_{1}

^{*}

A magneto-electric field appearing in a laboratory due to moving charges has unusual properties. In particular, such a field of kinematical origin does not obey the wave equation with a non-relativistic velocity instead of light speed; so, its movement resembles that of a rigid body. In this paper the field of a uniformly charged sphere moving at constant velocity is considered. Relativistic axiom, implicitly used in the derivation of formulas describing a kinematic deformation for the proper spherical field from the point of view of a fixed observer, is revealed. A discrepancy was found between the generally accepted idea of the configuration of a deformed field and its real geometry. It is shown that the correct interpretation of known formulas leads to a logical contradiction, which cannot be eliminated within the framework of the theory of relativity. A scheme of a decisive experiment is proposed.

The Coulomb field of a stationary point charge, which has the simplest spherical symmetry with a strength decreasing inversely proportional to the square of the distance from the source, undergoes a very exotic deformation, if a charge (or an observer) is forced to enter a state of uniform rectilinear motion. It was first pointed out by Oliver Heaviside [

The chronological continuity between the relativistic concept of electromagnetism and Maxwell’s theory is well known. It was clearly manifested in the study of the motion of a point charge, whose field, from the point of view of a stationary observer in the laboratory, loses its Coulomb configuration. In this case, the vector field of electric strength E satisfies the wave equation

∇ 2 E − 1 c 2 ∂ 2 E ∂ t 2 = 0 .

As a parameter, it includes the speed of light c. The accompanying magneto-kinematic field of vectors H is subject to the same equation

∇ 2 H − 1 c 2 ∂ 2 H ∂ t 2 = 0 .

Consequently, the displacement of a field in the laboratory space with a velocity v has a non-wave nature; rather, it is akin to the motion of a rigid body for the non-relativistic case [

The following demonstrates the fact that in deriving formulas describing the kinematic deformation of a field for a stationary observer, a special relativistic axiom is implicitly applied. At the same time, a discrepancy between the generally accepted idea of the configuration of a deformed field with respect to its real geometry is revealed. The field of a uniformly charged sphere moving at a constant speed is considered. It is shown that the correct interpretation of known formulae leads to a logical contradiction, which cannot be eliminated within the framework of the theory of relativity. A scheme of the experiment is proposed which is to confirm or refute the generally accepted formula for a deformed electrostatic field of a charge moving at a constant velocity.

Recall how the expressions for the fields of a moving point charge are derived. Using the four-dimensional formalism of the special theory of relativity (STR), the authors of the book “Field Theory” [

“According to the formulas for retarded potentials, the field at the point of observation P ( x , y , z ) at time t is determined by the state of motion of the charge at the earlier time t ′ , for which the time of propagation of the light signal from the point r e ( t ′ ) , where the charge was located, to the field point P just

coincides with the difference t − t ′ . Let R ( t ) = r − r e ( t ) be the radius vector from the charge e to the point P; like r e ( t ) it is a given function of the time. Then the time t ′ is determined by the equation

t ′ + R ( t ′ ) c = t . (*)

For each value of t this equation has just one root t ′ ( [

“Passing now again to three-dimensional notation, we obtain the following expressions for the field potentials of an arbitrarily moving point charge:

φ = e ( R − v ⋅ R c ) , A = e v c ( R − v ⋅ R c ) , (**)

where R is the radius vector, taken from the point where the charge is located to the point of observation P, and all the quantities on the right sides of the equations must be evaluated at the time t ′ determined from the previous equation. The potentials of the field, in the form (**), are called the Lienard-Wiechert potentials” ( [

Let us turn to the case of uniform and rectilinear motion of a point charge along the trajectory r = r 0 ( t ) with a constant velocity v . We introduce the notation R ′ and R for the vectors R ( t ′ ) and R ( t ) , respectively (

E = − 1 c ∂ A ∂ t − grad φ , H = rot A , (1)

which yields the expressions

E = e 1 − v 2 / c 2 ( R ′ − R ′ v c ) 3 ( R ′ − v c R ′ ) ; H = 1 R ′ [ R ′ E ] . (2)

Here, the symbol [ .. ] denotes vector product.

