Electrodynamics are considered to be a beautiful specimen for other sections of physics. But two points look unsatisfactory:
1. electrodynamical forces do not satisfy the third Newtonian law,
2. some experiments produced by scientists in Russia, USA, Austria and other countries show existence of electrodynamical forces which are not described by modern electrodynamics.

Generalized formulas for Lorentz force and Maxwell equations are proposed. These formulas satisfy the third Newtonian law, predict classical forces for classical cases and some nonclassical forces (in their number all the forces appearing in experiments known to the author). They also predict appearing forces of the third rank smallness with respect to light velocity c which apparently are manifested in electro-weak interaction.

 INTRODUCTION

Last years Nikolaev in Russia (Tomsk University) [1], Marinov in Austria (Institute for fundamental physics in Graz) [2], Spencer et al. In USA (University of Connecticut) [3] produced experiments which showed existence of nonclassical forces in electrodynamics. Experiments of this kind were produced earlier for instance by Kennard [4], Sansbury [5], Edwards [6], but they were not paid necessary attention. All the authors proposed their own theoretical explanation for the facts. This paper is in their number. On historical reasons modern electrodynamics opened to be a combination of two similar but different approaches. In 1846 one of these approaches was formulated by W.Weber who proposed a generalization of Coulomb law for the case of moving charges. Electromagnetic interaction was understood by him as interaction of charges and no other understanding could be because Coulomb law was generalized. Characteristic of such an interaction was force and up till now all the devices measuring electromagnetic field in fact measure force. The second approach was proposed by Maxwell. This approach partly excluded Weber’s one by the end of the century. Maxwell’s idea was founded on the concept of field tensity which was formally realized as a solution of certain system of differential equations, but its physical essence was not clear enough. Lorentz proposed an idea which was to combine Weber’s and Maxwell’s approaches. He proposed a force formula in which not two charges (that would correspond to Weber’s approach) and not two field tensities (that would correspond to Maxwell’s approach) but a charge and electromagnetic field figured. Lorentz believed that electric field originated by one charge directly interacts on static other charge, and magnetic field originated by the first charge interact on the charge moving with certain velocity. This belief was fixed in tradition and now textbooks often use Lorentz formula as a definition for electric and magnetic fields tensities i.e. tensities are defined as forces. Nevertheless it became clear soon that Lorentz formula does not cover all practical cases and it is necessary to often use other methods to find force. Let us cite characteristic opinion in Feynman lectures [7, p.54]: “Two possibilities: “a loop moves” and “a field changes” are undistinguishable in formulation of “flow rules”. Nevertheless in order to explain the rule in these two cases we use two quite different methods: Lorentz formula for moving loop and Faraday law for changing field. We know in physics no other example when a simple and accurate law needs for its real understanding an analysis in terms of two different phenomena. Usually such a beautiful generalization opens to issue from a common deep basic principle. But in this case some very deep principle is not seen. We are to perceive a rule as a mutual effect of two completely different phenomena.” Let us say in addition that Lorentz formula does not imply the third Newto9nian law.

The proposed generalization for Lorentz formula and Maxwell equations is coordinated to the third Newtonian law, “works” in all cases known to the author, predicts classical effects in classical cases and predicts existence of the third rank small with respect to light velocity c force which apparently is essential for electro-weak interaction.

 1. MAIN FORMULA

Let rectangular right hand coordinate triple is fixed in three-dimensional Euclidian space. Let be a point in this space, t be time and i, j, k be orts. Let be electric charges 1 and 2. be their velocities and accelerations. If other assertion is not declared, charges are considered to be evenly distributed in a ball of radius . Let be electric and magnetic fields’ tensities originated by the charges in space. Double index from below means field tensity created by the charge whose index goes first in the point where the charge whose index goes second is situated. For instance means electric field tensity created by the second charge in the point where the first charge is situated. Let be the radius-vector from charge 2 to charge 1, r be its modulus, , and be electric constant.

Generalized formula for Lorentz force. Charge 2 produces the following force on charge 1:

Gradient is calculated with respect to passive charge 1 coordinates. Here and everywhere below , where is light velocity. This quantity is called pseudoscalar light velocity.

