"JOURNAL
OF RADIOELECTRONICS" N 10, 2000 |

A PROPOSED EXPERIMENT OF DIRECT DETECTING OF THE VECTOR POTENTIAL WITHIN CLASSICAL ELECTRODYNAMICS

**V.
Onoochin
"Sirius", 3A Nikoloyamski lane, Moscow 109004, , Russia**

*An experiment within the framework of classical
electrodynamics is proposed, to demonstrate Boyer’s suggestion of a change in
the velocity of a charged particle as it passes close to a solenoid. The moving
charge is replaced by an ultra-short pulse (USP), whose characteristics should
depend on the current in the coil. This dependence results from the exchange of
energy between the electromagnetic field of the pulse and the magnetic field
within the solenoid. This energy exchange could only be explained by assuming
that the vector potential of the solenoid has a direct influence on the pulse.*

PACS 41.10

1. Introduction.

It has been assumed for some time that electromagnetism is completely described by electromagnetic fields. Potentials, especially in classical electrodynamics, are treated as merely a convenient mathematical tool for calculations concerning these fields. However, Aharonov and Bohm (A-B) have shown that in the quantum description potentials rather than fields play the essential role [1]. Unfortunately, the requirements placed on experiments which might determine the existence of the AB effect are so stringent that other interpretation of the obtained results are possible. The Aharonov-Bohm interpretation has therefore been questioned [2, 3].

We are able now to describe an experiment to detect the
vector potential directly, whithin classical electrodynamics. The essential
advantage of this experiment is that it requires less accuracy in the
measurements of the currents and fields. The concept of the experiment
originated in the work of Boyer [4]^{ }which
predicts a change to the velocity of the particle as it
passes near the solenoid:

Here, *p* and *q*
are the particle momentum and charge. According to this model, the particle
slows when approaching the solenoid and then speeds up when leaving. Below we
point out such a possibility (we follow the consideration presented in Ref.5).

The total energy *W *of the system,
considering both the solenoid and the charged particle, is

where *B _{s}* is the fixed
magnetic field due to a current in the solenoid,

Since the *B _{s}*

The aim of this paper is to describe an experimental method
of demonstrating the above within the framework of classical electrodynamics. It
should be noted that the charge is not needed to show its velocity change, it
may be replaced by a current *qv*. Therefore, in the real experiment, it is
proposed to use a single ultrashort pulse (USP) to acts as this current.

The experiment is described in Section 2. In Sec. 3 we point out some electromagnetic properties of the USP which allow to achieve expected result. In Sec.4 we calculate the energies of the fields and currents involved in the process, giving the expression for the interaction that can influence the characteristics of the USP. Finally, in Sec.5 we discuss the possible results of the experiment in both cases, firstly in the situation when the characteristics are assumed to change, and secondly when they are assumed not to do so.

**2. Design of experiment.**

The arrangement of the proposed experiment is shown in Fig.1 (top view). Two wires form a dipole antenna. The antenna is placed in an anechioc chamber, in order that only the original signal will be observed by the spectrometer used to observe signals from different parts of antenna. The diameter of the wires is chosen simply to suit the requirements of mechanical rigitily; 1m wires can be fixed only at the point where the coaxial cable from the pulse generator connect to the dipole, and must remain straight.

Two solenoids (noted by the letters *s* in Fig.1)
are placed in the chamber. These are designed to be equivalent to those used for
the A-B experiments, i.e. the pulses in the wires can be influenced only by the
vector potential of the solenoid, but not by its magnetic field. Because if the
solenoid is formed of the cylindrical coil it always exists some “leakage”
of the magnetic field through the insulation layer of the wires independently of
how close the adjacent turns of the coil are mounted each to the other. So
actually some nonzero magnetic field created by the current in the coil does
exist outside the solenoid. To eliminate this factor, it is possible to use the
solenoid formed of the cylinder made of soft magnetic materials. Using of such
soft materials allows to control the magnitude of the magnetic momentum inside
the solenoid, and at the same time, there will be no magnetic field outside the
cylinder. To eliminate the magnetic field caused by the edge effects, it is
possible to use the magnetoguides which form jointly with the solenoids the
closed circuit for the magnetic flux.