“Indeed, at constant speed, the difference

R ′ − v c R ′ = R ′ − v ( t − t ′ ) (***)

there is a vector R from charge to observation point at the very moment of observation. It is also easy to verify by checking directly that

R ′ − 1 c R ′ v = R 2 − 1 c 2 [ v R ] 2 = R 1 − v 2 c 2 sin θ t , (****)

where θ t is the angle between R and v ” ( [

Now you can substitute the right side of the Formula (***) in the numerator, and the right side of the Formula (****) in the denominator of the first term in the expression (2):

E ( θ ) = e ( 1 − v 2 / c 2 ) R R 3 ( 1 − ( v 2 / c 2 ) sin 2 θ ) 3 / 2 .

We have arrived at a formula that describes the distribution in space of the electric intensity in a field of a point charge, which moves relative to the observer along a straight line at a constant speed. The authors of the book [

E = q ( 1 − v 2 / c 2 ) 4 π ε 0 r 3 ( 1 − ( v 2 / c 2 ) sin 2 θ ) 3 / 2 r , (3)

where the designation of the individual charge of an electron is replaced by the universal letter .

As you can see, in the SRT there is complete continuity in what is concerning the terminology of retarded potentials. This way of reasoning leaves behind the scene an arbitrary assumption of an incessant outflow of a field from its source, the charge. In fact, since the potential propagates from the field source at the speed of light “c” regardless of the kind of movement the charge performs, this outflow also takes place when the charge moves with a constant velocity v . And since the speed of light does not depend on v , the outflow of potential from the charge remains unchanged when the speed v tends to zero. Therefore, we have to agree that an unceasing extrovert flow from a charged source exists in any electrostatic field. This statement should be explicitly introduced into the axiomatic basis, when the SRT is transferred to the realm of electromagnetism.

The expression (3) can be obtained within the framework of a pure STR, without resorting to the Lienard-Wiechert potentials. Section 20 of the textbook [

φ = φ ′ + v c A ′ x 1 − v 2 c 2 .

On page 256 we read: “The scalar potential φ has a constant value on the surface of the ellipsoid

( x − v t ) 2 + ( 1 − v 2 c 2 ) ( y 2 + z 2 ) = c o n s t .

This ellipsoid is obtained from the sphere by compressing it in the direction of the x-axis into 1 : 1 − v 2 c 2 times”. We apply the formalism used there to the field of the conducting sphere of unit radius, over which the charge q is uniformly distributed.

This sphere has a centre at the origin ( x ′ , y ′ , z ′ ) of a primed IRF' (Inertial Reference Frame), moving in a straight line with a constant speed v = ( v , 0 , 0 ) relative to the non-primed (laboratory) IRF with coordinate axes ( 0 x , 0 y , 0 z ) parallel to corresponding axes of the primed coordinate system. For the origin of time, the moment is taken when the origins of spatial coordinate systems coincide; therefore, the position of the centre of the sphere in the non-primed IRF is the point with coordinates x = v t , y = 0 , z = 0 . The scalar potential outside the sphere in its own IRF' coincides with the Coulomb potential (in this section, formulas are written in the Gaussian system of units)

φ ′ = q r ′ .

On the sphere itself and inside it, it has a constant value φ ′ 0 = q / 1 . According to the Lorentz transformations for electromagnetic fields, the scalar potential in a non-primed system is

φ = γ φ ′ = γ q r ′ , (4)

since the vector potential here is zero. The Lorentz transformations for coordinates give the expression of the primed radius-vector from the point of view of the non-primed IRF in the form r ′ = γ 2 ( x − v t ) 2 + y 2 + z 2 . Substitute this expression in (4):

φ = γ q γ 2 ( x − v t ) 2 + y 2 + z 2 = q ( x − v t ) 2 + 1 γ 2 ( y 2 + z 2 ) .

