Download SR Theory of Electrodynamics for Relative Moving Charges

SR Theory of Electrodynamics for
Relative Moving Charges
By James Keele, M.S.E.E.
October 27, 2012
Special Theory of Relativity (SRT)
Basic Postulates
1. Relativity Principle(RP): “all inertial frames
are totally equivalent for the performance of
all physical experiments”
2. “light travels rectilinearly with constant
speed c in vacuum in every inertial frame”
Logic applied to the 2nd Postulate: Constancy of
the Speed of Light.
1
o
If Physicists accept the law, c o    , and believe
o o
that  o and  o are constant in all inertia frames then c o
must be constant in all inertial frames. Otherwise, if  o
and/or  o are/is not constant in all inertial frames,
because c o is not constant, then Postulated 1 is invalid
because the electrodynamics laws of nature contain
these constants.
Other Basic Considerations
1. The charge of an electron or proton is
mathematically considered herein to be a
point charge and does not have a finite size.
2. An uniform electric field exists about a
stationary electric point charge, pervading
the space equally in all directions and falling
off in intensity at 1/r2.
3. Velocities are relative between interacting
particles.
Other Basic Considerations Cont’d
4. The electric field of a moving charge pervades all the space of
its inertial frame, thus having instant reaction with a charge in
contact with it. Acceleration of a charge creates a new velocity
that changes the electric field that spreads out over the new
inertial frame at the speed of light.
5. The charge value, q, is invariant from one inertial frame to
another.
6. A positive sign on the overall magnetic force represents
repulsion while a negative sign represents attraction.
7. A negative sign must be entered into the equations for
negative charges such as electrons. A positive sign must be
entered into the equations for positive charges such as protons.
This makes the direction of the overall force appear correctly as
in 6 above.
SRT Formalism Employed
1. Lorentz Transformation
2. Three Vectors: a(a1, a2, a3) (lower case)
3. Four Vectors: V(V1.V2.V3,V4) (Upper case)
Four vector force formula: F   v  f , f  v 


2
 c 
4. The four vector force is good for transforming
force between inertial frames and creating
force laws.
5.  v   1 / 1  v 2 / c 2
F
Mathematical Setup for Analysis of Force
Between Relative Moving Charge Particles
Lorentz Force Law—Starting Point for
the Four-Force SRT Transformation
Relativist’s Starting Point Lorentz Force Law (Wrong)
vh 

f  q e  2 
c 

(1)
Keele’s Starting Point (Right)
f  qe 
f=3-force, q=charge, e=electric field, v=relative
velocity, h=magnetic field, c=velocity of light
(2)
Why Relativist’s Starting Point is
Wrong
1. When we are just trying to derive the force between a stationary charge and a moving charge
the Lorentz Force Law that contains the magnetic field is the wrong starting point. If we include
the term that has the magnetic field and assume it comes from the moving charge itself, then we
would be doing a transform from two different inertial frames which is a “no-no” in SRT. If the
term involving the magnetic field arises from a separate source other than the moving charge,
then the transform would work for the transform of magnetic field. But we are not doing that, so
we can expect the magnetic force to fall out of the transform of just the moving charge’s electric
field. This was found to be the case. The is the most important point of this whole presentation
that shows where errors are made.
2. The regular Lorentz Force Law could be applicable in Cathode Ray Tube or accelerators where
the source of the magnetic field is separate from the “magnetic field” created by the moving
charge. You don’t start out in two different inertial frames.
3. An experiment performed by Keele with his results shows that his is the correct starting point.
4. The magnetic field h in (1) is derived from the Biot-Savart Law which was derived before the
Old Ampere’s Law from current flowing in a wire. The magnetic field from a moving isolated
charge is different from the magnetic field of a current element as will be shown in a later slide.
Transformation Results
Relativist’s Results
1
h  ve
c
e

qr
 r 1 - v / c sin 
3 3
2
2
2

3/2
(3)
Keele’s Results
(4)
Length Contraction of e-field of Moving Electron
as seen by a Stationary Particle
Magnetic Force Between Relative
Moving Isolated Charges
Total electric field and magnetic forces between the
relative slow moving charges (after mathematical
manipulations of (4) using the Binomial Theorem and
2
2
v
/
c
eliminating higher orders of
):
kq1q 2 r12  v 2
(5)
2 


f12 
1  2 0.5  1.5cos  
3
r


c
Subtracting the stationary electric field force from (5)
we have the magnetic force:
f m12

kq1q 2 r12 v 2
2

0
.
5

1
.
5
cos

3
2
r
c

(6)
Current Element
Moving Electrons --- showing length contraction of each
charge’s field and the length contraction of the spacing
between the charges in the current element
Stationary Electrons
Stationary Protons
Length contraction of the spacing between charges in the current element
increases the charge line density of the current element as seen by a stationary
observer (γσ).
The Magnetic Force Law Between a Stationary
Charge and a Stationary Current Element
Include one more relativistic effect to (4) for a current
element (current flowing in a short piece of a conductor):
This effect is length contraction of the spacing between the
current carrying electrons in the current element as seen by
the stationary charge. This has the effect of increasing the
charge density of the current carrying electrons in the
current element. The result is:
f m12
kq1ds 2 r12 v 2
2

