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Feb 8, 2007, 10:03:27 AM2/8/07

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Gauss' law for electricity states that the electric flux out of any

closed surface is proportional to the total charge enclosed within

the surface. The law is also valid if the closed surface is replaced

by two parallel planes. The flux out of each plane is then half the

charge enclosed between the planes.

closed surface is proportional to the total charge enclosed within

the surface. The law is also valid if the closed surface is replaced

by two parallel planes. The flux out of each plane is then half the

charge enclosed between the planes.

Let us imagine an electron at x = 10 nano-light-seconds (nLS = 0.3 m)

of a coordinate system (y = 0, z = 0). The flux through the y-z-plane

(x = 0) divided by the permittivity of space is therefore equal to

half the elementary electron charge.

Now let us assume that the electron at rest is captured at the time

t_0 = 0 by a fast moving neutral particle and that afterwards the

electron moves at v = 0.9 c to the origin of the coordinate system.

A change in electric field of the plane can occur at the earliest at

t_1 = t_0 + x_0 / c = 10 ns at the orgin of the coordinate system.

At t_1 the electron has reduced its distance from the plane to

x_1 = 1 nLS. When an effect from x_1 propagating at c can reach

the plane, the electron's position is x_2 = 0.1 nLS. When an

effect from x_2 can reach reach the plane, the position is already

x_3 = 0.01 nLS, and so on.

Time in Position Duration Effect radius

nano-sec. in nLS until crossing in nLS

t_0 = 0 x_0 = 10 dT_0 = 11.11111 r_0 = 4.8432

t_1 = 10 x_1 = 1 dT_1 = 1.11111 r_1 = 0.48432

t_2 = 11 x_2 = 0.1 dT_2 = 0.11111 r_2 = 0.048432

t_3 = 11.1 x_3 = 0.01 dT_3 = 0.01111 r_3 = 0.0048432

t_4 = 11.11 x_4 = 0.001 dT_4 = 0.00111 r_4 = 0.00048432

The duration of the movement from x_n to the cross-point x = 0 is

dT_n = x_n / v. During this time dT_n the electric field and flux

on the plane can only change within a circle around the cross point

with the "effect radius" r_n = sqrt ((dT_n * c)^2 - x_n^2).

So whereas the electric field at the origin of our coordinate system

continuously increases to infinity until the electron crosses the

plane, the region where the flux no longer can change increases

continuously in the same way. At least superficial reasoning leads

to the conclusion that the flux through the whole plane increases

to infinity instead of remaining constant (and changing sign at

crossing time).

In the original theory of Coulomb, there is no paradox, because the

flux increase near the cross-point is instaneously compensated by a

flux decrease in more distant regions because of decreasing angles

of incidence.

Feb 9, 2007, 2:05:43 PM2/9/07

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On Feb 8, 4:03 pm, "Wolfgang G. Gasser" <z...@z.lol.li> wrote:

>

> In the original theory of Coulomb, there is no paradox, because the

> flux increase near the cross-point is instaneously compensated by a

> flux decrease in more distant regions because of decreasing angles

> of incidence.

>

> In the original theory of Coulomb, there is no paradox, because the

> flux increase near the cross-point is instaneously compensated by a

> flux decrease in more distant regions because of decreasing angles

> of incidence.

There's no paradox either in standard EM. Charge is relativistic

invariant and Gauss' law works just as fine for a moving charge.

Consider the electron to be at rest and now move the plane

instead. The flux through the plane is that of half an electron

always. This is in the rest frame.

Now consider the viewpoint from the plane. The EM fields

transform according to Special Relativity:

E'x = Ex

E'y = gamma * (Ey - v/c Bz)

E'z = gamma * (Ez + v/c By)

Now only E'x goes through the plane, while E'y and E'z are

parallel to the plane. Thus: The flux is again that of half an

electron.

We can go a bit further and look at the plane through which

E'y goes. We forget Bz because it's zero in the electron's

rest-frame. The electric field E'y as seen by the moving plane

is now increased by a factor gamma. However, the plane sees

the world Lorentz contracted by a factor gamma as well.

So, again: The flux through the plane is that of half an electron.

Regards, Hans

Feb 10, 2007, 10:42:49 AM2/10/07

to

I can suggest another argument that gives the same result as Hans'.

It is a bit classical and not as rigorous, but is maybe a bit more

intuitive to understand:

It is a bit classical and not as rigorous, but is maybe a bit more

intuitive to understand:

If you think of the electric field in terms of the old fashioned

concept of lines of force, then the electron has radiating from it one

unit charge worth of these lines, uniformly distributed about 4pi

steradians. As the charged particle moves towards the plane these

lines of force will tilt since the changes in the fields propagate at

the speed of light. There will thus be a "kink" in the lines that

propagates outward from the time/place where the charge suddenly

starts moving.

Since the fields propagate at the speed of light, but the charge must

move at less than the speed of light, the lines of force will always

have the same direction of tilt towards or away from the plane, just

the magnitude of the slope changes. That is, a line of force that is

crossing the plane initially will always be tilted towards the plane.

