### Asymmetries in Maxwell`s Electrodynamics

```Asymmetries in Maxwell’s
Electrodynamics
W. J. Wilson
Department of Engineering & Physics
University of Central Oklahoma
Edmond, OK 73034
Web: www.physics.uco.edu/wwilson
Email: [email protected]
ON THE ELECTRODYNAMICS OF MOVING BODIES
By A. EINSTEIN
“It is known that Maxwell's electrodynamics - as usually understood at the
present time - when applied to moving bodies, leads to asymmetries which
do not appear to be inherent in the phenomena.
Take, for example, the reciprocal electrodynamic action of a magnet and a
conductor. The observable phenomenon here depends only on the relative
motion of the conductor and the magnet, whereas the customary view
draws a sharp distinction between the two cases in which either the one or
the other of these bodies is in motion. For if the magnet is in motion and the
conductor at rest, there arises in the neighborhood of the magnet an electric
field with a certain definite energy, producing a current at the places where
parts of the conductor are situated. But if the magnet is stationary and the
conductor in motion, no electric field arises in the neighborhood of the
magnet. In the conductor, however, we find an electromotive force, to which
in itself there is no corresponding energy, but which gives rise - assuming
equality of relative motion in the two cases discussed - to electric currents of
the same path and intensity as those produced by the electric forces in the
former case.”
Engineering & Physics
2/12
Motionless Magnet/Galvanometer
When there is no relative
motion, there is no current
indicated by the
galvanometer:
B = Const.  ∂B/∂t = 0  E = 0.
Engineering & Physics
3/12
Moving Magnet
Moving bar magnet generates time
change of magnetic field.
B
0
t
Variable magnetic field, in turn,
generates induced electric field
according to the Maxwell's
equation:
B
E  
t
inside the coil that moves free electrons of the coil producing
induced current through the galvanometer, whose needle is moving
left and right.
Engineering & Physics
4/12
Moving Coil
When bar magnet stands still, there is
no time variation of magnetic field
and there is no induced electric field,
according to Maxwell's equations:
B = Const. 
This is obviously
inconsistent being that
the two last cases are
equivalent and they
should be described by
same fundamental
equations.
Engineering & Physics
∂B/∂t = 0
E=0
However, the galvanometer needle is
equally moving left and right. This
phenomenon is now explained by the
Lorentz's force:
F=qvxB
that acts on free electrons of the coil.
5/12
Scientists were aware of this asymmetry of Maxwell's equations at
the beginning of the 20th century. Hertz proposed a variant of
Maxwell's equations (H.R. Hertz, Electric Waves), that differed slightly
from Maxwell's original form (partial derivatives were substituted by
total ones), and that did not "lead to asymmetries" as mentioned by
Einstein and are invariant to Galilean transformations.
  E  4
  E  4
B  0
B  0
1 B
E  
c t
4
1 E
B 
J
c
c t
1 dB
E  
c dt
4
1 dE
B 
J
c
c dt
Maxwell Electrodynamics
Engineering & Physics
Hertz Electrodynamics
6/12
Moving Frames
Let's now append a truck to the previous
experiment. Let's take into consideration
that part of the case when bar magnet
SN moves toward the coil, say by the
velocity v1 = 3m/s. For the blue observer
O1 on the illustration, the galvanometer
needle turns due to induced electric
field that pushes electrons in the coil.
Let's now take that this observer with all
his gadgets is inside a truck that moves
by the opposite velocity
v2 = -v1 = -3m/s.
For another observer O2 outside the
truck, the bar magnet is standing still,
while the coil is moving toward the bar.
For that observer, there is no electric
field and the galvanometer needle turns
due to Lorentz force.
Engineering & Physics
In other words, the same physical event
is described by two different equations,
i.e. by two different physical laws.
Namely, in the first case by the low of
electromagnetic induction, and in the
second case by the low of Lorentz force.
Thus, it is violated one of the
fundamental postulate of physics:
The laws of physics are the same in all
inertial frames!
7/12
Moving Truck and Special Relativity
Using Special Relativity, we'll start
with green observer O2, since E2 = 0 in
the coil for him Now get the electric
field in the coil inside the reference
frame of the blue observer O1.
Applying relativity transformations on
E2, we get:
E1 = E2 + v2×B/c,
Since: E2 = 0, we get, finally:
E1 = v2×B/c.
Engineering & Physics
8/12
From these results we can deduce the conclusions
Expressions for relativistic transformation of electric and
magnetic field:
E1 = E2* + v2×B/c, B1 = B2* − v2×E/c,
it may be seen that they comprise just those parts missing
from Maxwell's equations that are included in Hertzian
equations:
dB
B
 B

 
  v   B  
    v  B
dt
t
 t

1 dE
1  E
1 E 1

  B  0J  2
 0J  2    v    E  0J  2
 2    v  E
c dt
c  t
c t c

E  
If the bar magnet moved by a non-uniform velocity, it
would be inappropriate to use special relativity for it is valid
just for uniform transformations.
Engineering & Physics
9/12
d
Copper Disk
Permanent Magnet
M
Engineering & Physics
the rotation of the
copper disk above the
magnet influences the
generated potential.
The rotation of the
permanent magnet does
not affect the magnitude
of generated potential at
all!
10/12
Action-Reaction Asymmetry
1 q2 v 2  Rˆ
B2 
c
R2
q1q2
 F12  2 2 v1  v 2  Rˆ
cR
1 q1 v1  Rˆ
B1  
c R2
q2 q1
 F21   2 2 v 2  v1  Rˆ
cR
v1
F12  q1  B2 ,
c

v2
F21  q2  B1 ,
c



v1
1
R
2
F21   F21
v2
Engineering & Physics
11/12
References
H.R. Hertz, Electric Waves: Being researches on the Propagation
of Electric Action with Finite Velocity through Space, Cornell
University Press (1893) ISBN: 1429740361
Petrovic Branko, “Lorentz's Force”,
http://www.angelfire.com/sc3/elmag/files/MaxLor.html
H. Aspden, “Electromagnetic Reaction Paradox” , Lettere Al
Nuovo Cimento 39, 247 (1984).
Engineering & Physics
12/12
References
H.R. Hertz, Electric Waves: Being researches on the Propagation
of Electric Action with Finite Velocity through Space, Cornell
University Press (1893) ISBN: 1429740361
Petrovic Branko, “Lorentz's Force”,
http://www.angelfire.com/sc3/elmag/files/MaxLor.html
H. Aspden, “Electromagnetic Reaction Paradox” , Lettere Al
Nuovo Cimento 39, 247 (1984).
Engineering & Physics
13
```