1

1
.
5
cos

3
2
r
c


Where σ is the line charge density.
Notice there is a small magnetic force on a stationary
charge with respect to the current element.
(7)
The Magnetic Force Law Between Stationary Current
Elements (Found to Be the Old Ampere’s Law)
Eq. (7) is applied three times to the cross combinations of charges in
the two current elements and then the resulting forces are added. The
charges in (7) are replaced by I ds/v' s resulting in the Old Ampere’s
Law:
f m12
kI1I 2 ds1ds 2
 rˆ12
2dsˆ 1  dsˆ 1  3dsˆ 1  rˆ12 dsˆ 2  rˆ12 
2 2
c r
(8)
A study of this law reveals that successive current elements with
current in the same direction repel each other. This fact has been
demonstrated by experiments of Peter Graneau. This is an effect the
Biot-Savart Law does not allow.
Eq. (8) obeys Newtons 3rd Law whereas most other similar laws do
not. They generally have to be integrated around a closed loop to have
any applicability.
Equivalent Mathematical Form of Old Ampere’s
Law
Eq. (9) below is mathematically equivalent to Eq. (8) in the
previous slide:
kI1I 2 ds1ds 2
(9)
ˆ
2sin sin cos  cos cos 
f
 r
m 12
12
c 2 r12
2
1
2
1
2
where r̂12 = unit vector in the direction of r12 and r12 =
magnitude of the vector r12 joining the two current
elements. The constants are k = 1/40 (0 = permittivity of
free space) and c = speed of light. The I1 and I2 are current
magnitudes and ds1 and ds2 are current element lengths.
The angles are: 1 = angle between ds1 and r12; 2 = angle
between ds2 and r12;  = angle between the plane of ds2
with r12 and the plane of ds1 with r12.
Experiment With the Old Ampere’s
Law
A simple experiment was performed on various
shapes of one-turn coils. The inductance L of
these coils was computer calculated using
Ampere’s Law. Then the Inductance was
measured by determining resonance of the
inductor with a calibrated capacitor. The
measured value of L is then compared with the
computed value.
Method employed to measure inductance (Lm) of a coil
Sine Wave
Generator
One-turn coil
Frequency
Meter
2
I
1
Formulas: E  L , L 
m
2
2f 2 C
Scope
Resonance
Calibrated Capacitor
One-turn solenoid set up for calculating its inductance
using the old Ampere’s Law
Math showing how inductance may be calculated using
a computer
The energy
and
stored between two current elements
is:
(3)
If both differential lengths
and
are imagined to move
outward with r to keep angle variables constant, then all the
variables in the (…) function are constant and only r varies.
The (…) function can then be moved outside the integral sign,
and the result of the integration is then:
(4)
Math for one-turn solenoid inductance cont’d
Inserting Eq. (1) into (4), we get:
(5)
From
Fig.
1
we
know
,
,
. To set up (4) for computer integration,
we further set
1 degree, so
(6)
For the present case,
get:
. Substituting (6) into (5), we
(7)
where K is a constant.
Math for one-turn solenoid cont’d
Then the total energy
stored in the solenoid is:
(8)
where the summation limits prevent counting of infinite self
contributions.
Computing
for our one-turn solenoid with radius of
0.254m:
(9)
The value of L computed from (9) compares
favorably with the measured value; i.e. within 2% of
2.205 micro henries.
Coil Forms
Experimental Results
•
•
•
•
•
•
•
Name
Triangle
Circle
Square
3-D Square
3-D Tetra
Measured and Calculated Results (Wire dia. = 0.635)
ds (mm)
Lm (microhenries) Lc (microhenries)
2.4
1.914819
1.909496
2.7
2.399168
2.396962
2.8
2.717203
2.717502
3.2
3.222207
3.223535
3.1
3.760989
3.765264
Δ%
-0.28
-0.09
0.01
0.04
0.11
Since no self-inductance relationship of ds was employed in the
calculation of Lc , it is deduced quite appropriately from the
above table that for the measured and calculated values to agree
so closely, the internal inductance of the current element ds
must be zero. Varying the ds length on either side of the ones
presented above will produce a calculated value of Lc that will
be above or below the measured value of Lm.
Computer modeling of a current element to show zero
energy stored in its inductance
Current Element
End Views of current element
The self-inductance of the current element was
not added in the computer model for calculating
the inductance of single-turn inductors of the
various shapes, so the self-inductance needed to
be verified to be zero.
A square cross-section is used
in the model to approximate
the round cross-section area of
the wire current element to
make the math easier. Areas
are made equal. The current
element is divided up into
many rectangular
parallelpipeds.
Current Element Length
A computer model of the current element using the inductance (energy storage)
formulation of Ampere’s Law was created. It determined the length of the current
element so that there would be no energy storage in it. (The energy stored in each
current element was not added in the computer calculations of the previous slide.)
The following table shows the comparison between the required length of ds for a
match between the calculated values of L with the measured values of L.
•
•
•
•
•
•
•
Circle coil. Wire dia. a parameter.
Wire dia. (mm)
Lm (microhenries)
0.254
0.635
2.134
2.747
2.399
1.944
ds (mm) required ds (mm) calculated
for Lc match
for zero inductance in
current element model
1
1.08
2.8
2.71
12
9.1
Some Conclusions From Experiment
1. The old Ampere Law is the correct one.
2. The other magnetic laws derived above: force
between relative moving charges, and the force
between a charge and current element are correct
since they preceded in derivation of the Old
Ampere’s Law.
3. The SRT derivations of these laws are correct.
Some Applications
1.
2.
3.
4.
Arc Welding
Radio Wave Propagation
Study of Elementary Particles
Gravity
Arc Welding
A study of the plasma current flow in the arc gap
suggests the relative moving positive and
negative ions will be pulled together greater
than when stationary, and thus will emit energy.
(This is analogous to an electron in a hydrogen
atom falling from a higher orbit to a lower
orbit.) It may be possible to “burn” the material
of the plasma this way and thus emit more
energy than is output from the welding
machine.
Radio Wave Propagation
Eq. (7) appropriately modified appears to be a natural for the expression of
the magnetic field of a propagated radio wave.
f m12
kq1ds 2 r12 v 2
2