Hence, the surface of level φ = a = c o n s t is described by the equation

( x − v t ) 2 + 1 γ 2 ( y 2 + z 2 ) = q 2 a 2 . (5)

Here it is logical to investigate the question on the deformation of the charge carrier—the conducting sphere of a unit radius—upon transition to a non-primed IRF. From the point of view of the non-primed IRF, the carrier of the moving charge loses its spherical shape. Consider the section of the sphere by the plane z = 0 (

x 1 = γ ( x ′ 1 + v t ′ ) , y 1 = y ′ , t 1 = γ ( t ′ + v c 2 x ′ 1 ) ;

x 2 = γ ( x ′ 2 + v t ′ ) , y 2 = y ′ , t 2 = γ ( t ′ + v c 2 x ′ 2 ) .

For the time interval t 2 − t 1 = γ ( v c 2 x ′ 2 − v c 2 x ′ 1 ) = γ v c 2 ( x ′ 2 − x ′ 1 ) the second end of the chord managed to drive off at a speed v to a distance γ v 2 c 2 ( x ′ 2 − x ′ 1 ) , which should be subtracted from the abscissa difference x 2 − x 1 to get chord length in non-primed IRF:

l = ( x 2 − x 1 ) − γ v 2 c 2 ( x ′ 2 − x ′ 1 ) = γ ( x ′ 2 − x ′ 1 ) − γ v 2 c 2 ( x ′ 2 − x ′ 1 ) = γ ( 1 − v 2 c 2 ) ( x ′ 2 − x ′ 1 ) = l ′ γ

By virtue of the arbitrariness in the choice of the chord and in the choice of the section by the plane passing through the abscissa axis, we conclude that the entire sphere undergoes longitudinal compression by γ times and turns into an ellipsoid of rotation. The half-axis 0A turns out to be γ times shorter than the radius of the sphere, that is, its length is 1 / γ . Just the same deformation is mentioned in the cited above excerpt from the textbook [

On the unit sphere itself, the potential is φ ′ 0 = q / 1 , and this spherical surface of level a = q / 1 becomes, according to [

( x − v t ) 2 + 1 γ 2 ( y 2 + z 2 ) = 1 (6)

in terms of the non-primed IRF. If you believe the textbook’s statement [

To put it mildly, strange misunderstandings are associated with the field configuration of a uniformly moving charge. For example, on the behaviour of vectors E we read in ( [

As for the picture of the equipotential surfaces, we are faced with fata morgana, which for decades has been unconditionally accepted by all readers for an objective reality (“When there is no real life, they live in mirages. Still, better than nothing.” A.P. Chekhov). The fact is that the notorious equation in the quotation from ( [

In

γ 2 ( x − v t ) 2 + y 2 + z 2 = 1.

The second is the equipotential surface φ 0 = q / 1 of the electrostatic field, objectively existing in the same IRF, with Equation (6). The second ellipsoid encloses the first, and between them is a layer of variable thickness. In the longitudinal direction, the layer thickness is ( γ − 1 ) / γ , and in the transverse direction, the thickness is ( γ − 1 ) , so that with increasing speed v , the longitudinal thickness approaches unity, while the transverse one increases without limit.

What lies within this layer? In accordance with the relativistic tradition, an ellipsoidal charge carrier retains a spherical potential, and in the layer under consideration everywhere we have

E = − 1 c ∂ A ∂ t − grad φ = 0 ,

because the scalar potential φ = q / 1 = c o n s t in a finite region, on the boundary of which it is constant, and the vector potential A = φ v , as a result, is also constant in this region and the partial derivative with respect to time is zero. It turns out that the charged ellipsoidal shell is paradoxically immersed in a volume free of the electrostatic field, and only from the outer boundary of this volume does the space, penetrated by the field (3) with equipotential surfaces of type (5), is starting.

The Ostrogradsky-Gauss theorem does not work in the resulting bizarre electrostatic field. This becomes apparent when the test closed surface surrounding the charge carrier is selected entirely inside the neutral layer. The flux of vector E through this surface is zero, while the integral charge inside is non-zero.

