# Time delay/magnetic fields/coloumb fields.

• andrewr
In summary," );if ( f==2.601392e-45 ){ printf( "The force of a magnetic field is " ); printf( "within the order of magnitude of what was expected." );} else { printf( "The force of a magnetic field is " ); printf( "much larger than expected." );}return 0;}

#### andrewr

I have seen several papers indicating that A magnetic field can be considered a time delayed electric field; So, I began attempting order of magnitude checks on the idea to see if it could be understood in a classical reference frame with a propagation media having the non-classical speed of light issue envisioned by Einstein. I have good results, but also an unexpected result that I don't think is realistic -- and am wondering if anyone can spot an error that would explain why it is so large.

The thought/experiment:
So, as a simple problem, I wanted to compute the difference in repulsion of two electrons ~1 meter apart that are moving parallel to each other at 1 [m/s]. In essence, this is a problem of two parallel infinitely long wires carrying a current of 1/q coulomb/second each past any given point in the vicinity of the electrons.. Ideally, I have the two electron paths oriented strictly in the "y" direction and the separation of the two wire/electron paths in the "x" direction with no Z components of either one.
It's just two "wires" in a plane.

From my E&M book, the problem works out to an attractive force developing between the wires of about 5.13x10-45N when the currents are flowing in the same direction.

Essentially, since the electrons are moving (but any protons of the wires are NOT moving) -- the proton E&M field is static but the electron E&M field is not. Therefore, the electrons experience an attraction to the protons which is independent of their speed, but they feel a repulsion from each other which depends on the propagation delay of the field from where it was emitted -- to where it is received because the electrons are in motion.

For the first problem, I assumed the two electrons to be at the same "y" height vs t. eg: y1(t)=y2(t); dy/dy=1[m/s]. The result of the calculation is that the path from where the first electron emitted it's field to the place where the second one receives it is *very* slightly longer than the 1m distance between the two electrons (low velocity = no relativistic correction needed!).

I took the static force between two electrons at 1m to be: 2.307077129049784e-28 [N]
the speed of light to be: 299792457.999458 [m/s] (codata).
And did a calculation yielding:

t0=3.335641e-09 t=3.335641e-09 Dt=1.857925e-26 l0-1m=0.000000e+00 l-1m=5.637851e-18 Dl=5.637851e-18
(1.000000,0.000000) force=2.601392e-45 [N]

Which is of the right order of magnitude to be a magnetic force. I am not concerned about being off by less than a factor of 10x the correct value; so this result is acceptable.
IF I separate the electrons by 1meter + 1mm; the force is essentially within 1% of the previous value.
However, when I try to offset the electrons from each other in the y direction, by as little as 1mm; the force suddenly computes out to over a million times stronger than expected.
t0=3.335643e-09 t=3.335643e-06 Dt=3.332307e-06 l0-1m=4.999999e-07 l-1m=5.033411e-07 Dl=3.341204e-09
(1.000000,0.001000) force=1.541681e-36 [N]

I thought at first perhaps it was round off error; so I made sure to use a long double (C language, 128 bit floating point nr; around 32 significant digits.) and did some analysis; but I find that the results are still way too large; and when I do some analysis on the result, I come to the conclusion that (y+0.001)**2 term used in the Pythagorean theorem causes this event to happen and that it isn't purely a math-coprocessor/program issue. So, I wanted to know if others come to the same conclusion; The program written in "C" follows; I would expect that increasing the distance between electrons by 1mm (in X) would have a "BIGGER" effect on the change in force than would having the "y" distances offset by "1" mm, for the hypotenuse of a right triangle must be shorter than the sum of its two legs -- and the static force on an electron goes as "k*q1*q2 / d" between them.
Can anyone explain, reasonably, why the force is over a million times more different in the two cases?