Fig. 1

The USPs from the signal generator pass through the coaxial cable and appear as electromagnetic radiation from the antenna, and the physical form of the wire determines a path of the USP near the solenoid and the radiative losses from the signal. Losses can be significant at bends in the conductors, but if their radius of curvature is much greater than the diameter of the wire, the losses will be small since there is little deceleration of the current at the USP.

The generator can produce USP with duration of order of 10^{-10}
sec., by the way, the repetition time of generation of each separate pulse is of
order of 10^{-5} sec. [6, 7], so we
are able to assume the pulses as single ones. The characteristics of the USP are
determined after the radiation and after the pulse from the ends of the antenna
and returning to the generator. The ideal matching of the antenna is achieved in
case of full absence of the reflected pulses under the condition of zero current
in both solenoids (in performed experiments with similar design, the magnitude
of reflected current pulse typically was higher 95% of the USP loading the wire
antenna).

Two solenoids are used to achieve more matching of the wires characteristics, by the way, they are placed under the wires (see Fig. 1) for eliminating the influence of displacement current generated between the wires. So when the USPs pass in dipoles of the antenna this current can influence a radiation region but not the region where the solenoids are placed.

Fig. 2

Each wire has two parts. At the first one (noted as *ab*),
each USP moves transversally to direction of the vector potential created by
solenoid (along the *x* axis, see Fig.
2) so the interaction term between the USP and the solenoid

should be equal to zero. Here,
is the current density at the USP and *A* is the
vector potential of the solenoid, *V _{w}* is
the volume of the wire at the part

3. The electromagnetic properties of the USP.

Here, we give more explanation on the EM properties of the
USP because we have seeming contradiction since it is well known that the EM
pulse radiated by the antenna is always of plus and minus polarity. It is caused
by the fact that the radiated field *E* is
proportional to the time divergence of the current
at the pulse. Therefore, due to plus and minus polarity the total contribution
of
to the
interaction term can be equal to zero so we have no expected effect. However,
such a differentiation of
takes place only at the ends (points *d*) of the antenna so on the arc *bc*
each USP still remains of the only one polarity. One can object that when the
reflected USP returns from the ends of the antenna, it has the polarity opposite
to polarity of incoming USP, and because amount of the energy of reflected USP
is about (and more) 95% of the amount of the energy of incoming USP, the total
sum of the interaction term during the time of travelling separate USP through
the wire is quite equal to zero. But we do not intend to measure the energy
characteristics of the USP, we intend to measure changing the parameters of the
radiated signals which are separated in time for incoming and reflected USP, and
we intend to measure the changing the duration of the USP.

It is accepted that propagation of electromagnetic field inside the metal is described by the following wave equation [8]

Here, and are conductance and dielectric constant of the metal (the material of the wire).

However, one must note that the above equation is able to describe the transversal components of the EM field only, but the EM field of the UPS is of longitudinal type, i.e. the vector of the electric field is directed along the vector of the current (otherwise, i.e. if the electric field of the USP is of transversal type, the current pulse would radiate the longitudinal EM waves which is impossible). So here we have an example of the longitudinal EM field propagating with the velocity comparable to the speed of the light. To the author’s knowledge, unfortunately, the correct equation describing behavior of the USP is unknown in scientific literature. But we will not need to know the explicit functional dependence of the electric field of the USP on the spatial and time variables.

Thus, the USP moves along the wire and its magnetic field penetrates inside the solenoid. Therefore, according to the Eq.(1) the total energy of the system changes. However, to detect this changing, we face some problems:

- the magnetic field of the UPS induces the vortex electric fields in the coil so the latter fields create secondary magnetic field which must be taken into account while calculating the total energy;
- the problem considered here has complex boundary conditions for the fields; these conditions must be included into the calculations too;
- changing of the energy takes place, according to the Eq.(1) only inside the solenoid; - it is difficult to detect the changing of the energy of the magnetic field there.

