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Notes: Measuring propagation speed of Coulomb fields

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Source: Arxiv.orgarxiv.org
Local copy: Measuring Propagation Speed of Coulomb Fields 1211.2913v2.pdf
See also: Notes on the Coulomb field

November 11, 2014

R. de Sangro, G. Finocchiaro, P.Patteri, M. Piccolo, G. Pizzella

Abstract

The problem of gravity propagation has been subject of discussion for quite a long time: Newton, Laplace and, in relatively more modern times, Eddington pointed out that, if gravity propagated with finite velocity, planets motion around the sun would become unstable due to a torque originating from time lag of the gravitational interactions.

Such an odd behavior can be found also in electromagnetism, when one computes the propagation of the electric fields generated by a set of uniformly moving charges. As a matter of fact the Liénard-Weichert retarded potential leads to a formula indistinguishable from the one obtained assuming that the electric field propagates with infinite velocity. Feyman explanation for this apparent paradox was based on the fact that uniform motions last indefinitely.

To verify such an explanation, we performed an experiment to measure the time/space evolution of the electric field generated by an uniformely moving electron beam. The results we obtain on such a finite lifetime kinematical state seem compatible with an electric field rigidly carried by the beam itself.

Introduction

We remark that an intriguing behavior of electromagnetism occurs when the field of an electric charge moving with constant velocity is computed. One finds that in such a case the electric field at a given point P(x, y, z, t) evaluated with the Li´enard-Wiechert (L.W.) potentials is identical to that calculated by assuming that the Coulomb field travels with infinite velocity.

This is a direct consequence of the velocity field, the part of the L.W. potentials independent of the charge acceleration, being a static field. This feature has been strassed [sic] by several authors e.g. [5] [6]. The Feynman [3] interpretation is based instead on the assumption that the uniform motion lasts indefinitely and that an observer would see an angular acceleration of the approaching charge.

The only way to shed some light on this problem, either the Feynman interpretation or the static Coulomb field carried rigidly by the charge is by means of an experiment.

To verify if the Feynman interpretation of the L.W. potentials holds in case of a charge moving with constant velocity for a finite time, we have performed an experiment to measure the time evolution of the electric field produced by an electron beam in our laboratory; such kinematic state has obviously a finite lifetime.

It is well known that a sizable number of instrumentation devices (e.g. beam position monitors) are based on effects produced by electric fields carried by particle beams. The effects and the propagation of such fields, however, have never been studied in details: the main point exploited by these devices is that the field effects are contemporary to the particles passage and that the signal size obtained, for instance, on a pair of striplines inside a vacuum pipe yields a measurement of the transverse position of the beam itself. The experimental situation for those devices is quite complicated to understand, as all the fields are inside a conductor and the transverse distances exploited are always small. We, on the contrary, tried to carry out our experiment in a clean environment: the electron beams used were propagating in a vacuum like environment. We covered a wide range of transverse distances w.r.t. the beam line (up to 55 cm). Such range leads to explore time and space domains for the emission of the detected field far outside the physical region.

The data we collected, as shown in the following, are compatible with the hypothesis of a Coulonb filed [sic Coulomb field] carried rigidly by the moving charge

[...]

Discussion

longitudinal distances between two sensors [cm] expected [ns] experimental a5 [ns] experimental a6 [ns]
(552.5-329.5) 223.0±1.5 7.43 ± 0.05 7.28 ± 0.02 7.52 ± 0.04
(552.5-172.0) 380.5±1.5 12.68 ± 0.05 12.62 ± 0.04 12.84 ± 0.05
(329.5-172.0) 157.5 ±15. 5.19 ± 0.05 5.21 ± 0.03 5.17 ± 0.04

Table 1: Timing measurements. The expected differences are calculated for 500 MeV electrons.

With reference to table 1, we notice that the longitudinal time differences are completely consistent with the hypothesis of a beam traveling along the z axis with a Lorentz factor γ ≈ 1000.

Such an occurrence agrees with the Li´enard-Weichert model. Retarded potentials, however, predict that most of the virtual photons [10] responsible for the field detected at coordinates z and y be emitted several hundred meters before the sensor positions and at different times according to the detectors transverse distances. Conversely, assuming that such virtual photons are emitted in a physically meaningful region (between the beam exit window and our detectors), the amplitude response of the sensors should be several order of magnitude smaller than what is being measured (cfr. Fig.2). Our result, obtained with a well definite set of boundary conditions (longitudinal and transverse distance between beam line and sensors, details of the beam delivery to the experimental hall etc.) matches precisely (within the experimental uncertainties) the expected value of the maximum field calculated according to L.W. theory, that is also the value calculated with Eq.4 when the beam is at the minimum distance from the sensor.

We again point out that the consistency of our measurement and eqn 8 has been obtained without any kind of normalization.


Conclusions

The data we have discussed in the previous paragraphs led us to assume that the electric field of the electron beams acts on our sensors only after the beam itself has exited the beam pipe and that Cerenkov and/or transition radiation effects are negligible.

Our results agree with the prediction of L.W.formula, however if equation 1 is intended as if the fields were launched at an earlier time with respect to the sensors’ response, such response would be orders of magnitudes smaller than the ones we measure.

The Feynman interpretation of the L.W.formula for uniformly moving charges does not show consistency with our experimental data. [bold mine] Even if the steady state charge motion in our experiment lasted few tens of nanoseconds, our measurements indicate that everything behaves as if this state lasted for much longer.

To summarize our finding in few words, one might say that the data do not agree with the most common interpretation of the Li´enard-Weichert potential, while seem to support the idea of a Coulomb field carried rigidly by the electron beam.