#include <stdio.h>

static long double f = 2.307077129049784e-28; /* Force due to static repel @ 1[m] for two electrons.*/
static long double c = 299792457.999458; /* codata speed of light */

/* Compute a square root of a long double, accurately! */
long double sqroot( long double s ){
long double est=s/2.;
int i;
for(i=0; i<100; ++i){
est = (long double)0.5*(est + s/est);
}
return est;
}

/* Compute force *change* due to time lag */
long double force01( long double x, long double y ) {
long double t0,t,l0,l;

l0=sqroot( y*y + x*x ); /* Static length between electrons (simultaneous) */
t0=l0/c; /* Time to traverse static length */

/* The distance traveled at 1[m/s] in t seconds is t meters.
* Therefore, the total vertical distance traveled is y+t meters.
* since the total horizontal distance is always x [m], the hypotenuse distance is:
* is l = ( xx + (t+y)**2 )**0.5 meters.
* Since the length of the hypotenuse is also l=Ct, for speed of light;
* substituting in...
* tC=(xx + (t+y)**2)**0.5
* ttCC = xx + (t + y)**2
* ttCC = xx + tt + 2ty + yy
* ttCC - tt - t* 2y - ( yy + xx ) = 0
* tt(CC-1) + t*-2y - ( yy + xx ) = 0 ;; for h=0, t=sqrt( xx/(CC-1) )
* tt + t( -2y/(CC-1) ) - (yy+xx)/(CC-1) = 0
* t = (2y/(CC-1) +- sqrt( 4yy/(CC-1)**2 + 4(xx+yy)/(CC-1) ))/2.
* t = y/(CC-1) +- (y/(CC-1))*sqrt( 1 + (CC-1)(1+xx/yy) )
*/
if (!y) t=sqroot( x*x/(c*c-1.) ); /* Prevent divide by zero error */
else t = (x/(c*c-1.))*( 1. + sqroot( 1. + (c*c-1)*(1.+(x*x)/(y*y)) ) );

l = sqroot( x*x + (t+y)*(t+y) ); // More accurate than l=t*c for some reason... but can use either.

printf( "t0=%e t=%e Dt=%e l0-1m=%e l-1m=%e Dl=%e\n",
(double)t0, (double)t, (double)(t-t0),
(double)(l0-1.), (double)(l-1.), (double)(l-l0) );

return f * (1.0/(l0*l0) - 1.0/(l*l) );
}

/* Take 1 electron on each "wire" 1 meter apart, moving at 1meter/second.
* That means that through 1 meter of wire a single electron flowed (1/q coloumb)
* in one second. That can be used to compute the effective force between two
* such wires. Adjusting for relativity, the length change may be used to
* convert an electrostatic force into a magnetic one; eg: the difference in
* electrostatic force due to motion, is a ficticious "magnetic" force.
*/

int main(void) {
long double x=1.0;
long double y=0.001;

long double fi = 5.1339389934080655e-45; /* Expected Force due to current */

// Calculate and print results...
printf( "(%f,%f) force=%e [N]\n", (double)x, (double)y, (double)(force01( x,y )) );
return 0;
}

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Early in you post you talk about the repulsive force between 2 electrons and then you talk about two infinite current carrying wires. The B field of a moving point charge would be calculated different than an infinite wire. If you considered an infinite moving line charge moving at speed v. you could use amperes law to compute the B field, and Gauss's law to compute the E field and then analyze the situation that way, which would be pretty straight forward.

cragar said:
Early in you post you talk about the repulsive force between 2 electrons and then you talk about two infinite current carrying wires. The B field of a moving point charge would be calculated different than an infinite wire. If you considered an infinite moving line charge moving at speed v. you could use amperes law to compute the B field, and Gauss's law to compute the E field and then analyze the situation that way, which would be pretty straight forward.

Right; The actual field I am computing is for two discreet electrons found in charge neutral wires with no other electrons moving. I pre-suppose two electrons are free, the others are all trapped and effectively at "absolute zero" and therefore ignorable. The solution still should be on the same order of magnitude as that of a continuous distribution of electrons moving on the same wires but with a much lower average velocity.

Whenever a real problem is done, it is a superposition of many electrons or distributions of space charge; in theory, classically, one ought to be able to take the position of many electrons and protons and apply the same discreet rule to all of them pair-wise and sum the results to get an "exact" answer
.
Whenever the integrals used in Biot Savart law, Gauss, etc. they are approximations which replace large numbers of electrons and protons with a continuous distribution of charge. Since a typical wire has on the order of 10**23 electrons present that are moving (drifting) in a net direction, such approximations are justified.