1

1
.
5
cos

3
2
r
c


(7)
Modified by:
Remove q1 to go back to the magnetic e-field. Also the following:
I 2 ds 2
ds 2 
,
v
I2 I  I
2
max2 sin(wt  r,/c)
v ,

kL I 2max sin 2 wt  r / c r12
2
em 
1

1
.
5
cos

3 2
r c
This magnetic field e m acts on a charge like an electric field.
2

ds 2  L

Further development is beyond the scope of my present intentions.
(10)
E-field of Dipole Antenna
E-field is like the e-field between capacitor plates.
Dipole
Antenna
e-field
Study of Elementary Particles
Eq. (4) can be used in a Bohr Atom like model for
elementary particles since the equations is good
for up to the speed of light:
(4)
Gravity
If the gravity can be considered to be a field, and if the
gravity field has similar characteristics to an electric
field (but much smaller in amplitude), and if the
heavenly bodies are treated as mathematical points,
then the gravity field of a heavenly body will have the
four-force Lorentz Transformation given by Eq. (11).
e m2 

Gm2 r
 r 1 - v / c sin 
3 3
2
2
2

3/2
(11)
Gravity Cont’d
For slow relative moving heavenly bodies (earth
in orbit about the sun for example) then Eq. (11)
may be reduced to:
2

Gm1m2 r12
v
2 
f12 
1  2 1  1.5cos  
3
r
 c

This is a new law of gravity.


(12)
Gravity Cont’d
Eq. (12) was tested in a computer program to
see if the excess force (in addition to the
Newton gravity law) would produce the Mercury
Perihelion Advance.
f excess
Gm1m2 r12

r3
 v2
2 
 2 1  1.5cos  
c



(12)
Gravity Cont’d
It did. Table below show the computed results:
Planet
Calculated Results
Observed Results
Mercury
46.328
43.11
Venus
9.0634
8.624, 8.4
Earth
4.0230
3.38, 5.00
Expressed in arc seconds per 100 Earth years.
(If Mercury’s advance was measured relative to Earth’s then
Mercury’s Advance would be 42.3 calculated according to
the computer model.)
Uses of New Gravity Law
1) May explain why rockets launched vertically travel further
than predicted by Newton’s gravity law.
2) Use in accelerating universe ongoing studies.
3) Weight of a vertically rotating disc will be non-uniform: The
top and bottom will be heavier, and the sides will be lighter.
A horizontal spinning disc should weigh more than the same
disc spinning vertically.
Summary of Derived Laws
Magnetic Force Between:
Relative Moving Isolated Charges:
2
f m12 

kq1q 2 r12 v
2
0
.
5

1
.
5
cos

3
2
r
c

Stationary Charge and Current2Element:
kq1ds 2 r12 v
2

f m12 
1

1
.
5
cos

3
2
r
c
Two Current Elements (Old Ampere’s Law):
kI1I 2 ds1ds 2
f m12  rˆ12
2dsˆ 1  dsˆ 1  3dsˆ 1  rˆ12 dsˆ 2  rˆ12 
2 2
c r
Total Gravity Force BetweenTwo Masses:
Gm1m2 r12  v 2
2 
f12 
1  2 1  1.5cos  
3
r
 c



Reasons that these Laws are correct:
1) The old Ampere’s law was derived from SRT and found to be
the correct one from the experiments performed.
2)
The other two magnetic laws are fall-outs of the steps to derive the
old Ampere’s, therefore they can be taken as correct.
3) The gravity law was derived by analogy with SRT applied to efield of an isolated charge. The excess gravity force thus
created was found by calculation to be responsible for the
perihelion advances of the planets by computer computation.
Therefore this law is correct.