Coulomb’s Law Revisited:
This is the equation of the electrostatic potential I derived. It differs from the classic potential by the multiplicative exponential term. The classic potential is just k/r and this goes to infinity as r goes to zero. This behavior is termed “singular.” My potential does not go to infinity and is called a “nonsingular” potential. It is this nonsingular character of my potential, both in the electrostatic and gravitational potentials that really allow my work to predict things current physics cannot do. For instance, the compact reactor is a direct result of this potential form. The best form for playing with the equation should be
because phi is typically used to denote a gauge potential.
Coulomb’s Law Revisited
Dan Ross
I cannot remember when I first became aware of Coulomb’s Law of electrostatics, but it was probably a few years after my first studies of electricity and magnetism in the mid1930s and before my first highschool course in physics in the late 1930s. I do remember that I was greatly puzzled, almost from my first awareness of Coulomb’s Law, about the following question, “What holds the electron together?” I know that I have not been alone in puzzling over this question. This Law says that the force of repulsion between two like charges q1 and q2 is given by
which leads to tremendous repulsive forces on each charge for separations small compared to the dimensions of an electron. If the electron were considered as being split into two parts, q1 and q2, how can believers in Coulomb’s Law assume that the two parts would continue to be a stable particle and not fly apart with huge accelerations in opposite directions?
; Even after studies of electrostatics as an undergraduate student of electrical engineering in the early 1940s and as a graduate student of classical electromagnetic theory in the late 1940s, my puzzlement remained on the obvious stability of the electron in clear defiance of Coulomb’s Law. My later studies of modern theoretical physics and its history fascinated me. Nevertheless, my puzzlement concerning Coulomb’s Law continued and even became more intense.
I did begin to think about my problem more from the point of view of the electric potential Φcabout a central charge q, where the subscript C identifies the Coulomb Potential. If I wanted to consider the force field, I could easily find it by differentiating Φc with respect to r. I also became aware of the great interest of the theoretical physicists during the 1920s and early 1930s toward the development of various potential functions to counteract the tremendous repulsions dictated by Coulomb’s Law encountered in their studies of atomic nuclei. Their development of the concept of the strong nuclear force did not satisfy my continued puzzlement about the electron and Coulomb’s Law. I had frequent occasion to think about this puzzle while teaching electrical engineering at Purdue in the late 1940s and at USMA, West Point, in the early 1950s, but much less frequently while serving IBM Federal Systems Division, 19511969, as an engineering manager and in my own business in Washington, DC, 19691979. IBM sent me to Johns Hopkins University to get my Doctorate in Engineering, which I received in 1964. In the late 1960s, I continued to serve JHU as a parttime Adjunct Professor.
In July 1969, I left IBM to start a small consulting engineering firm specializing in telecommunications and, in 1971, we added a second and larger business to develop cabletelevision systems in the midAtlantic area. By the mid1970s, these enterprises were well established and growing rapidly; and I started planning my return to my former university life of teaching and research and to retire from active management of my business perhaps as early as 1980. I sent letters to universities where I was well known (Purdue, MIT, Johns Hopkins, VPI, GWU, USMA, USNA) stating my future plans for return to the halls of ivy.
One day in early July 1979, the Chairman of the Electrical Engineering Department of USNA, Annapolis, inquired as to my possible interest in coming to USNA in late August as a Visiting Professor. If that call had come from any university far from Washington, I would have had to decline such an opportunity. But the proximity of Annapolis to Washington would facilitate my ability to complete my affairs with my business. So I agreed to visit USNA the following Friday to give a lecture to some of the USNA Faculty.
The lecture on my ideas on starting the teaching of calculus as a branch of algebra was well received. After lunch at the Officers and Faculty Club, I accompanied the Chairman to his office, where he offered me the position and I accepted. By that time on a Friday afternoon, it was too late to expect to accomplish anything worthwhile in my office back in Washington, so I decided to spend the rest of the day at Nimitz Library. After perusing the stacks on the upper floors, I went down to the Ground Floor where the research reports were kept. I found a card file of items produced by USNA Faculty, which I proceeded to examine. I found many pieces of excellent work but nothing that attracted my immediate interest until I reached the W's. There I found a card concerning a research report by Pharis E. Williams, LCDR USN accompanied with the extremely interesting title, “The Dynamic Theory: A New View Of Space And Time.”
I realized immediately that the author considered himself to be some sort of neoEinstein. I had studied much of Einstein’s work for more than forty years. I was astonished by the abstract of Williams’ paper, which indicated that he had claimed to derive all of physics from three fundamentals of thermodynamics which he took as axioms for his mathematical derivations!