- in the experiments concerning Aharonov-Bohm effect, it is assumed that the trial charge does not influence the current in the coil of the solenoid so the magnetic field of the latter is not distorted while performing the experiment. If we use the USP instead of the trial charge the electromagnetic fields of the pulse cause induction current in the coil so the additional magnetic field appears outside the solenoid and, therefore, we cannot state that the magnetic field is equal to zero outside the solenoid and only the vector potential is nonzero in this area.

The most difficult problem is the the problem of correct
describing the penetration of the magnetic field of the USP into the solenoid [9].
To overcome this problem, we use the method presented by the author in Ref. 10,
i.e. we assume that the original magnetic field of the USP penetrates into the
solenoid without any changing but this magnetic field induces some current *i
*in the coil of the solenoid, the current *i *creates
the additional magnetic field which penetrates into the solenoid too, and the
true magnetic field penetrating into the solenoid, except own magnetic field of
the solenoid is the algebraic sum of the original and induced magnetic fields.

We should note that we are not able to calculate the additional magnetic field, it is too combersome problem [9] but we are able to avoid such calculations in the following way:

We perform two experiments under quite identical conditions
but with the only difference, namely, during the first experiment, the value of
the initial current in the coil of the solenoid is *I*
, and during the second experiment, the value of the initial current in the coil
of the solenoid is zero.

Because of the principle of superposition in the
electrodynamics, value of the current *i* induced in the coil of the
solenoid by the current of the USP does not depend on the value of the initial
current in the solenoid, so the value of the current *i* must be the same
in both experiments. Therefore, by substracting the results of the second
experiment from the results obtained in the first experiment (experimentally by
comparing the results of both experiments one to another), we eliminate all
terms, containing the current *i,* from our
calculations. It allows to resolve all problems noted above. We discuss the
procedure of substracting the terms with *i* in more
details in the Sec. 4.

**4. Calculation of the EM energy of the system.**

It should be noted that the exact calculations of the total energy of the system require taking into account the initial and boundary conditions and, therefore, too cumbersome. As it is noted above, we are going to avoid all the problems in the following way:

It is proposed to perform two repeating experiments under
identical initial and boundary conditions except the only difference, namely,
the first experiment must be performed with the given value *I* of the
initial current in the coil of the solenoid, and the second one must be done
with *I *= 0. Obviously, for ideal solenoid, both initial and boundary
conditions for the electrical and magnetical fields must be equal one to another
in both experiments. So, under assumption that the magnetic field of the
solenoid does not influence the USP, the latter must pass along the wire under
identical conditions in both experiments, with given value *I* and with *I*
= 0. Therefore, subtracting the values of the conditions of the first experiment
from the corresponding values of the second one, we are obliged to obtain exact
zero, i.e. the boundary and initial conditions can be omitted from our
consideration. . So while performing calculations, we limit ourselves with the
expressions for the total energy of the system which do not consist of the above
conditions in explicit form.

Since on the part *ab* of the wires the plus and minus
pulses are weakly connected to one another, we are able to limit our
consideration, without loss of generality, only by one wire and one solenoid.

Here, before any calculations, one has to point out on
discrepancy between the proposed experiment and the experiments on AB effect. It
is shown in [5] (App.A) that, in the latter case, changing of
the charge's velocity does not take place. This is caused by performing of the
condition *I*(*t*) = *Const* i.e. the solenoid's supply source
compensates each changing of the current induced by the
magnetic field of the USP.

In this experiment, however, the velocity of passing of the
USP in the wire is comparable to the velocity of the light so we have adiabatic
regime when the external source cannot provide *I*(*t*) = *Const*.

So let us define *I* as an initial current that does not
change its value during passing the USP at the antenna and consider how the USP
influence the solenoid.

When penetrating into the latter, the magnetic field of the
USP induces in the coil, on the background of *I*, a current *i*; by
the way, *I* and *i* are independent of one another because the
physical cause of inducing of *i* is the electric field

where is a vector potential of the USP and it is independent of the external source.

