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Divergence of Electric Field of Continuous and of a Point Charge for Relativistic and non-Relativistic Motion

In this paper we considered divergence of electric and of magnetic fields for four cases: classical point charge, classical continuous charge, relativistic point and relativistic continuous charges. Results for classical and relativistic point charges are the same as in literature, i.e. Gauss's law is valid. However results for time-varying classical and relativistic distributed charges indicate that divergence of electric field is not zero even for volumes of space where no charges are present. For these cases original Gauss's law might require modification. Divergence of electric field seems to be far-field type scalar anisotropic field, which is generated by time-varying electric charges or currents. Results indicate that for these effects to be sufficiently large to be experimentally observable the time variation of electric charges and/or of currents should be very fast. Divergence of magnetic field is zero for all cases.


I. INTRODUCTION
Maxwell equations describe relations between electric and magnetic fields and electric charges ∇ · E = ρ ε 0 (1a) where E and B are electric and magnetic force field strengths, J and ρ are electric current and electric charge densities, ε 0 and µ 0 are permittivity and permeability of space, · and × denote dot and cross product. Using (1b) and vector calculus identity that divergence of curl of arbitrary continuous vector field is zero, magnetic vector potential field A is introduced: If (2) is substituted into (1c) and vector calculus identity, that curl of gradient of arbitrary continuous scalar field is zero, is used, then electric scalar potential field V is introduced: Even though E and B are force fields, while V and A are potential fields, for simplicity we will refer to the former as fields, while to the latter as potentials. Historically electric and magnetic fields were discovered (or introduced), and therefore were used, before the vector and scalar potentials. However there is a point of view, supported even by founders of classical electrodynamics, like Maxwell and Faraday, that these vector and scalar potentials are more fundamental than electric and magnetic fields 1 . This assumption becomes apparent when we derive Liénard-Wiechert potentials and fields for a point charge. For example, in his classical textbook "Introduction to Electrodynamics" Professor D. Griffiths argues that for a point and continuous charges it is correct first to extend the classical potentials to relativistic motion and then from these extended potentials to find relativistic fields.
If we try simply to extend the non-relativistic fields to relativistic case, then we get the wrong result. In other words, classical electrodynamic fields lead to classical electrodynamic potentials, which are extended to relativistic electrodynamic potentials, which in turn lead to relativistic fields. This relationship between electrodynamic potentials and fields is often encountered in science. If a fact or event X in all cases leads to Y (X → Y ), it doesn't necessarily mean that Y in all cases leads to X (Y → X). In other words, if Y is more fundamental than X, i.e. it is generalization or extension of X, then Y only in special cases leads to X. In this work we simply want to go one step further and check if divergence of electric and of magnetic fields, calculated from electrodynamic potentials, still satisfies the Gauss's law. For example, if it turns out that electrodynamic fields, obtained from potentials, do not satisfy the Gauss's law, then it would mean that potentials are more fundamental than fields. If, however, fields do satisfy the Gauss's law, then it would mean that potentials are not more fundamental than fields.
Using (2) and (3) four Maxwell equations in (1) can be reduced to the following two where L is defined as ∂t 2 denotes d'Alembertian operator. For a vector valued function A Laplacian operator can be written as ∇ 2 A = ∇(∇ · A) − ∇ × (∇ × A). Often using gauge transformation, L is set to 0, which is called Lorentz (Lorenz) gauge condition. For simplicity we will refer to L as Lorentz gauge function. In Lorentz gauge condition (4) become In this work we also will check if Lorentz gauge condition is satisfied for electrodynamic potentials for different cases.

II. THEORETICAL METHODOLOGY
In order to calculate the divergence of fields, we have to first calculate the fields themselves. However as mentioned above, fields are obtained from scalar and vector potentials 1 .