My astonishment arose from knowledge that the worldfamous physicists had traveled around the world in the 1920s giving lectures at major universities on the quantum theory and relativity. At each lecture, some young physicist would be bound to ask the lecturer about which theories he regarded as the ‘Rock Of Gibraltar’ and which he regarded as subject to future revision or, in some cases, complete overthrow. I do not know what Bohr, Heisenberg, Rutherford or Schrödinger said in reply to such questions, but I do know what Einstein said. Einstein always said that all of the theories of physics were subject to future revision, or in some cases, complete overthrow, except for the fundamentals of classical thermodynamics. On some occasions, he would also say that he could not even imagine a process through which the basic principles of thermodynamics could begin to be modified.
Einstein is well known, not only for Special Relativity and General Relativity, but also for his lifetime search for a unified theory of physics. Knowing of his high opinion of classical thermodynamics, it is surprising that he never used thermodynamic axioms as a point of departure for the development of a unified theory and that is exactly what Williams has done with his spare time for the past twentyfive years!
All of these facts from the history of science flashed through my mind while I was reading the abstract of Williams’ paper. I checked out the paper from the Librarian and took it to my home in Bethesda to study during the weekend. I was amazed at what I read, particularly Williams’ derivation of eight partial differential equations in five dimensions (independent variables). Five of these PDEs were clearly generalizations of the four Maxwell equations of classical electrodynamics plus the equation of continuity of charge. I had never even imagined that anyone could derive electrodynamics by starting from thermodynamic axioms!
On the following Monday morning, I did not go immediately to my office in Washington. Instead, I went back to Nimitz Library at USNA and checked out copies of all of the papers by Williams that the Library had. That weekend in July 1979 changed my whole life. During my year of teaching at USNA, plus the following year as the first Visiting Professor of Electrical Engineering at USMA, I spent all of my spare time studying Williams’ Dynamic Theory, particularly his use of tensor analysis in deriving his eight PDE’s from thermodynamic axioms (conservation of energy; the law of entropy; the absolute zero).
Before I arrived at USNA, Williams had left to begin work on the nuclear weapons program at Los Alamos National Laboratory (LANL). He continued work on his Dynamic Theory, which he had started in 1974 at the Naval Postgraduate School in Monterey. All of his work on the Dynamic Theory has been done on a parttime basis in addition to his fulltime assignments as a naval officer and as an engineering executive at the Energetic Materials Research And Testing Center, New Mexico Tech.
I was invited by LANL to join Williams as his mathematical assistant, June 1981 to August 1982. Upon my arrival at LANL, I started a thorough check of Williams’ derivations, starting immediately after his thermodynamic arguments and including all of his steps involving tensor analysis. I paid particular attention to the derivation of his eight PDE’s in five independent variables, 3 space variables, t, γ, where γ is Williams’ symbol for mass density. At first, it seems strange to consider mass density as an independent variable. However, the fact that the Dynamic Theory considers mass density to be an independent variable in general does not prevent it being dependent on the other four dimensions in special cases. Just as we have become accustomed to Newton’s 3dimensional space and time becoming Einstein’s 4dimensional spacetime, we must now become accustomed to Williams’ 5dimensional spacetimematter. Other authors have proposed 5dimensional systems for theoretical physics, but these authors do not identify the 5th dimension with any particular physical unit.
A few weeks after my arrival in Los Alamos, I discovered an error in the eight PDE’s: in one of the 24 terms, there was a reversal in sign. As soon as I had convinced Williams that he had indeed made an error, he started spending every evening and weekend in correcting all of the Dynamic Theory from the point where the error had occurred. In this process, he was able to derive a replacement for Coulomb’s Law, which he calls the neoCoulombic Potential, but I will call it
Prior to this remarkable discovery in the Summer of 1981, Williams and I had always considered the first five PDE’s to be generalizations of classical electrodynamics, and he had considered the remaining three PDE’s to have some connection with the strong nuclear force concept developed by theoretical physicists over the previous halfcentury. After the discovery of the replacement for Coulomb’s Law, a different picture of the latter three PDE’s emerged!
The constant k is still the same Coulomb constant that we have known for more than two centuries; and λ is a small number depending on which particle is the source of the charge q located at r = 0. The remarkable difference between and Coulomb’s Law is that has no singularities anywhere!
Immediately, I realized that Williams had found the answer to the puzzle that had plagued me for forty years. His nonsingular potential, instead of rising without limit as r approaches zero, peaks at r = λ, and then goes to zero at r = 0. In other words, there is no longer any need to postulate a strong nuclear force. In a ‘like charge’ situation, the force field is repulsive for r > λ and closely Coulombic for all values of r much greater than λ; but for r between 0 and λ the force field is attractive!
Since there was no longer a need for the strong nuclear force concept, the next question is: “What is the interpretation of the latter three of Williams’ PDE’s? “ If Williams is going to be actually successful in developing his Dynamic Theory into the unified theory of physics that Einstein and his successor’s have sought for many years, the answer to the question has to be that the latter three PDE=s must be generalizations of Einstein=s gravitational equations! So, ever since the Summer of 1981, Williams has spent a large fraction of his spare time in correlating our knowledge of gravitation with his eight PDE’s. The first publication of the Dynamic Theory including his nonsingular potential discovery was in a LANL report dated May 1983, The Possible Unifying Effect of the Dynamic Theory LA9623MS.