When no electrons are moving, there is no magnetic field; and the problem reduces to a completely balanced (no net force) static system between the two wires. So, I am reasoning that the *change* in force due to motion of electrons caused by the propagation delay for the "E" fields to travel from a point of origin to that of a "detecting" electron must in fact equate to that of a magnetic field. (Classic physics often assumes this time is instantaneous, and we know that isn't right).

It was my Physics Professor, Dr. Young from University Of Portland, OR; USA; who pointed out that these time delays are not accounted for in classical physics and may be equivalent to the mathematics of magnetic fields.

In reading papers on special relativity; I have found many others who agree with my physics prof; and so I am simply testing his idea out to see if it is at least qualitatively correct.

When I compute the force for two electrons based on time delay (not based on Gauss' law, etc) I get the correct answer for an infinite line charge within a factor of about 2. That's excellent agreement showing that indeed the force of magnetism might be nothing more than time delay in a crude calculation.

However, when I try to offset the electrons by as little as 1mm perpendicularly to each other -- something totally counter-intuitive happens; the difference in the static vs. dynamic force for two discreet electrons becomes close to a million times stronger. I have spend several days checking the math, and sharing this with a few friends, and none of us is able to find a mistake in the program that would account for such a change in results. I'm hoping someone with sharp eyes might notice something about the discreet problem that I don't or some kind of mistake in the mathematics (which are outlined in the program's comments.).

I know how to get the "right" answer using magnetic fields; But this is just my attempt to explore it from a different perspective. Are you up to the challenge? Thanks!

I probably can't answer your question but when you say time delayed electric field, what do you mean by this. And you say when you offset them by 1mm you get a field strength of a million times stronger. When you offset them why can't you just find the new distance between them and use that, or unless I am missing something simple. When I learned about E and B field in different frames, we talked about how moving in one frame, we would see length contraction because the electrons were moving towards us at a speed v and the protons were at rest, so the wire now had a net charge and an E field in that frame. While in the other frame it just had a B field, is this similar to what your talking about.

Normally the force between charges is said to depend only on the distance between them:
Feynmann's lectures on physics, see section 12-4.
http://student.fizika.org/~jsisko/Knjige/Opca%20Fizika/Feynman%20Lectures%20on%20Physics/Vol%201%20Ch%2012%20-%20Characteristics%20of%20Force.pdf

In particular, the information about "where" a particle was in the past can only travel at the speed of light (Einstein's prediction). So that, one ought to compute the "force" based not on the simultaneous position -- but rather based on the "retarded" position.

When you offset them why can't you just find the new distance between them and use that, or unless I am missing something simple

Nope; you are thinking identically to me and others here. When I calculate the new distance (retarded) for particles moving parallel to each other (in y), I get a very reasonable result; the distance is only longer by fractions of a micron -- and the change in apparent distance (retarded at the velocity of light) causes a force change (repulsion decreases since the "distance" between particles increases).
Working the math out for static particles separated by a slightly longer distance, they have less repulsion; and therefore the net CHANGE in force is the same as attraction. The magnitude of this change happens to be the same as the force calculated for a magnetic field due to motion.
Feynmann, and others, using relativity (Lorentz transformation) show that a Lorentz contracted electric field will produce the correct results for a magnetic field; and therefore it is un-necessary to compute both the electric and magnetic fields; one may simply compute electric fields and lorentz transforms to get the same result.

See section 13-6:
http://student.fizika.org/~jsisko/Knjige/Opca%20Fizika/Feynman%20Lectures%20on%20Physics/Vol%202%20Ch%2013%20-%20Magnetostatics.pdf

I have found that in many ways, one may compute the same results without changing reference frames -- for example, the Lorentz contraction of an object moving away at relativistic speeds can be calculated by computing the Doppler shift of a fixed frequency bouncing off that object -- and then computing the length where the two different frequencies (The outgoing higher frequency and the lower reflected frequency) produce an electric field of zero. There is a fixed distance from the moving object where the incident and reflected wave will cancel -- and the length happens to be exactly the length contraction predicted by special relativity.

The interesting issue, here, is that when I calculate the hypotenuse distance for the retarded distance between particles; I get length changes which are millions of times too *LARGE*. I am wondering if I made an algebra mistake, or if my math coprocessor is giving bad results (Intel has done it before!), or if I did all these things right, but am still just getting the "wrong" answer. (Which would suggest a need to revisit the idea of the pythagorean theorem...!)

