A COMPARISON OF THREE ELECTRODYNAMIC EQUATIONS

Domina Eberle Spencer

University of Connecticut

Storrs, Connecticut, 06268, U.S.A.

Uma Shama

Bridgewater State College

Bridgewater, Massachusetts, 02325, U.S.A.

Philip J. Mann

University of Connecticut

Storrs, Connecticut, 06268, U.S.A.

Abstract Three electrodynamic equations for the force between moving charges will be compared. The first is the Weber equation1 which is a function of relative velocity but assumes that the velocity of light is infinite. The second is the classical equation2 which is expressed in terms of absolute rather than relative velocity and is based on the Einstein postulate that the velocity of light is a constant. The third is the new Gaussian equation3 which is expressed in terms of relative velocities and the universal time postulate on the velocity of light. Equations will be derived for the force produced by a current element on a moving charge and for the force between two current elements. Applications will be made to three configurations and conclusions will be drawn on the validity of the three equations.

Introduction

According to the last chapter4 of Maxwell’s great treatise on electricity and magnetism, the keystone of electromagnetic theory is an equation for the force between moving charges. The number of such equations which have been proposed is very large. However, there are only three equations which are believed by a significant number of scientists today. These three equations will be studied in this paper. Three experiments will be described which will enable us to draw significant conclusions on the validity of the three equations.

1. The Weber equation

The first equation for the force between moving charges is the Weber equation1 which was first published in 1846. This equation was independently rediscovered by Bush5 in 1926. Today, it is employed in the research of Andréé Assis, Peter and Neal Graneau, P.T. Pappas, Tom Phipps and J.P.

Wesley. The force per unit charge FW produced by a charge Q which moves at velocity v acting on a unit test charge at point P, Fig. 1, which moves with velocity u is

 

 

 

 

 

 

 

 

 

 

 

(1W)

The Weber equation has several peculiarities. In the first place, Weber assumed that the force must be in the radial direction like the previously developed equations: the Newton force between two masses and the Coulomb force between two charges. It is uniquely determined by the four Ampere experiments6 if one postulates that the force must be in the radial direction. It satisfies the first Gaussian criterion7 that the force must be a function of the relative velocity of source and receiver, u – v. But it does not satisfy the second Gaussian criterion that it must take into account the fact that the velocity of light is finite. The Weber equation implies that the velocity of light is infinite.

2. The Classical Equation

The equation for the force between moving charges which is consistent with the Maxwell equations and the work of H.A. Lorentz and Einstein is called the classical equation Suppose that the charge Q is at the point x(tI),h(tI)),z(tI) at time tI, where tI is the instant of emission, Fig. 2. The unit test charge is at point P(x,y,z) at time t and moves with a velocity u. The radius rI is measured from where the charge Q was at time tI to where the receiver is at time t in accordance with Einstein’s postulate on the velocity of light, Postulate I*. Likewise, the unit vector aI is directed along rI., Fig. 2. Then it can be shown8 that, if the electric and magnetic vectors are defined in terms of scalar and vector potentials in the classical fashion, the force per unit charge FC is defined by the equation,

(1C)

 

The classical equation does not satisfy both of the Gaussian criteria. It is not defined in terms of relative velocities but in terms of absolute velocities u and v. It does take into account the fact that the velocity of light is finite, but does this using Einstein’s postulate on the velocity of light, Postulate I*, which Moon, Spencer, Shama and Moon9 have shown to be experimentally invalid.

3. The New Gaussian Equation

 

The new Gaussian equation for the force between moving charges was derived by Moon, Spencer, Mirchandaney, Shama and Mann10. This equation utilizes both a scalar and a vector potential like the classical equation. It differs from the classical equation in three ways. In the first place, it is defined in terms of relative velocities rather than absolute velocities. In the second place, it utilizes the universal time postulate on the velocity of light, Postulate III* which is in accord with all of the experimental data hitherto analyzed9. Finally, it employs a slightly different definition of the electric field strength in terms of the scalar and vector potentials, the most important difference being replacing a partial derivative with respect to t by a partial derivative with respect to tIII. The resulting equation10 for the force per unit charge FG is