In the eight PDE’s of the Dynamic Theory, the dependent variables of classical electrodynamics are easily identified and appear mainly in the first five PDE’s, but they also appear to a limited degree in the latter three PDE’s. Conversely, the nonelectrodynamic (gravitational) dependent variables appear mainly in the latter three PDE’s, but they also appear to a limited degree in the first five PDE’s. It is interesting to note what happens to the eight PDE’s when all of the terms involving partial differentiation with respect to γ are cancelled. The system then divides into two independent parts:
(1) five PDE’s with only electrodynamic dependent variables, and
(2) three PDE’s with only gravitational dependent variables.
This mixing of gravitational terms with the classical electrodynamic equations led me to suggest to Williams that the Dynamic Theory might explain some of the large discrepancies between electromagnetic measurements and classical electromagnetic theory. Several such large discrepancies appear in the book by Townes and Schawlow on Microwave Spectroscopy in connection with propagation of microwaves through matter.
Another suggestion that I made to Williams was to apply the Dynamic Theory to the explanation of the quasars and their huge redshift. Perhaps a very large part of the redshift is due to large gravitational potentials at the sources. If only a small part of the redshift is due to Doppler Effect, then the recession velocities and corresponding distances from the Earth are much less than presently assumed. Such changes would possibly avoid the unexplainably large energy output of the typical quasar according to the present theory of these very interesting objects. Williams has developed an equation linking four quantities for a single quasar:
(1) distance of the quasar from the detector,
(2) gravitational potential at the source,
(3) gravitational potential at the detector, and
(4) the observed redshift.
After the typical measurement from a spectroscope on Earth, the observer is left with one equation in two unknowns: (1) and (2). Perhaps future measurements of each quasar from two spectroscopes, one on the Earth and a second on the Hubble Space Telescope (or on the Moon) will yield two equations in two unknowns that will lead to a better understanding of quasars.
Williams has spent several years of parttime work in writing a book on his Dynamic Theory. My publishing venture, Los Alamos Books, plans to publish Williams’ book as soon as it is finished, which we expect will be in 2000. We hope that this work will stimulate many young physicists throughout the world to start experimental investigations of the many revolutionary ideas that Williams has introduced into theoretical physics.
So, Williams’ work has many facets, not just his neoCoulombic potential function that resolved my puzzlement over Coulomb’s Law. On that subject, another chance meeting, this time in Santa Fe, led to still another revisitation of Coulomb’s Law. In late 1997 or early 1998, I met James Keele, a retired electrical engineer, with a continuing interest in electromagnetic theory. In our first conversation, I illustrated my association with Williams’ Dynamic Theory by mentioning his variation on Coulomb’s Law. Keele immediately responded that he was also intensely interested in this subject and had developed his own modification of Coulomb’s Law! I was most amazed to learn that Keele had also introduced the same multiplying factor exp (λ/R)
that Williams had found!!!!
I told Keele how to get a copy of Williams’ May 1983 paper from the INTERNET, and Keele gave me a copy of his paper entitled “AThe Modification Of Coulomb’s Law” which he had written in March 1988. Except for the fact that both authors employ the same multiplying factor that removed the Coulomb singularity at r = 0, the two force formulas are quite different in form and in their development. Keele had apparently based his choice of multiplying factor because of some interesting properties of the exponential function, while Williams had derived his neoCoulombic potential through a long mathematical derivation from thermodynamic axioms. Keele goes on to apply his modification of Coulomb’s Law to: (1) Hydrogen Spectra;
(2) Pair Annihilation; and (3) Nuclear Structure. I have not had time for careful study of this work, but I have noted that it includes some interesting new ideas.
Williams used his nonsingular radial potential to develop new models of the neutron and of the stable isotopes of the light elements up to and including oxygen. He was drawn to this range of atomic number because of the wide discrepancies between the existing theory and the well established experimental measurements of binding energy in this range. Williams’ calculations based on his nuclear models check the measurements very closely indeed!
In conclusion, the neoCoulombic potential has not only resolved my puzzlement over the stability of the electron and other charged particles, but Williams’ entire theory has attracted much of my spare time for twenty years. I have been privileged to work with Williams and to have helped in small ways. More importantly, I hope to help in larger ways over the next several months in publishing
THE DYNAMIC THEORY: A NEW VIEW OF SPACETIMEMATTER
which has now grown to a book of more than 400 pages. If this monograph is studied and finally accepted by the physicists of the world, Williams will have reached the objective that Einstein sought from 1915 until his death in 1955! Such attainment of a unified theory of physics through thermodynamic axioms would be the ultimate irony because, as far as we know, Einstein never took this approach despite his extremely high opinion of the fundamentals of classical thermodynamics!!
Dan C. Ross
6 December 1999
(Sadly Dan C. Ross succumbed to the ravages of cancer shortly after penning these thoughts.)