What do you think? Am I making sense now?

andrewr said:
I have seen several papers indicating that A magnetic field can be considered a time delayed electric field;

There's your problem right there. That's not the case.

For the first problem, I assumed the two electrons to be at the same "y" height vs t. eg: y1(t)=y2(t); dy/dy=1[m/s].
Perhaps a typo: dy/dt=1m/s?
The result of the calculation is that the path from where the first electron emitted it's field to the place where the second one receives it is *very* slightly longer than the 1m distance between the two electrons (low velocity = no relativistic correction needed!).
Electrons which are static in their common rest frame cannot possibly have a different force acting upon them then the usual static coulomb force. They can only have a different force on say you as an observer having a velocity wrt those electrons.

You could perhaps arguing that these electrons are moving wrt the +ve ions but since these ions form a continuous line, even then I cannot see a possible longer y path.

Perhaps my main objection with all so called “magnetism is a result of Lorentz boost” theories is that a magnetic field is always present round a current carrying conductor. There are no moving (or non moving) charges elsewhere needed. I can calculate the magnetic energy in any piece of volume away from such a conductor without the help of any extra charge. In addition, I know that magnetic energy is there because the power supply supplied this extra energy.

Per-oni,
Yes, that's a typo. it is supposed to be dy/dt.

Electrons which are static in their common rest frame cannot possibly have a different force acting upon them then the usual static coulomb force. They can only have a different force on say you as an observer having a velocity wrt those electrons.

Yes, that's the point. I'm wanting to compute the force from an inertial frame which is different from the rest frame of the electrons.

You could perhaps arguing that these electrons are moving wrt the +ve ions but since these ions form a continuous line, even then I cannot see a possible longer y path.
The positive ions are not moving. Only two electrons are moving.

In earlier computations assuming an "ether" (Michaelson Moorley); the correct answer can be gotten by assuming a "PHYSICAL" length contraction of the device happens in the direction of motion through the ether. The "ether" becomes a non-falsifiable element, and therefore is discarded -- but the point is that the calculations can be carried out as if an ether existed and length contraction happens.

Electrons are "point" sources, and therefore, no contraction need be calculated for them. Contraction does need to be computed for distributed charges; though; *BUT ONLY IF THEY ARE MOVING*. That's the point of Feynman's analysis of retarded time and relativity.

In the thought experiment I have set up; the distributed charges (both electrons and protons) are not moving; and therefore length contraction may be ignored. The point charges of two electrons *are* moving, but as they have no length --therefore length contraction may be ignored as well.

Even though the moving electron charges are a fixed distance apart in their common rest frame; Light and static coloumb fields do not travel with the common rest frame when viewed from the lab frame.
Rather light, and therefore the electrostatic field, spreads out in space from its source with the amplitude dropping inversely with radius (and since the speed is "c") equivalently with the change in "time" from when and where it was emitted

One may reason either with an ether which has length contraction; or from the constancy of the speed of light in "all" inertial frames; but the result is the same. The effective time/length traveled in the lab frame is longer than that when observed from the frame of the moving electrons.

Per Oni said:
Perhaps a typo: dy/dt=1m/s?
Perhaps my main objection with all so called “magnetism is a result of Lorentz boost” theories is that a magnetic field is always present round a current carrying conductor. ...

Magnetic fields do not act on charges which are not in motion.

There's your problem right there. That's not the case.

OK, am I to understand that you don't agree with Feynman's position on retarded position that I already linked to; or do you mean something else?

I am open to relativity being "wrong"; however, I find it surprising that I get the right answer with the basic calculation based on the constancy of the speed of light in all inertial frames;(Feynman observes the same relationship) -- but my experiment only fails with a test where the geometry is slightly skewed from the simplest case..

Right now, I am going to re-try the experiment using Python's decimal class which would verify that the Intel math co-processor on my machine isn't at fault; and that the schwartz inequality holds for the pythagorean theorm when using this many digits.

I look forward to hearing clarifications as to what you mean...

Feynman is not talking about a time delay. He's talking about a relative velocity.