Speed
Classical Relativistic Scalar and vector potentials are different for point and for distributed charges. For a point and/or distributed charge, potentials also differ depending on relativistic or non-relativistic cases. Non-relativistic case is often called classical. In relativistic motion we can not ignore retardation effects, or we have to consider finite speed of information propagation in the field. While in classical case retardation effects can be ignored, or in other words the speed of propagation of information can be taken as infinite. Therefore in this paper we will consider four cases: classical and relativistic point charge, classical and relativistic continuous (distributed) charge. Table I summarizes and presents the four different cases of electrodynamic potentials 1,2 . From mathematical point of view, these vector potentials represent solution of non-homogeneous wave equations in (6) 3,4 . This will be our starting point. For these four different cases, we will re-derive electric and magnetic fields, divergences of these fields and we will check whether Lorenz gauge condition holds for them, i.e. if Lorentz gauge function is zero (L = 0) and if (6) are satisfied.
In equations presented in Table I, r denotes radius-vector of a point, at which we consider the fields at current time t, relative to origin of our arbitrarily chosen reference frame. r denotes radius-vector of a point, at which field is originated at retarded time t r , relative to origin of our arbitrarily chosen reference frame. r = r − r denotes position of a point, at which we consider fields, relative to a point, at which the field is "originated" or "emitted". r andr are magnitude and unit vector of r . q and I denote electric charge and current, ρ and J denote electric charge and current densities, while v is velocity of point charge or of distributed current flow. The following coordinate notation will be used for position vectors, are Cartesian unit vector of our chosen reference frame. For all derivations in this paper we assume that our point of observation r is stationary relative to our reference frame, i.e.
Additionally in this and in all following equations nabla operator is expressed in spatial partial derivatives of coordinates associated with current time t Therefore two following identities, which will be valid for all derivations in this paper, can be derived for r ∂ ∂z x y z = 0 (9b) Equations (7), (9a) and (9b) constitute our assumptions about point r, which can be called "global" assumptions, because they apply for all cases that will be considered.
In order to derive electrodynamic fields from potentials in Table I, we need to consider the following 2 general conditions. First classical case will be considered.

A. Classical cases (v c)
For a given time t retarded time t r is a function of r only: In other words, our current time t is taken as constant, because we can arbitrarily set and/or pick its value, and so gradient of retarded time is For gradient and time derivative of r we have In the derivation of (12) the assumption, that partial derivatives of x , y , z with respect to x, y, z are zero, i.e. ∂x /∂x = ∂y /∂x = ... = ∂y /∂z = ∂z /∂z = 0, was made. In other words, for classical case source at position r moves at low speed compared to the speed of propagation of field disturbance (or information) c, so that in this case even for a moving point or moving distributed charge r can be considered as almost static and constant for a given r and t. Thus electromagnetic fields at r and t can be considered as "emitted" by almost "static" (or slowly moving compared to c) charges. This assumption also implies that ∂x /∂t = ∂y /∂t = ∂z /∂t ≈ 0, which were used to obtain (13). In essence, as mentioned above, in classical case retardation effects can be neglected. For classical case above mentioned assumptions about x , y , z can be summarized as, in terms of r : We should give a word of caution here. Original assumptions ∂x /∂x = ∂y /∂x = ... = ∂y /∂z = ∂z /∂z = 0 lead to (14(b, c)), but (14(b, c)) don't lead to these assumptions.
Let's substitute back (12) into equation for gradient of retarded time t r , (11) By applying assumptions in (7), (9), (14) to the definition of r , the following identities can be found If we differentiate (10) with respect to t using (13), then we get the identity which is the same as 1 From (12) the following relation is obtained for gradient of r in power n ∇ r n = n r n−1 ∇ r = n r n−1r where n ∈ Z. For time derivative we have ∂ r n ∂t = 0 (20) Next using (16a)-(16c) and (19) the following general identities can be found ∇ · ( r nr ) = (n + 2) r n−1 , n ∈ Z, n = −2 The exception for the (21a) is a case when n = −2, at which point (21b) becomes valid 1 .
In (21b) δ 3 ( r ) = δ(x)δ(y)δ(z) is three-dimensional Dirac delta function, with the following For arbitrary vector function P(r , t r ) and a scalar function S(r , t r ), using (15), the following useful identities can be derived: whereṖ = ∂P ∂tr notation is used, i.e. dot means differentiation with respect to retarded time t r . For time derivative we have, using (18) Additionally the following two relations will be useful for us later Derivation of equations for the relativistic case is similar in strategy to derivation performed in classical case. However there is major difference in a fact that now we can't assume that "source" at r and at time t r moves with low speed compared to c. Equations (10) and (11) will be valid in relativistic case too, but there will be difference in ∇ r : During the derivation of (30) the assumptions for x , y , z , summarized in (14), are no longer used. Instead, we take that ∂x ∂x = ∂x ∂t r ∂t r ∂x = 0 and in the same way for y and z , and due to these terms gradient of retarded time t r appears in (30) (compare with (12)). Let's substitute (30) into (11), and after small algebraic manipulations we can find gradient of retarded time where v is velocity of motion of a "source" at retarded position at retarded time, i.e. a "source" was moving with velocity v, when it was "emitting" fields, which we measure later If we substitute back (31) into (30), then we get If we differentiate (10) with respect to t, then we have Next we apply differentiation with respect to t to the r where the following assumptions were used for x , y , z , but we write only for x : From the (34) and (35) we obtain the following relations The above mentioned assumptions about x , y , z can be summarized in terms of r as following, using (31) and (37) Using global assumptions (7), (9) and assumptions (39) and the relation r = r − r , the following identities are derived for r ∂ r From (33) the following relation is obtained for gradient of r in power n ∇ r n = n r n−1 ∇ r = n r n−1 c r and using (38) for its time derivative Next using (38), (40a)-(40c), (41) the following general identities can be found For arbitrary vector function P(r , t r ) and a scalar function S(r , t r ), using (31), the following identities can be derived: where againṠ = ∂S ∂tr notation is used. For time derivative, using (37), we have Additionally the following three relations will be useful for us later where in the last equation T(r , t r ), like P, is an arbitrary vector function of r and t r . In (50) we should not directly apply the transformation as, for example, in (51). Instead we should proceed with differentiation of r nr with respect to x, y, z and later apply that transformation. The following two identities will be useful for us during the derivation where a is acceleration of a "source" at retarded position at retarded time Lastly, next two identities will be necessary for us where b is time derivative of acceleration, or boost, of a "source" at retarded position at During the derivation we often will use vector calculus identities, but in order to save space, we will not present them here.

