric flu
x
of ch
arg
e
.
.
E
lim
⇢
∯
ds
∯
ds
γ
∯
ds
(6.3)
Usi
n
g
di
ve
rge
n
ce t
h
eo
rem
t
o
get electric fl
ux.
We
ha
ve
∯
E ds
μ
qv
γ
(6.4)
∯
E ds
γ
(6.5)
Pu
ttin
g v
a
l
u
e
of
g
a
mma an
d
after so
lv
i
n
g,
we g
e
t
∯
E ds
(6.6)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 2, A
p
ri
l
20
14
:
23
1 – 2
3
6
23
6
Th
is is th
e poin
t
form
o
f
M
a
x
w
ell’s equ
a
t
i
o
n
o
r
Gau
sses law in
i
n
tegral fo
rm
.
W
h
ere
′
ρ
′
is th
e
relativ
istic ch
arg
e
d
e
n
s
ity, wh
ich
v
a
ries wit
h
th
e v
e
l
o
city o
f
ch
arg
e
s d
i
st
ribu
ted
i
n
defi
n
ite vo
lu
m
e
d
u
e
to
ch
arg
e
v
a
riation
an
d leng
th
con
t
raction
acco
r
d
i
ng
to sp
ecial relativ
ity.
7.
CO
NCL
USI
O
N
In t
h
i
s
st
udy
,
we ha
ve i
n
v
e
st
i
g
at
ed a M
odi
fi
ed
Electro
m
a
g
n
e
tic th
eo
ry
that m
odifies quant
u
m
electrody
nam
i
cs and classical electro
m
a
g
n
e
tic th
eo
ry. Th
is led
u
s
to
m
o
d
i
fy to
Lo
ren
t
z fo
rce law, Maxwell’s
eq
u
a
tion
s
of po
in
t ch
arg
e
th
at g
i
v
e
s co
rrect
resu
lt in
rest fra
m
e
will g
i
v
e
d
i
fferen
t
resu
lt in
m
o
v
i
n
g
frames as
di
scuss
e
d i
n
t
h
i
s
pape
r. T
h
e
p
a
per t
h
en
di
sc
u
ssed a
b
o
u
t
t
h
e
ori
g
i
n
of el
ect
r
o
st
at
i
c
and el
e
c
t
r
om
agnet
i
c
f
o
rces
bet
w
ee
n c
h
ar
g
e
s.
Th
e in
teresti
n
g
resu
lt fou
nd fro
m
cou
l
o
m
b
’
s law is th
e fact
o
r
qc
which is t
h
e
actual
cause
of
attraction a
nd
repulsion
between two c
h
arges in s
p
ace as
mc
is th
e actu
a
l cau
s
e
o
f
g
r
av
itatio
n
a
l fo
rce t
h
at
appea
r
s i
n
s
o
m
e
Ei
nst
e
i
n
’s
fi
el
d eq
uat
i
o
ns
t
h
at
desc
ri
be
m
a
ss di
st
ort
s
space
-t
im
e cur
v
at
u
r
e i
n
t
h
eory
of
Gen
e
ral relativity.
ACKNOWLE
DGE
M
ENTS
I am
deepl
y
inde
bt
ed t
o
D
r
.
Am
eer Al
i
Lagha
ri
, E
n
g
.
M
a
ji
d
Al
i
Lag
h
ari
a
nd Za
hi
d
Al
i
Lag
h
ari
wh
ose s
u
ggest
i
ons a
nd c
o
nt
i
n
u
o
u
s enc
o
ura
g
em
ent
rel
a
t
e
d t
o
t
h
e a
b
o
v
e
wo
rk
hel
p
e
d
m
e
imm
e
nsel
y
t
o
investigate a
n
d
fu
rthe
r re
fi
n
e
th
e conj
ectu
r
e
p
r
esen
ted in
this p
a
p
e
r. Th
e
d
i
scu
ssi
on
s I
had
with th
em
wh
il
e
th
ey were rev
i
ewing
th
e work
led
m
e
to
d
e
v
e
lop
a fo
rm
id
ab
le set o
f
m
a
t
h
em
at
ical
m
o
d
e
ls to
b
e
tter rep
r
esen
t
th
e said conj
ectu
re.
REFERE
NC
ES
[1]
JC Maxwell. “A D
y
namical Th
eor
y
of
th
e El
e
c
tr
om
agnetic
Field
”
.
in W
.
D. Niv
e
n (ed
.
),
The S
c
ientific Papers of
James
Cler
k
Ma
xwell
, N
e
w York, Dover
.
1865
.
[2]
Laghar
i
, Asif Ali. “Asif’s Equation of Ch
arge Variation
and Sp
ec
ial Re
lat
i
vit
y
”
.
Journal of Applied Physics
. 2013;
4(3): 01-04.
[3]
Wald, Rob
e
rt M
.
“General relativ
i
ty
”.
Univ
ersity
of Chicago pr
ess. 2010
.
[4]
P Lorrain
and
DR Corsou. “Electroma
gnetic Field
s
and Waves”. S
a
n Franci
sco: Freeman. chs. 5
an
d 6, 1972.
[5]
CC Su. “Modifications of
th
e Lor
e
ntz force l
a
w in
vari
an
t under Galilean
transform
a
tions”. in
this Di
gest.
[6]
RE E
llio
tt. “Electrom
a
gneti
cs, Hi
stor
y
,
Th
eor
y
, and Applications”. (classic reissue). 1993
.
BI
O
G
R
A
P
HY
OF
A
U
T
HO
R
Asif Ali Laghari, he is undergrad
u
ated studen
t
of
Electronics Engineering in Mehr
an University
of Engin
eerin
g and Technolog
y
,
Jamshoro Pakistan.
He has been do
ing research
in
the fields of
El
ectrom
a
gne
tic
t
h
eor
y
and R
a
diating S
y
stems,
theor
y
of Speci
a
l
and Genera
l re
lativ
it
y, Quan
tu
m
Electrod
y
n
a
m
i
cs, Math
em
atic
al m
odeling of
phy
s
ica
l
s
y
ste
m
s from la
st 5-y
e
a
r
s.
He is the first who introduced
th
e conc
ept of ch
a
r
ge
variation in
his first research
paper that has
been pub
lished
b
y
Intern
ation
a
l
Organizati
on
of
Scientif
ic Resear
ch in
july
2013.
Em
ail:
asifa
li_l
e
ghari@
y
ahoo
.co
m
Evaluation Warning : The document was created with Spire.PDF for Python.