This is a nice exercise, but I got a feeling it’s leading into a blind alley.
However, when I try to offset the electrons from each other in the y direction, by as little as 1mm; the force suddenly computes out to over a million times stronger than expected.
t0=3.335643e-09 t=3.335643e-06 Dt=3.332307e-06
This second t and therefore subsequent Dt are very different here than in your first calculation. Since the length difference in y is only 1mm I can’t see why.

A small point, I wrote:
You could perhaps arguing that these electrons are moving wrt the +ve ions but since these ions form a continuous line, even then I cannot see a possible longer y path.
You replied:
The positive ions are not moving. Only two electrons are moving.
Then surely these electrons are moving wrt the +ve ions just as I said.

Also you stated:
Rather light, and therefore the electrostatic field…..,
One doesn’t follow the other. It is true that when charges are accelerated a “kink” in the electrostatic field will be traveling with the speed of light but to say that an electrostatic field travels out of the point charge at all times? I don’t see that. For instance: What is replenishing this field?

There’s another point: you wanted to extend your 2 pair model to approach many electron conduction. So let’s first add one electron. I can place this electron in such a way that the distance after t0 has elapsed is shorter than before. So in that case there will be a net force in the opposite direction to the one you calculated first. These 2 results should therefore cancel and no magnetic force will be the end result.

Per Oni said:
This is a nice exercise, but I got a feeling it’s leading into a blind alley.

This second t and therefore subsequent Dt are very different here than in your first calculation. Since the length difference in y is only 1mm I can’t see why.

BINGO!
That's *exactly* what's bothering me. I can't see why either, and I don't see a mathematical mistake.
Regardless of whether the "idea" as a whole is right or wrong, I don't understand why the math comes out so different between the two cases; That's the reason I'm posting it -- did I goof on the math, or is it really that different for a 1 mm difference in y vs. in x?

A small point, I wrote:

You replied:

Then surely these electrons are moving wrt the +ve ions just as I said.

The electrons are moving with respect to them; as you said; but those positive ions are actually in a charge neutral situation (there are non-moving electrons around them as well) -- therefore the net effect is as if the wires do not exist except to constrain the electrons in motion to not move in the 'x' direction but only the 'y'.
Even if switching reference frames, to move with the two maverick electrons, the wires are still charge neutral, and therefore any length contraction in a Lorentz sense -- is going to be canceled.

At best, the charge neutral situation can only be disturbed locally by the two moving electrons -- but even then, we know from lab experiments that wires can carry a non-neutral charge (static electricity and a gold leaf veined charge-meter are good examples of objects with excess electrons.) So that I believe I am justified in claiming that the wires can be charge neutral with everything that is *NOT* moving.

If the two excess electrons in the thought experiment were not (or rather stopped from) moving; then there is only a static repulsion to account for (I compute that to be: 2.307077129049784e-28 Newtons.) between the two excess electrons. The force is so small as to be easily exceeded in reality on statically charged wires from scuffing ones shoes on the carpet on a dry day when touching one of them.

That Force, however, is the default electrostatic (not magnetic) repulsion;
All the rest of the electrons, and protons, are distributed uniformly along the wire; For that reason (and the reason that they are not moving) the electrostatic fields are constant parallel to the wires (and being neutral should be 0 anyway); therefore THERE IS NO change in the static field with a change in the Y direction.

According to classical physics, then, any repulsion or attraction that the moving electrons would feel due to the electroSTATIC force of all other electrons and protons will be *constant* vs. Y (and still should be zero); Again, all the protons and electrons in the charge neutral wires (since they are not moving) will not be affected (in the slightest) by the magnetic field set up by the two maverick electrons that are in motion. Charges that are NOT in motion are NOT affected by magnetic fields.

Magnetic attraction, Dr. Young speculated (and perhaps it's an empty speculation) might be affected by the time delay it takes for information to travel from one electron to the other across the gap between the wires. So, I am checking it numerically as a hypothesis.

Also you stated:

One doesn’t follow the other. It is true that when charges are accelerated a “kink” in the electrostatic field will be traveling with the speed of light but to say that an electrostatic field travels out of the point charge at all times? I don’t see that. For instance: What is replenishing this field?

One of the great questions of physics -- the electron is replenishing the field; and a proton someplace else is draining it just as fast as the electron creates it. That "kink" is a change in the flow at a location.

