Journal of Theoretics


Author:  Bibhas R. De 

P. O. Box 21141, Castro Valley, CA 94546-9141


Maxwell’s equations are shown to have a second consequence beyond the electromagnetic (EM) wave: the source-free static magnetic field structure of finite extent in empty space. This has possible relevance to many current research areas such as field theory and plasma theory. When the result of the present paper is combined with a previous result (that a static magnetic field constitutes a mass), this relevance extends to a number of other fields: particle theory; string theory; dark matter; unified field theory; the ultimate nature of mass and matter; gravitation; cosmology etc. Additionally, the source-free structure may have implications for future technology.

KEYWORDS: magnetostatic field, field theory, plasma theory, particle theory, string theory, unified field theory, dark matter, gravitation, cosmology, future technology.


Maxwell’s equations have two distinct solutions in empty space: the static solution, or the source-free static magnetic field structure of finite extent; and the dynamic solution, or the electromagnetic (EM) wave. To date, and for more than century, any attempts to construct the first solution have been forbidden. This paper presents that solution. The deduction is on the same solid footing as that of the EM wave – requiring no assumptions or approximations.

A source-free magnetic field structure is considered forbidden by certain theorems in conjunction with the uniqueness theorem (Cf. [1], Ch. 1; [2], Ch. 1). It is shown below that these considerations do not fully encompass the issue of source-free fields. Specifically, this theorem excludes a priori field structures (or a class of solutions) corresponding to sources at infinity.2 It is shown below that in excluding such structures, useful information may have been discarded.


In cylindrical coordinates with no -dependence, the Maxwell’s equations .b = 0 and xb for the magnetic field b can be expressed as:

2bz/r2 + (bz/r)/r = -2bz/z2                                   (1)

2br/r2 + (br/r)/r - br/r2 = -2br/z2                          (2)

for which an exact solution o can be found in terms of the Bessel functions Jo and J1: 

Clearly, this may be seen as a structure with (unspecified) sources at infinity.


One can devise a method of exploring this structure further, instead of discarding it. By taking o with the minus sign (of the exponent) in the upper half space and with the plus sign in the lower half space, a “truncation” structure 1 is found:

with s = z / |z|  a shorthand notation, and an arbitrary, positive real quantity. At the z = 0 plane - the boundary plane - the field bz1 is continuous, but br1 flips sign. Thus the act of truncation generates a sheet current (with the permeability of free space.):



As a separately standing problem, the exact solution for the field structure 2 generated by a prescribed source current I can be calculated in terms of the potential A2 = A2a, and is found to be:1

Here k2 = 4rR/(R2+ r2 + z2 + 2rR), and K(k) and E(k) are the complete elliptical integrals of the First and the Second kind.3 Since I has a delta-function behavior and since the above solution is exact at every point in space for which z 0, it follows that br2(r, z 0) = br1(r, z = 0), and that therefore bz2(r, z 0) is a finite quantity. This can also be verified by numerical computations, integrating around the pole at (z = 0, R = r). Thus:

The structures 1 – created outside the scope of the uniqueness theorem - and 2 – created within the uniqueness theorem – are now shown to be unequal.


The potential A1 for 1 is found from br1 =  -A1/ z:

This can be rewritten in the following ad hoc form by invoking, for the upper half space, a mathematically provable identity: 3, (11.4.44), §(10.2.17), §(6.1.9):

Since for  for all values of R, but F2(R) remains unchanged, it follows that the two potential distributions are generally not equal. However, the inequality of the vector potentials does not necessarily prove the inequality of the field structures. To establish the latter inequality, one may readily verify that for r = 0, both potentials are zero regardless of . Thus, traveling from the axis outward, the two potentials are first in agreement, then begin to disagree. Therefore the same comment applies to the two field structures. 


A superposition of 1 and 2 causes I to vanish identically, leaving a source-free solution :

with bz (r, z = 0) finite and continuous through the z = 0 plane, and br (r, z = 0) = 0. No singularities or discontinuities remain across this plane (Cf. Eqs. (14) and (15). It should be noted that the structures 1 and 2 were only mathematical steps used in arriving at , and have no physical significance.

The standard argument invoked to preempt any discussion of source-free structures may now be revisited. Let the magnetic fields in Eqs. (21)-(23) derive from a scalar potential . Then consider Green’s first identity:

The left-hand side is zero in a source-free volume. Consider then the surface integral for the upper half space: On the z = 0 plane, can be taken to be 0 since here br = 0. In absence of any sources at infinity, the surface term should go to zero there as well. Thus the integral of the positive term ()2 is zero, meaning that is a constant over all space, so that the magnetic field is zero everywhere. However, depending on the field structure being examined, there are three possible outcomes relative to the surface integral as it is expanded to infinite distances: (a) It approaches zero (the above case), (b) It approaches infinity (unphysical case), or (c) It remains constant. The last case would allow the source-free structure, and the constant would be reflective of the finite energy of the structure. 

With the help of results developed earlier, it has been shown that the energy in a static magnetic field far away from its source can be drawn down without prior communication with the source.4-7 A conceptual process of creation of the source-free structure by removing a source of magnetic field from a location, without removing the field "energy," would accordingly present no theoretical difficulties. Thus the structure would not contravene any conservation principles. 


The study of magnetized and magnetically confined plasmas requires a full understanding of the basic nature of magnetic field. The present paper suggests that this understanding may not yet be complete. Instabilities in plasmas cause the magnetic field to assume new structures, including force-free structures. Such structures may further transition to the source-free structure

It has been suggested recently that magnetic field in empty space constitutes a mass.8 This suggestion, when combined with the present result, defines an entity in empty space made entirely of magnetostatic field and having a mass. In the limit of the smallest size, is then essentially a point particle. Whether a point particle or a macroscopic structure, such an entity would have implications for the current studies in gravitation, cosmology, unified field theory, the ultimate nature of mass and matter, particle theory, string theory, dark matter, etc. Additionally, if the structure can be created in the laboratory, it would have implications for future technology.

[1] Jackson, J.D. Classical Electrodynamics. New York: Wiley, 1972.
[2] Panofsky, W. K. H., and M. Phillips. Classical Electricity and Magnetism. 
Reading, Mass.: Addison-Wesley, 1962.
[3] Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions.
New York: Dover, 1972.
[4] De, B.R. J. Phys. A 26 (1994): 7583.
[5] De, B. R. Phys. Fluids 22.1 (1979): 189.
[6.] De, B. R., in Plasma and the Universe. Ed. Falthammar, C. G. et al.
Dordrecht-Holland: Kluwer Academic Publishers, 1988. 99.
[7] De, B. R. J. Phys. A 27 (1994): L431
[8] De, B. R. Astrophys. Space Sci. 25 (1996): 239.


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