Two notions of force are used in modern physics: idea inherited after Newton and Descartes as an impulse derivative with respect to time and idea inherited after Huygens and Leibnitz as an energy gradient. (1.1) combines both ideas. Every charge creates electric and magnetic fields in space. Scalar product of passive charge magnetic field and active charge electric field describes interactive energy density originated by the charges. Its integral is in square brackets in the first item. Vector product of charges’ magnetic fields defines interactive impulse density. Its integral is in square brackets in the second item.

One needs to find the fields originated by the charges. One can get it from equation describing the fields. Maxwell equations is such a system in classical theory. But Maxwell equations should be modernized in order to be coordinated to formula (1.1).

Generalized Maxwell equations. Electric charge q distributed in the space with density , originates electric and magnetic fields which are solutions of following system:

Let us begin our explanations with the equation (1.5).

where v is the charge velocity. The first item in the right hand part of (1.6) generalizes the idea of a current in classical theory and comes to it if E satisfies some additional conditions.

(,

where j is current density. So right hand part of (1.5) contains a rotor component in addition to classical one. This item is manifested for instance in a light wave.

(1.4) means that equations (1.3) – (1.5) generalize the idea of magnetic field. Magnetic field B that is the solution of (1.3) – (1.5) possesses not only rotor but divergent components as well. Divergent component of B is defined by pseudoscalar electric charge (usual electric charge divided by mixed product of orts and light velocity). B opens to be pseudovector just like in classical theory.

Right hand part of (1.4) may be considered as “another incarnation” for electric charge because existence of electric charge is necessary and sufficient for its existence. One may consider it as a “magnetic charge”as well. But it is necessary to emphasize that such a “magnetic charge” does not coincide with Dirac’s monopole. Let us pin-point some of the differences.

1. Such a “magnetic charge” is pseudoscalar, i.e. its sign changes when right hand coordinate triple is changed for left hand one.
2. It is c times less than electric charge, correspondingly its dimension differs from electric charge dimension.
3. And last but not least (1.1) implies that two static “magnetic charges” do not interact, because second item in (1.1) responsible for magnetic field’s interaction is zero in this case. I ask reader to pay attention to this fact because “ordinary physical mentality” usually identifies field and force.

So (1.3) differs from classical one in that it includes gradient derivative of B originated by electric charge (and correspondingly “magnetic charge”) movement with velocity v.
Classical theory associates the appearance of magnetic field just with the movement of electric charges but does not include the originating movement into (1.3) equation.
(1.2) coincides with classical one.
E and B in (1.2) – (1.5) may be defined by means of potentials.

Let A be vector and be scalar potentials of electric field and satisfy the following equations

(1.10) means that A is rotor of a certain vector function and is independent with respect to t . If is imagined as a density of a certain “electric liquid” and A defines velocity of such a liquid, then the first part of (1.10) opens to be a continuity equation for and the second part of (1.10) becomes a condition of incompressibility for .

If to define

B= +grad /c +rot A (1.11)

E= , (1.12)
then (1.8) – (1.10) comes to (1.2) – (1.5).

1.

4.

5.

If the time of signal's lagging behind is not essential with respect to the problem's conditions then the derivatives are calculated at the same time t. Otherwise the active charge velocity and acceleration should be calculated at the previous time.

One gets finally: the force which the second charge produces on the first one is

One gets another form for the force when vector triple products are revealed:

Let us write out additional form of (3.2) using explicitly the angles between vectors. Let

be angle between and

be angle between and

be angle between and

be angle between and

be angle between and

be angle between and

be angle between and

be angle between and .

All derivatives here are calculated with respect to laboratory frame of reference for “nude charges” and with respect to conductors for currents in neutral conductors. Let us return to function (2.5) and (2.6). The second item in their right hand parts define static component and is manifested only for “nude charges”. The first one defines dynamic component and is manifested not only for charged but for neutral currents as well. This quality is inherited when these components are multiplied and when derivatives are calculated in formula (1.1). For instance the first item in (3.1) – (3.3) is got as a gradient of static components’ product. Therefore it is valid only for “nude charges” (Coulomb force). On the contrary the first square bracket is a result of dynamic components’ product. So it is valid for neutral currents as well. One can easily see that this square bracket is a symmetrization of classical Lorentz force. The first two items correspond to this classical case and the second two ones work in symmetrical cases.