III. RESULTS
Here we present results of our derivations.

A. Classical point charge
For a point charge we assume that its charge is constant q = const, i.e. doesn't change with time t r or with space coordinates r . This assumption is due to fundamental property of nature that a single charged particle, like electron, can not be divided (or multiplied).
Potentials for this case are given in Table I in  can't generate steady current 1 . Therefore current I should be called pseudo-current, but for simplicity we will continue to call it current. For classical case we assume that charge is moving slowly, i.e. v ≈ 0 and (14). Therefore even from strict mathematical point of view we need to take I = 0 and A = 0 when v ≈ 0, as long as v c, we can assume that there is small nonzero current, which gives rise to nonzero A. Due to the fact that we can ignore retardation effects, we assume that current is constant I = const. In other words, motion of a charge, at the moment when it "emits" the fields, does not affect the fields. Thus current does not depend on t, t r , r, r . Using (19) and (21b), we can find gradient and Laplacian of Using (13) the following two identities are found: Substituting (59b) and (60b) into (6), we get Thus Lorentz gauge assumption in terms of scalar potential V is satisfied. Now we do the same analysis for vector potential A(r, t) using (19), (21b), (21c) and some vector calculus identities.
We see that strictly speaking, Lorentz gauge condition is not satisfied, but if we take exact cases when I = 0 or when I is perpendicular tor , then it is satisfied. Let's now find fields.
By substituting (59a), (62c), (63a) into (2) and (3) we get These are electrostatic and magnetostatic fields of a point charge. As mentioned in 1 , (66b) is approximately correct for a point charge when we can neglect with retardation, which is precisely the case we are considering here. Finally, in order to see if Gauss's law is satisfied for these electric and magnetic fields, let's find divergence of fields by using (19), (21b), (21c), (66a), (66b) These results give us back original equations from electrostatics and magnetostatics. In  Table I, and we present them here also There is slight difference in variables used in vector potential A given in (58) and in (68), in that in the former there is current I, while in the latter there is velocity v, and I = qv.
Taking into account (44), (52) and (55), we find for scalar potential Using (38), (53) and (56), and after some algebraic manipulation, we can find time derivatives of scalar potential If we substitute (69b) and (70b) into (6), then we get This result shows that Lorentz gauge condition for V in this case has trivial solution.
Using (44), (45), (49)-(51), (52), (55) and some vector calculus identities, we find for vector potential A Now we need to compute time derivative of A, using (38), (43d), (47), (48), (53) and (56), we get If we substitute (72b), (72d) and (73b) into (6), then we get ativistic point charge. Let's now use (70a) and (72a) to find Lorentz gauge function in (5): which tells us that relativistic point charge satisfies Lorentz gauge condition exactly. Equations (69a), (72c) and (73a) are used in order to find fields Equation (76b) is the same as (72c). If we introduce variable u = cr − v, then equations for E and B can be re-written in more familiar form as Next divergence of electric and magnetic fields is found using (52), (55), (76a), (76b) and some vector calculus identities As it can be observed, in Lorentz potential divergence of electric and of magnetic field is exactly zero no matter how a charge moves. Again Gauss's law is satisfied, nothing is new in this case.