There’s another point: you wanted to extend your 2 pair model to approach many electron conduction. So let’s first add one electron. I can place this electron in such a way that the distance after t0 has elapsed is shorter than before. So in that case there will be a net force in the opposite direction to the one you calculated first. These 2 results should therefore cancel and no magnetic force will be the end result.

:!)

I agree with your intuition here; there does seem to be a way to place moving electrons to cancel the pseudo magnetic force out; but then, in reality -- electrons moving the same direction in parallel wires will cause them to "attract", whilst electrons moving in opposite directions on two wires will cause the wires to repel. Hence if three wires were placed parallel to each other (two of them practically on top of each other) it is physically possible to cause the forces to cancel exactly as you are commenting -- and to test that in a lab environment.

Let's work that problem out after figuring why my off by 1mm makes such a big difference mathematically. If you'd like to detail out your experimental arrangement, I'd be happy to think about it.

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Feynman is not talking about a time delay. He's talking about a relative velocity.

Hmmm..
Secton 1-34: right after equations 34-3:
"of course we realize that the coordinates must be measured at the retarded time. ... what time is the retarded time? ... the time (tau) ... is delayed by the total distance that light has to go, divided by the speed of light."

He talks about both as far as I can see...

He DOES take that relative velocity of the Coloumb (electroSTATIC) field when doing the derivative -- but he also requires a time delay for the propagation of the *information* about that field reaching a far away target (eg:) changes in the (Coloumb field) according to Einstein can't arrive everywhere at once (for that is *spooky* action at a distance argument); Einstein's hypothesis naturally extends to an ElectroSTATIC field's change taking a certain time (and therefore diminuation in amplitude) to arrive at a remote point from where and when it originated.

I would think that velocity implies continually changing time delays...

Is this just a preferred vocabulary issue, or did I miss something substantial?

AHA.

When I convert the program to Python, the magnitude of the force becomes reasonable for the y offset case.

The new result is still magnitudes larger than the y=0 case, but **many** magnitudes smaller than the baffling +1mm case in my earlier program (and the new program still isn't corrected for x forces vs. y forces; or third fourth and any other electrons others would like to add to the melee! )

I'd appreciate it (still) if anyone has an AMD machine and could re-run the c-test program I posted above to see if it is just my Intel processor (A celeron SSE2 machine from 2004/5) just for my own information on isolating what kinds of math bug that was...!

In the meantime, here's a version of the program in Python running the same basic algorithm as the "C" program but which avoids the use of the math coprocessor all together. Unfortunately, the built in class "decimal" from Python is cumbersome to work with, and it will be a few days while I write wrapper functions before I can continue the conversation on how "wrong" the theory is quantitatively;
(Which win or loose, will be enjoyable...!)

I want to be able to try out other peoples ideas without having to fight the math accuracy issues, or re-doing all my thoughts in klunky object function call syntax for simple exact decimals like the number '1'. So, I'm going to write a personal library to avoid having this issue -- although this isn't a "Fast" library like "C", and doing more than about 100 electrons will be prohibitively slow.
Still, I'm pumped up!

Code:
#!/bin/env python
# Use the python decimal class to compute a pythagorean value
# and prevent roundoff/hardware error.

from decimal import *			# No roundoff error/intel coprocessor!
getcontext().prec=250			# BECAUSE *250* digit EXACT precision
getcontext().rounding=ROUND_HALF_UP	# American style chem/physics rounding
print getcontext()			# sanity! display context at runtime

# Constants pre-computed or gotten from NIST, etc.

f = Decimal( '2.307077129049784e-28' )	# Static electron repulsion [N] @ 1[m].
c = Decimal( '299792457.999458' )	# Codata for the speed of light [m/s]
one = Decimal( '1' )

def LukeForce( x0, y0 ):
"""
LukeForce returns difference in Force in [N]ewtons due to longer
travel paths of hypothetical force exchange photons between moving
particles emitting and receiving said hypothetical photons...

Computes distance of a path traveled from an emission electron @
{x,y}(t0) = 0,0        to an interception electron initially @
{x,y}(t0) = x0,y0
given that the two electrons are traveling at a uniform speed of
1[m/s] in the strict y+ direction, and that information about the
new location of the couloumb field travels hypothetically at the
speed of light.

Therefore both electrons moving at 1[m/s] will *also* have traveled
t meters in the y direction when the interception happens.