Let us calculate the energy *W* of the system when the USP is on the
part *ab* of the wire (we use Gaussian units):

Here, *B*(*I*) the magnetic field produced by the
current *I* in the volume
of the solenoid. For simplicity of calculations, we will carry out the
integration of the magnetic fields over all the space, however, keeping in mind
that the field *B*(*I*) does not equal to zero only in an internal
area of the solenoid; besides, both other fields and
*B*(*i*) do not equal to zero in the external area too.
is the total energy of the electric field *E* inside and outside the wire.

It is important to point out - and due to this fact the
expected effect is achieved - that when the USP is on part *ab* of the wire
the magnetic fields of the USP and of the solenoid are directed in such a way
that their term of interaction

equals to zero. Indeed, *B*(*I*) field is directed
along the axis of the solenoid; besides the*
*field circulates around the *x* axis coinciding with the part *ab*
of the wire and intersecting *z* axis at a right angle. So after
integrating over all the space of the solenoid, the term (3)
will be equal to zero (Fig.2).

Obviously, due to the same symmetry it will be equal to zero
the term with the current *i* induced by the USP

when the USP is passing along the part *ab* of the
wire.

Supposing the vector potential of the solenoid does not
influence the USP, i.e.
remains the same, we calculate the total energy when the USP is on the
arc *bc*:

Actually, the energy of the USP changes while the USP is passing along the wire. But this changing is caused only by the resistance and radiation losses of the current pulse. We describe this changing by the equiation:

where is the energy of the electric field of the USP when the peak current of the
pulse passes the middle of the arc *bc* and
is the loss of the energy of the pulse while the latter passes from point *a*
to the middle of the arc *bc*. The resistance losses (Joule heat) of the
secondary current *i* are included in the term
too.

The total energy of the system cannot change at the adiabatic regime so

Subtracting the Eq.(2) from the Eq.(4), we obtain

Here, we use the linear dependence of the magnetic field on the current creating that field

Indeed, the magnetic field defines as a curl of the vector potential; the latter is defined from the (linear) wave equation of which right-hand side dependes linearly on the current. So, the vector potential and, therefore, the magnetic field is defined from the integral but linear operator of the current, and the magnetic field of sum of the currents is the sum of the magnetic fields created by each current independently.

Now let us calculate the total energy of the system in the
absence of the initial current in the coil of the solenoid, i.e. at *I* =
0. When the USP is on the part *ab* of the wire,

When the USP is on the part *bc* of the wire the total energy of the
system is

Subtracting the Eq.(6) from the Eq.(7), we obtain:

One can easily see now that consequent carrying out of two
experiments, with given *I* and with *I* = 0, allows to eliminate the
count of the secondary current *i* to the effect of influence the vector
potential on the USP. It is achieved by comparing of the results of two
experiments: let us subtract the Eq.(8) from the Eq.(5):

Now it is sufficient to carry out the integration only over the internal area of the solenoid .

Thus, the end parameters of the USP must change due to existing of the last term.

4. Discussion.

Since there is no the energy exchange with the external
sources, this term cannot be equal to zero if only the current *i* does not
fully shield the field inside
the solenoid (for example, for superconducting winding of the coil).

Therefore, to detect the greater effect it is desirable to
use the coil with such inductance and resistivity that the value of *i*
would be reduced as much as possible.

Now we should estimate another possible causes of changing of
the USP characteristics. The main one is charging of the coil while current
passing, the lower end of the coil by the (+) sign and the upper end by (-) sign,
for example. However, we can use magnetic materials instead of convenient coil
to create the same magnetic field *B*(*I*) but now without charging
the external side of the solenoid. Obviously, all the above calculations must
not change.

Another possibility to change the characteristics is
nonidentity of the USP produced by the generator. However, if we find for series
of the USP the dependence of power of the radiated signals from the antenna on
the value of *I*, it is sufficient to conclude about reality of the vector
potential as a physical quantity.

It would be liked to make one more note about possible influence of the magnetic field which could present outside the solenoid due to nonperfect

design of the latter on the USP. It is known that the
component of Lorentz force caused by the magnetic field is always directed
transversally to the current, i.e. to the motion of the USP, in this case.
Therefore, even the magnetic field created by *I*
would exist outside the solenoid it would not accelerate or decelerate the USP.