C. Classical continuous charge
In this section we focus our attention on distributed continuous charge. For continuous charge we assume that charge density and current density are functions of retarded time t r and of space r : ρ = ρ(r , t r ) and J = J(r , t r ). In other words, the difference between a point and distributed charges is that for the latter we can no longer take the assumption that charge (or charge density) is constant. In other words, charge and current densities are assumed to vary in time and in space, as for example, density and velocity of compressible fluid. Electric and magnetic potentials for this case are given in Table I in the lower left corner where dτ = dx dy dz is infinitesimal volume of space, defined by r . For distributed charges we will often use Leibniz integral rule to interchange the order of differentiation with integration. Therefore we assume that boundary of connected region, where integration by dτ takes place, is fixed relative to global xyz frame and so does not move. This is the simplest case, in which terms in Leibniz integral rule related to boundary motion vanish. In other words, integration with respect to τ is decoupled and is independent from differentiation with respect to t or x or y or z. Thus using (19), (21a), (21b), (25) and some vector calculus identities, we can find divergence and Laplacian of scalar potential Using (13) time derivatives of scalar potential are found Let's substitute (80b) and (81b) into (6) where Dirac delta function will pick up those values of charge density, where r = r and t r = t, i.e. ρ = ρ(r, t). As we can see Lorentz gauge condition for V is satisfied. Now let's focus on vector potential A.
If we substitute (83b), (83d) and (84b) into (6), then where again Dirac delta function will pick up those values of current density, where r = r and t r = t, i.e. J = J(r, t). Here we again notice that Lorentz gauge condition is satisfied for vector potential A. Let's now check the value of the Lorentz gauge function (5), using (81a) and (83a) It seems that Lorentz gauge function is not zero, which is the same situation as in classical point charge (65). In derivation of (86) the following continuity identity is used In derivation of (89a) the continuity equation (87) and its time differential version are used We can get also time integral version of (87) which can be used to obtain the alternative version of (86) We can see that divergence of magnetic field for continuous charge is exactly zero. However interesting case happens for divergence of electric field, which has two components. One  Table I V (r, t) = c 4πε 0ˆρ (r , t r ) and using (37), (38), (53), (56) We can now check the Lorentz gauge condition for V by substituting (94b) and (95b) into Let's put (97b), (97d) and (98b) into (6) For vector potential Lorentz gauge condition is trivially satisfied. Lorentz gauge function for relativistic distributed charge is We can see that Lorentz gauge function is not zero and Lorentz gauge condition is not satisfied for relativistic continuous charge. As in the case for classical continuous charge, in derivation of (100) we used continuity equation The equation on right side of (101) is obtained if (48) is used for time differentiation and (44) is used for divergence operator. Let's now find fields using (94a), (97c) and (98a) These fields are relativistic versions of Jefimenko equations. Finally, we are ready to find divergence of electric and of magnetic fields In derivation of (103a) the continuity equation (101) and its time differential version are In order to obtain the time integral version of (101), it needs to be integrated with respect to time. However at this point in time we could not find analytical expression for integral of (101), because r and r are functions of time also, which we cannot neglect.
We can see that divergence of electric field for continuous charge in relativistic case is not zero. While divergence of magnetic field is exactly zero.

IV. DISCUSSION
Divergence of magnetic field for all four cases considered here is exactly zero, see (67b), Finally, for relativistic distributed charge the divergence of electric field is not zero (103a).
Divergence has some complex terms. However the main result is the same, which is that divergence of electric field, generated by time-varying sources (charges or currents), is not zero, even for volumes of space, which do not contain these sources. Therefore we can conclude that for time-varying electric charges and currents Gauss's law needs to be modified.
We believe that these results were escaping attention of scientists before because in most textbooks and papers only motion of a point charge is considered. As we have seen, charge value of a point charge does not change and is constant, and so divergence of electric field satisfies original Gauss' law, i.e. it is either constant or zero. However for continuous charge its charge value changes as a function of time and space, and so original Gauss' law does not hold and divergence of electric field itself becomes function of space and time. This result becomes even more clear if we consider the following "Gedanken" experiment. Imagine closed volume, isolated electrically from outside universe, with net zero charge inside at initial time. Therefore divergence of electric field is zero at initial time. Imagine that at some later time due to ionization and/or charging net nonzero charge is created inside. As this charge appears, its electric field lines start to spread in space of the isolated volume. The speed, with which electric field lines spread in space is finite, and so divergence of electric field will change from initial zero value to some nonzero value at later time.
We note that results for relativistic cases can be reduced to their corresponding classical cases by simply setting v and a equal to 0. It is also interesting that no boost b is present in the equations for E and for B and for their divergences.

V. CONCLUSION
In this paper we considered divergence of electric and of magnetic fields for four cases: ification. Divergence of electric field seems to be far-field type scalar anisotropic field, which is generated by time-varying electric charges or currents. Results indicate that for these effects to be sufficiently large to be experimentally observable the time variation of electric charges and/or of currents should be very fast. Divergence of magnetic field is zero for all cases.