The second square bracket in (3.1) – (3.3) is a product of dynamic and static components. So it is equal to zero between two neutral currents. It is valid if at least one of the currents is charged. This square bracket depends on the charges velocities’ difference and predicts some effects of Relativity theory. It also predicts a force produced on a “nude charge” at rest near a neutral current.

The third square bracket depends on the charges’ accelerations and describes field radiation. It is valid for all kinds of currents because radiated field should be considered as a “nude one”. It usually predicts the same results as classical theory but example 2 in section 4 shows that it predicts no radiation for an electron rotating around nucleus.

The last two items in braces have the third small rank with respect to light velocity c. They are apparently essential in electro-weak interaction.

2. EXAMPLES

EXAMPLE 1
Let N charges are evenly distributed along circumference of radius situated in the () plane with the center in the coordinate system origin. The charge is situated in circumference center. If is at rest, classical Lorentz formula and (3.3) formula predict only Coulomb force directed along radius. Let moves with constant velocity v along axis. Relativistic effects are predicted in this case by classical theory. They are considered to change Coulomb force magnitude but to preserve its radial character. This force is considered to be

where is angle between v and radius-vector to .

When is small enough and it is possible to decompose (4.1) in a row, one gets

When , (4.1a) predicts Coulomb force decrease with multiplicity of . When ( about ) additional to Coulomb force ( second item in (4.1a)) changes its sign. When it predicts Coulomb force increase with multiplicity of . When increases, other items in the row become essential, and when Coulomb force transverse deformation has multiplicity of .

Let us see predictions of (3.3) formula. Only the second square bracket is nonzero in (3.3) for the case. Two forces are predicted by this bracket: radial force and directed along velocity force .

One gets for radial force:

One can see that (4.2) predicts qualitatively the same but twice greater result for small in comparison with (4.1a). This difference in transverse direction decreases with increase and becomes zero when . When , and aims at double Coulomb force. Let us note that (4.2) is also valid when one of the currents is neutral ( for instance are distributed in a neutral conductor), while (4.1a) is zero for the case.

Velocity force

This force is maximum when ( longitudinal direction). When it decreases from to zero and when it goes on decreasing from zero to . Common force produced on charged circumference is sum

originates tangential to circumference force. If is a negative charge and circumference is neutral conductor, free electrons gather in the region where circumference cross axis. Correspondingly and circumference cross is charged positively. This charging goes on until mechanical moment because of Coulomb force equilibrates the moment transferred to the system by external forces which give velocity v to charge ( see details in section 7). If velocity is not constant i.e. moves with acceleration, an additional force ( the third square bracket in (3.3)) is produced on circumference charges. Its magnitude

If velocity and acceleration directions coincide then this force is maximal on the cross of circumference and axis (). It decreases not changing its sign on the intervals , .

Some deductions.

1. (3.3) predicts two ( in the case of accelerated movement – three ) forces produced on test charge.
2. Acceleration force coincides with classical one. Radial force is close to Relativity theory predictions in big range of velocities. But velocity force is not predicted by classical theory and may be used for experimental verification of the proposed scheme.

EXAMPLE 2

Let a positive charge is at rest, i.e. . A negative charge rotates around with constant velocity and constant centripetal acceleration . What effects does (3.3) predicts?

The first square bracket in (3.3) is zero because . The third square bracket is zero because ( one can see this especially clear in (3.1)). , i.e. . One gets finally:

. (4.5)

(4.5) predicts no force produced on because of centripetal acceleration, hence does not radiate. Such a radiation takes place only if is tangentially accelerated. (4.5) predicts radial force which “helps” to Coulomb one. This force leads to orbit rotation as a unit ( pericenter removal in the case of elliptic orbit). One may compare this assertion with example 4 of section 6.