Subsequently, the total dy distance traveled by the *information*
about the couloumb field is the hypotenuse of the x and y travel,
giving:

l = ( x**2 + (t+y)**2 )**0.5 meters.

Since the length of the hypotenuse is also l=Ct, for speed of light;
we have two equations in two unknows: substituting in...

tC=(xx + (t+y0)**2)**0.5
ttCC = xx + (t + y0)**2
ttCC = xx + tt + 2ty0 + y0y0
ttCC - tt - t* 2y0  - ( y0y0 + xx ) = 0
tt(CC-1) +  t*-2y0  - ( y0y0 + xx ) = 0

for y0=0, t=sqrt( xx/(CC-1) )
tt + t( -2y/(CC-1) ) - (y0y0+xx)/(CC-1) = 0

t = (2y0/(CC-1) +- sqrt( 4y0y0/(CC-1)**2 + 4(xx+y0y0)/(CC-1) ))/2.
t = y0/(CC-1) +- (y0/(CC-1))*sqrt( 1 + (CC-1)(1+xx/y0y0) )
"""

l0= x0*x0 + y0*y0	# Distance if electrons were NOT moving.
t0= (l0/c)		# Time to traverse non-moving length.
l0.normalize()
t0.normalize()

if y0.is_zero():
t=( x0*x0/(c*c-one) ) # **0.5
t=t.sqrt().normalize()
else:
disc=( one + (c*c-one)*(one+(x0*x0)/(y0*y0)) )
t = (x0/(c*c-one))*( one + disc.sqrt() )
t.normalize()

l = x0*x0 + (t+y0)*(t+y0) # **0.5
l = l.sqrt().normalize()

# Magnitudes...
print "deltaHypotenuseLength=",l-l0
print "deltaTime=",t-t0

print ( "Sanity check: tc=l :. tc-l=",
((t*c)-l).normalize()
)

print ( "The schwartz be with you, inequality, checking:"),
lPerimeter = x0 +y0 +t
if l.normalize() <= lPerimeter.normalize():
print "Sane schwartz! (whew!)"
else:
print "I've been schwartz changed?!? Conspiracy theory??!"

# Compute *difference* between the rest static force case and the
# motion case with propagation delay time according to F=const/l**2
# L0 is not always exactly one meter, so compute difference of
# ACTUAL length of static repulsion, vs. time/distance retarded one.

return f * (one/(l0*l0) - one/(l*l) );#) sugar,spam, & happy semicolon

# Take 1 electron on each "wire" 1 meter apart, moving at 1meter/second.
# That means that through 1 meter of wire a single electron flowed (1/q Coul.)
# in one second, by definition (1/q Ampere).
# That can be used to compute the effective force between two such wires
# ignoring the difference between a finite current element and a distributed
# one; but order of magnitude ought to be correct, none the less.

x=Decimal('1.0')
y=Decimal('0.001')
fi=Decimal('5.1339389934080655e-45');# Precalculated expected force

lf=LukeForce(x,y)
print "offset of (",x,y,") produces ",lf," delta [N]ewtons."

At best, the charge neutral situation can only be disturbed locally by the two moving electrons -- but even then, we know from lab experiments that wires can carry a non-neutral charge (static electricity and a gold leaf veined charge-meter are good examples of objects with excess electrons.) So that I believe I am justified in claiming that the wires can be charge neutral with everything that is *NOT* moving.
I can see what you mean, I’m ok with your idea to start with 2 electrons and then to go to many electrons. But your problem is that you do this in one step without considering whether your model is still valid. I say you cannot do that. Even adding just a simple 3rd electron can cause a problem. Also it really is not a good idea to add a third wire, your experiment is set up for 2 wires, if this fails then I’m afraid it’s curtains, no matter how many wires you introduce.
One of the great questions of physics -- the electron is replenishing the field; and a proton someplace else is draining it just as fast as the electron creates it.
This not an accepted theory, if so it’s new to me. Can you provide some references?

However, like I said its a nice exercise and you can learn a lot this way.

andrewr said:
Hmmm..
Secton 1-34: right after equations 34-3:
"of course we realize that the coordinates must be measured at the retarded time. ... what time is the retarded time? ... the time (tau) ... is delayed by the total distance that light has to go, divided by the speed of light."