We will try to get away from this electric monster by allowing the scalar potential φ 1 of the ellipsoidal charge carrier to be different from φ 0 : φ 1 < q / 1 or φ 1 > q / 1 . In this case, we are faced with another paradox. Suppose there are two identical and equally charged conductive spheres uniformly moving in the laboratory IRF to meet one another. Own IRF' of the first sphere moves with velocity v and own IRF'' of the second sphere moves with velocity ( − v ), relative to the non-primed IRF. At the zero moment of time ( t = t ′ = t ″ = 0 ), when the origins of all three IRFs coincide, the charge carriers touch each other by the top and bottom limit points respectively. From the point of view of the laboratory IRF, both ellipsoids have the same values of scalar potentials φ 1 = φ 2 ≠ q / 1 and the voltage between them is zero. But in the IRF' the picture is devoid of such symmetry, because the first sphere has the potential φ ′ 1 = q / 1 , whereas the second sphere, compressed into an ellipsoid, has the potential φ ′ 2 ≠ q / 1 . Between them there is a voltage U = φ ′ 2 − φ ′ 1 ≠ 0 , causing a discharge. An electrical discharge occurs also in IRF'', but only with oppositely directed current. The pulsed current arising from the discharge becomes a source of electromagnetic wave, and its objective existence does not depend on the inertial reference frame. Consequently, the EM-wave should also appear in the laboratory IRF, where there is no current at all, that is, radiation arises from literally nothing! A sceptic who doubts the “promptness” of conduction electrons can be calmed by a metal tape longitudinally stretched in IRF , which will sufficiently prolong the contact time of charged spheres (ellipsoids).

So, contrary to the generally accepted opinion of the complete compatibility of the SRT with Maxwellian electrodynamics, the field of a uniformly moving charge throws to physicists in general, and to experimenters especially, a serious challenge. Indeed, since the time of Heaviside’s guess, that is, for more than 130 years, the “relativistic” configuration of such a field has not been confirmed by experience. In [

Suppose there is a high-ampere electron beam supported in a rectilinear vacuum tube with a length of 2s (x-axis in

d E = 2 σ d x 4 π ε 0 ( 1 − β 2 ) sin θ ( x 2 + h 2 ) ( 1 − β 2 sin 2 θ ) 3 2 = σ ( 1 − β 2 ) d x 2 π ε 0 [ x 2 + ( 1 − β 2 ) h 2 ] 3 2

according to Formula (3). Here the letter σ denotes the linear charge density, sin θ = h / x 2 + h 2 , and β = v / c . The total electrical intensity is obtained by integrating over the length of the beam:

E = ∫ 0 s d E = σ ( 1 − β 2 ) 2 π ε 0 ∫ 0 s d x [ x 2 + ( 1 − β 2 ) h 2 ] 3 2 = σ 2 π ε 0 ⋅ s h [ s 2 + ( 1 − β 2 ) h 2 ] 1 2 .

Expressing the charge density through the electron velocity in the beam and the current strength in it I = σ v , we get

E ( h , s , β , I ) = I s 2 π ε 0 v h s 2 + ( 1 − β 2 ) h 2 = I s 2 π ε 0 β c h s 2 + ( 1 − β 2 ) h 2 . (7)

If the electric field of a moving point charge remains Coulomb at any speed, then the dependence of E on β will be different:

E 0 ( β ) = I s 2 π ε 0 β c h s 2 + h 2 . (8)

The well-known experiment on the detection of the electric field around a superconducting ring with current [

The experimental scheme proposed here is completely free from the main obstacle facing the authors of the work [

In the paper [

The author sincerely thanks Mr Michael Kyle for his valuable help while preparing this paper.

The author declares no conflicts of interest regarding the publication of this paper.

Leus, V.A. (2020) Relativistic Paradox of a Uniformly Charged Sphere Moving with Constant Velocity. Journal of Modern Physics, 11, 145-155. https://doi.org/10.4236/jmp.2020.111009