It should be noted that, independently of the end results,
the proposed experiment is of big physical interest. If the characteristics of
the USP coincide in both the cases of the given value of *I* in the coil
and of *I* = 0, it means that one must introduce into the energy equations
an unknown term which should fully compensate the count from the right-hand side
of the Eq.(9). However, the main problem is to give a physical explanation of
such a term.

If the USP loses its energy in some way, therefore, it should
exist a certain object which takes the energy from the USP and transfer to the
fields inside the solenoid. It can be solely the vector potential *A* that
forms the potential barrier for the pulse

where is the
volume of the piece of the wire at smooth joint from the straightline part *ab*
to the arc *bc*. If such a barrier exists in reality the USP should whether
reflect off or decelerate on it. We are able to detect experimentally both these
processes. In the first case, the reflected pulse can be detected in the
generator. In the second one, the decelerating pulse should radiate so the
receiver will register additional signal from the antenna.

It should be noted that one problem remains, namely, how our considerations corresponds to the gauge invariance. Unfortunately, the author does not know how to resolve this contradiction. The crucial argument in favour of reality of the vector potential as a physical quantity can be obtained from the experimental data. Only then any correspondence of the theory to the experimental data could be discussed.

It is interesting to note that in one experiment performed by Marinov [11] the obtained data can be explained by existence of the vector potential in the classical electrodynamics too. However, in the above experiment, it is very diffcult to perform the energy analysis similar to given one for the proposed experiment. Therefore, it is diffcult to state that changing of the velocities of the electrons in the rared gas tube surrounding the solenoid is caused namely by influence of the vector potential. In opposite to Marinov’s experiment the possible changing of the velocity of the charge is limited by the only cause.

Thus, independently, of the end experimental result, we have three possibilities:

- The end parameters of the USP do not change, therefore, we face deviation from the law of conservation of the energy;
- The end parameters of the USP do not change but it can be caused by the fact that we assume the form of the energy of free magnetic field
- The end parameters of the USP do change. It will mean that the vector potential influence the USP even in the region where the magnetic field corresponding to that vector potential equals to zero.

as incorrect;

To the author’s point of view, in the above experiment, the third case must be realized.

Final remark. It is accepted that in the classical electrodynamics not the EM potentials but the EM fields and, therefore, forces are the quantities describing behavior of the system. So it is interesting to analyse what force can be responsible to desceleration of the USP. Obviously, it is not Lorentz force. Mathematically such a force must be calculated as

using the vectorial identity

we obtain from the Eq.(11) four different terms including the term corresponding to the magnetic Lorentz force. However, in this case, the following term

is responsible for descelerating the USP. From the point of view of the convenient electrodynamics, this term is very unusual because

- it depends explicitely on the vector potential
*A*but not on its divergences, and - this “magnetic” force is of longitudinal type, i.e. is directed along motion of the charge, oppositely to convenient Lorentz force which is of transversal type.

References.

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**8**, 1679 (1973). - M. Peshkin and A. Tonomura ,
*The Aharonov-Bohm Effect*, (Springer-Verlag, 1989). - I. Grekhov, V. Kozlov, A. Rozkov, S. Shenderey and B. O'Meara, Power Picosecond Pulse Generators on the base of GaAs Devices, Record Abstract ElectroMed'99, April 1999, VA, USA, p.140
*Ultra-Wideband, Short-Pulse Electromagnetics*, eds. H.Bertoni et al., (Plenum Press, 1993).- L.D.Landau and E.M.Lifshitz,
*Electrodynamics of Contunuous Media*(in Russian), Ch.VII, (Moscow, Nauka, 1992) - T.H. Boyer, Amer. J. Phys.
**67**, 954 (1999). - V. Onoochin, in Proceedings of the 12
^{th}IEEE Conference on Pulsed Power, June 1999, Monterey, CA, USA. - S. Marinov, in Proceedings of the Scientific Conference “Space, Time and Gravity”. (St.Petersburg 1995.)

*Author:*

V. Onoochin, e-mail: a33am@dol.ru