Every line of force that initially passes through the plane will still

pass through the plane for any positon of the plane between the

original position and the particles location.

The only way you could get the "paradox" you are describing is if the

charge could move faster than the speed of light and thus get ahead of

the lines of force. Then the slope of some of the lines would change

sign (i.e. tilt the opposite direction) and then Gauss' Law would be

violated. This, of course, is never observed.

This argument is identical to Hans', but is a little more intuitive

perhaps.

Feb 14, 2007, 2:30:45 AM2/14/07

to

> In the original theory of Coulomb, there is no paradox, because the

> flux increase near the cross-point is instaneously compensated by a

> flux decrease in more distant regions because of decreasing angles

> of incidence.

> flux increase near the cross-point is instaneously compensated by a

> flux decrease in more distant regions because of decreasing angles

> of incidence.

In my first response I showed why Gauss's law also works for a

moving electron via special relativity.

I do see now why you get the wrong result. You let the electric

field E propagate away from the electron with c. To get the correct

answer one should let the electric potential V propagate away with

c and THEN differentiate to get the correct electric field E.

This was first recognized by Lienard and Wiechert around 1900.

With this method you can obtain the EM fields of an electron

making arbitrary motions and accelerations.

Regards, Hans

Feb 14, 2007, 2:35:15 AM2/14/07

to

Hans de Vries in news:1171020466.4...@p10g2000cwp.googlegroups.com :

> Wolfgang G. Gasser in news:eqffqq$t24$1...@atlas.ip-plus.net :

> Wolfgang G. Gasser in news:eqffqq$t24$1...@atlas.ip-plus.net :

>> In the original theory of Coulomb, there is no paradox, because the

>> flux increase near the cross-point is instantaneously compensated by

>> a flux decrease in more distant regions because of decreasing angles

>> of incidence.

>

> There's no paradox either in standard EM. Charge is relativistic

> invariant and Gauss' law works just as fine for a moving charge.

>

> Consider the electron to be at rest and now move the plane

> instead. The flux through the plane is that of half an electron

> always. This is in the rest frame.

In this case, the flux can change at all points of the plane at

the same time. In my example however, an electron is at rest at

[x=10, y=0, z=0] and starts at t = 0 moving at v = 0.9c towards

[0, 0, 0]. Until the electron crosses the plane [0, y, z] at

t = 10/0.9, the flux through the plane can only change within the

circle y^2 + z^2 <= r^2 with r = sqrt((10/0.9)^2 - 10^2) = 4.84.

The assumption seems reasonable that from t = 10 on (when the

information that the electron is no longer at rest at [10,0,0] can

reach the plane), a flux change through the plane spreads from

[0,0,0], reaching the circumference y^2 + z^2 = r^2 of the circle

at t = 10/0.9.

Any flux increase near the center [0,0,0] of the y-z-plane however

must be compensated by a flux decrease at more distant regions,

because the flux integral through the circle cannot change before

the electron crosses the center. So any flux increase near the

center must wait (how long?), until a flux decrease in more

distant regions is possible.

As far as I've learned now, the logical conclusion of such an

"infinite flux paradox" is avoided by the assumption that the

velocity of a moving charge leads to extrapolated/anticipated flux

changes: The instantaneous change of the flux integral through a

plane from -flux to +flux at the moment a charge crosses the

plane, is caused primarily NOT by local effects of the crossing

charge BUT by extrapolated effects of former positions of the

charge.

But does this solution not entail another problem: The charge

can be stopped just before crossing the plane. Because this

information cannot reach the whole circle of the plane where the

flux is about to change, the flux integral through the circle

and therefore through the whole plane will nontheless change,

thus leading to a contradiction with Gauss' law for electricity?

Isn't it astonishing that in dynamic situations, Gauss' law

is as well valid under "retardation with c" (Maxwell) as under

"instantaneous interaction" (Coulomb), despite very different

mechanisms/maths (very simple/transparent in the case of Coulomb,

and rather complicated/opaque in the case of Maxwell)?

Has the old reproach that Maxwell starts with not-mediated

actions-at-a-distance and ends with the claim that the effects

are produced without such actions-at-a-distance actually been

cleared up? According to Heinrich Hertz 1890 this procedure

leads to the unsatisfying feeling that either Maxwell's result

or his reasoning leading to this result should be wrong.

Cheers, Wolfgang

________________

"Maxwell geht aus von der Annahme unvermittelter Fernkräfte,

.. und er endet mit der Behauptung, dass diese Polarisationen

sich wirklich so verändern, ohne dass in Wahrheit Fernkräfte

die Ursachen derselben seien. Dieser Gang hinterlässt das

unbefriedigende Gefühl, als müsse entweder das schliessliche

Ergebnis, oder der Weg unrichtig sein, auf welchem es gewonnen

wurde. " (Heinrich Hertz, "Über die Grundgleichungen der

Elektrodynamik für ruhende Körper", Gesammelte Werke, 1894)

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