 

(1G)

 

4. The Force of a Current Element on a Moving Electric Charge

A current element Ids which is not in motion contains positive charge which is stationary and equal but opposite negative charge which moves in the direction opposite to ds. From Eq.(1), we calculate the force on charge Q2, which moves at velocity u, produced by positive charge Q1 for which v+ = 0. This must be added to the force produced by the equal and opposite negative charge – Q1 which moves at velocity v-. The resulting expression is simplified by introducing the fundamental equation

 

 

 

 

 

 

 

 

 

 

(2)

The three resulting equations are: for the Weber equation,

(3W)

for the classical equation,

(3C)

and for the new Gaussian equation,

 

 

 

 

(3G)

 

 

 

5. The Force between Current Elements

In order to calculate the force between current elements, Eq. 3 must be applied to both the stationary positive charge in the second current element and to the moving negative charge in the second current element. Adding these results together and employing Eq. 2, we obtain expressions for the force d2F between current element I1ds1 and I2ds2.

 

For the Weber equation, the terms involving the rate of change of the current cancel out and the result is exactly the expression first published by Ampere11 in 1823:

(4W)

Note that this expression vanishes when the angle between the current elements is 54.7o. This peculiarity has strange consequences.

 

Both the classical expression and the new Gaussian expression are expressed in terms of vector triple products. The use of the vector triple product was first suggested by Grassmann12 in 1845. The classical expression also contains a term involving dI1/dt. The velocities of the electrons in the conductors are much less than c so terms in v/c are neglected.

(4C)

According to the classical equation, the force on ds2 is always perpendicular to ds2. Tangential forces cannot exist.

The new Gaussian equation contains two vector triple products: one is perpendicular to ds1, the other is perpendicular to ds2:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(4G)

According to both the Weber equation and the new Gaussian equation, tangential forces can exist. However, these tangential forces are not identical.

6. Applications

In order to determine which of the electrodynamic equations is correct, it is necessary to apply each to many configurations and to compare the results with experiment. In this paper, we will consider three applications. The first two applications employ the force between current elements, Eq.4.

Two years ago, our research group13 presented a paper on the Hering furnace14, Fig. 3, in which we studied the operation in terms of the classical equation and the new Gaussian equation. We have recently extended this analysis to the Weber equation15. The conclusions are threefold. For the Weber equation, it is predicted that the Hering furnace could operate but the tangential forces are in the wrong direction. The rotation of the molten metal is in the opposite direction to that observed by Hering. According to the classical equation, the Hering furnace cannot possibly operate because there are no tangential forces. Only the new Gaussian equation predicts tangential forces in the correct direction.

Two years ago our research group16 also studied overhead welding from the point of view of the classical equation and the new Gaussian equation. This analysis has recently been extended to the Weber equation15. The welder’s electric circuit is shown in Fig. 4. The net force on the weld bead predicted by the Weber equation is a force of repulsion! Overhead welding, contrary to the experience of welders throughout the world, is predicted to be impossible! The classical equation on the other hand predicts that there is no force on the weld bead! Again the prediction contradicts the experience of the welder! Only the new Gaussian equation predicts a force in the correct direction. The weld bead is pressed tightly against the weld rod and the weld joint.

 

 

 

 

 

 

In the third application Eq. 3 is applied17 to the unipolar generator, Fig. 5. General expressions have been derived for the induced voltage when the copper disc rotates at angular velocity w and the outer current carrying wire rotates at angular velocity W_ Three special cases have been considered: Case I. W=0. The induced voltage, VI , is a linear function of_w; Case II. w=0 There is no induced voltage. Case III: w=W. The induced voltage is the same as in Case I. For the Weber equation the induced voltage between center and edge of the copper disc is a quadratic rather than linear function of w and W. This contradicts the predictions in all three cases. The induced voltages in Cases I and III are also predicted to be different. The classical equation predicts a linear function of both w and W. In Case I the form of equation is correct. But the predictions are incorrect in Cases II and III. The new Gaussian equation predicts that the induced voltage is a linear function of w and predicts the correct result in all three cases.