He talks about both as far as I can see...

He DOES take that relative velocity of the Coloumb (electroSTATIC) field when doing the derivative -- but he also requires a time delay for the propagation of the *information* about that field reaching a far away target (eg:) changes in the (Coloumb field) according to Einstein can't arrive everywhere at once (for that is *spooky* action at a distance argument); Einstein's hypothesis naturally extends to an ElectroSTATIC field's change taking a certain time (and therefore diminuation in amplitude) to arrive at a remote point from where and when it originated.

I would think that velocity implies continually changing time delays...

Is this just a preferred vocabulary issue, or did I miss something substantial?

I haven't read through this thread because I would point out that Vanadium is correct, your original conjecture is not true and it would probably be best that you correct it before you expend any large amounts of effort and discussion on problems based off of it. The magnetic field is not a time-delayed electric field, it can be seen to be a Lorentz transformed electric field. The electric field undergoes a Lorentz transformation between two frames of references that are in motion relative to each other. The transformed field results in a magnetic and electric field component in the other frame. That is, let us take two electrons moving in opposite directions at equal speeds in the lab frame. In the frames of the individual electrons we only see an electric field due to the frame's respective electron (ignoring the contribution from the other electron for the moment). In the lab frame, if we Lorentz transform the Coulomb field from the electron's frames we get the associated magnetic and electric fields.

We can also transform the electric field from the frame of one electron to the frame of the other electron. What we find is that in the frame of the other electron there is an electric field that exerts the same force as the magnetic force seen in the lab frame.

Actually the above is a bit of a simplification since with a single electron we have a combination of electric and magnetic forces but I have chosen to talk as if the transformed field from electron frame to other electron frame results in a purely electrical field which may not be the case. That is the case however if we assume infinite lines of current (which we know in the lab frame only has magnetic fields). This is a case that is discussed in several textbooks like Griffiths or Purcell and you can read up on that to see how the magnetic field can be thought of as a transformed electric field.

In terms of your code and results, I am rather worried that you get different results between C and python and that you have written your own square root function. You shouldn't have to write your own square root function (unless the language does not provide one) to get a more accurate result. In addition, you shouldn't need to switch from C to python to get better results. My guess is that there is a numerical instability or error in your algorithm that is more at fault. It would be easier if we had a pseudocode or explanation of the algorithm that you are doing here as opposed to the code itself.

andrewr said:
Secton 1-34: right after equations 34-3:
"of course we realize that the coordinates must be measured at the retarded time. ... what time is the retarded time? ... the time (tau) ... is delayed by the total distance that light has to go, divided by the speed of light."

The result of the calculation is that the path from where the first electron emitted it's field to the place where the second one receives it is *very* slightly longer than the 1m distance between the two electrons (low velocity = no relativistic correction needed!).

Sorry to bring up an old thread but it hit me earlier that the difference in length is equal to either $\gamma_{(v)}$ or $\gamma^{-1}_{(v)}$ since we basically have replaced the mirrors of the common thought experiment with electrons, but c and v have their same meaning as in that thought experiment, so it seems that either $\gamma_{(v)}$ or $\gamma^{-1}_{(v)}$ should be the ratio of the x distance between wires and the distance at an angle determined by v. So given that we are considering gamma to be close enough to 1 to not need relativistic corrections, it should be close enough to 1 that the change in distance is negligible?

Per Oni said:
This not an accepted theory, if so it’s new to me. Can you provide some references?

However, like I said its a nice exercise and you can learn a lot this way.

It's been over a year... and I finally have some time to pursue it.. although, it might be better to just start over; I figure I'll post here first just to let those who already posted have first whack.

As to that comment I made, it's generic.

Faraday spoke loosely about "force", "source", and "sinks" for the motion of objects.
In some of his correspondence to Maxwell etc., he talks about it... and is usually being picked on by others for what they perceived as the imprecision of what he said ; although, in the end -- Faraday's "force field" prevailed.

If history interests you, get "The correspondence of Michael Faraday" around volumes 4 or 5.

I still have no idea what exactly is "in" the Field in Faraday's mind, but ... it goes from positive charges to negative when you draw an arrow for a line of force. :)

In electrostatics, the teachers generally speak of the field originating on one charge -- and sinking (draining) on the other.