EXAMPLE 3

Let charges and of the same sign move along parallel straight lines with equal constant velocities, i.e. , , and only the first square bracket is nonzero.

(4.6) implies that in addition to Coulomb force radial force ( the second item) and directed along velocity force ( the third item) are produced on charge 1.

When ( approximately and ), radial force is zero. When and , is positive and “helps” Coulomb force. When , it is negative and “weakens” Coulomb force.

Velocity force is equal to zero when i.e. charges fly “side by side”. When ( the first charge is behind the second one), is directed along the first charge velocity and accelerates it ( the second charge “helps” its partner). When ( the first charge is before the second one), is directed against the first charge velocity ( the second charge brakes the first one). Modulo equal and oppositely directed force is produced on the second charge, so equilibrium point for the charges is going “side by side”.

Let is distributed with constant density along axis. This means that boundary conditions (1.2) and (1.4) are changed. (1.2) looks as follow:

are the points where the charges are situated.

Equality (1.2a) should be integrated over topological product in order to find . Here b(r) is a circle of radius .

Volume integral in the right hand part of (1.2a) comes to integral with respect to d over . One can get details of passage to the limit for instance in [8, p.21].

One gets finally

Where r₀ defined by boundary va;ue on x _3. Instead of p.2 of section 3

One gets instead of p.5 of section 3 just in the same way

If calculations of section 3 are repeated one gets

The same formula looks as follows being expressed with the help of angles between vectors:

EXAMPLE 1

Let charge moves parallel to with the same velocity as charge along, i.e. .

All the square brackets in (3.3a) are equal to zero except the first one in which , .

One gets finally

(6.1) coincides with Lorentz formula predictions.

EXAMPLE 2
Let in the previous example . The first and the second square brackets in (3.3a) are nonzero for the case, .

(6.2) differs from Lorentz formula predictions with coefficient in the second item. The reason is that Lorentz formula ( the first square bracket in (3.3a)) does not distinguish charged and neutral currents interaction. Meanwhile (3.2a) and (3.3a) takes this difference into account with help of the second square bracket. It is equal to zero for two neutral currents and coefficient (1-4lnr) comes to 1.

 EXAMPLE 3

Let the first charge moves perpendicular to axis from it along radius-vector. The first two square brackets are nonzero in (3.3a), .

The force produced on is

The last square bracket is not predicted by Lorentz formula.

EXAMPLE 4

Let is steady neutral current and “nude” charge is in rest, i.e. . Classical theory predicts no force produced on . But the second square bracket in (3.3a) is nonzero, it predicts

(6.4) can be used for experimental verification of the proposed theory.

Let us clear up mechanical qualities of the two charges’ system in consideration. Let us emphasize that (3.1) – (3.3) suppose that external forces which induce charges’ velocities and accelerations are produced on the system. formulas contain noncentral items, therefore classical mechanical theorems can not be transferred directly on the system under our consideration. But the principle force vector

One gets integrating this identity with respect to time and along a voluntary trajectory in space

Equalities (7.2) and (7.3) mean validity of two theorems.

Theorem 1. Internal forces do not change the system impulse.

Theorem 2. Internal forces do not produce work.

Let us find internal forces momentum. Let O be voluntary point in space, be radius-vector from O to and be radius-vector from O to . Internal forces’ principal momentum with respect to O is

(7.4) means validity of

Theorem 3. Force momentum transferred to the system by external forces does not depend on the point of its application and comes to force couple.

EXAMPLE 1

Let us find force momentum produced on the charges in example 3 of section 4. The force is defined by (4.6).

(7.5) means that both arms work.

EXAMPLE 2

Let us find force momentum produced on the charges in example 3 of section 6. The force is defined by (6.3).

Only the first equality here is valid in accordance to Lorentz force, i.e. only one arm works in classical case.

J.G. Klyushin, ph. d., St-Petersburg, Russia, State University, Faculty of applied mathematics.
Russia, 192242, St-Petersburg, Budapest str. 5-3-241, tel. (812) 174-88-48 (home),
(812) 246-25-07 (office), E- mail: echoice@a33spb.su