7. Conclusions

The paper has begun the task of comparing the three most promising electrodynamic equations: the Weber equation, the classical equation and the new Gaussian equation. We have studied the Hering furnace, overhead welding and unipolar induction. In all of these applications, the only electrodynamic equation which always gives correct results is the new Gaussian equation

 

References

1.W.Weber, “Elektrodynamische Maassbestimmingen uber ein allgemeines Grundgesetz der elektrischen Wirkung”, Abh. Leibnizens Ges., Leipzig, p.316, 1846.

2.J.A.Stratton, Electromagnetic Theory, McGraw- Hill. New York, 1941

3. P. Moon, D.E. Spencer, A.S. Mirchandaney, U.Y. Shama, P.J. Mann,

“A Gaussian equation for the force between moving charges”, Problems of Space, Time, and Gravitation, 4th International Conference, 1996, St, Petersburg, Russia

4. J.C. Maxwell, A Treatise on Electicity and Magnetism, Vol. II, Third Ed.. Oxford, Clarendon Press, 1904, p. 490.

5. V. Bush, “The force between moving charges”, J. Math.. Phys., Vol. 5, 1926, p. 129.

 

 

 

 

 

 

 

 

6. P. Moon and D.E. Spencer, “Interpretation of the Ampere force”,

J. Franklin Inst., 257: p.203-220.

7. K.F. Gauss, Zur mathematischen Theorie der elektrodynamischen Wirkung, Werke, G_ttingen, 1867, Vol. V, p. 602.

8. P. Moon, D.E. Spencer, A.S. Mirchandaney, U. Shama. P.J. Mann,”The Derivation of a New Gaussian Equation for the Force Between Moving Charges from Fundamental Postulates”, to be published.

9. P. Moon, D.E. Spencer, U.Y. Shama and E.E. Moon, “The Universal Time Postulate on the Velocity of Light”, 4th International Conference on Problems of Space, Time and Gravitation, St. Petersburg, Russia, 1996, p. 57.

10. P. Moon, D.E. Spencer, A.J. Mirchandaney, U.Y. Shama, P.J. Mann, “A Gaussian Equation for the Force between Moving Charges”, 4th International Conference on Problems of Space, Time and Gravitation, St. Petersburg, Russia, 1996, p.188.

11. A.M. Ampere, “Memoire sur la theorie mathematique des phenomenes electrodynamiques, uniquement deduite de l’experience”, Mem. De l’Acad, des Sci. VI (1823), p. 175.

12. H. Grassmann, “Neue Theorie der Elektrodynamik”, Ann. D. Phys., 64, 1845, p. 1.

13. D.E. Spencer, G. Coutu, W.W. Bowley, U.Y. Shama, P.J. Mann, “The experimental verification of the new Gaussian equation for the force between moving charges”, 4th International Conference on Space, Time, Gravitation, St. Petersburg, Russia, 1996 p. 280.

14. C. Hering, “Studies in Electrodynamics”, J. Franklin Inst., 37, 1923, p. 124; “Electrodynamic Paradoxes”, J. Am., Inst. Elect. Eng., 42, 1923, p. 139.

15. D.E. Spencer, G. Coutu, U. Shama, P.J. Mann, “The Electrodynamic Formulation of the Hering Furnace and Overhead Welding: Three Viewpoints”, paper presented at meetings of the Natural Philosophy Alliance, Grand Junction, Colorado, U.S.A., May 21, 1998.

16. D.E. Spencer, G. Coutu, W.W. Bowley, U.Y. Shama, P.J. Mann, The Experimental Verification of the new Gaussian equation for the Force between Moving Charges: Overhead Welding”, Problems of

Space, Time and Motion, 4th International Conference, September 23-29, 1996, St. Petersburg, Russia, p. 43-52.

17. D.E. Spencer, U. Shama, P.J. Mann, “An Analysis of Electromagnetic Induction”, 5th International Conference on Space, Time, Gravitation, St. Petersburg, Russia, 1998

 

 

 

 

 

Fig. 1 The Weber equation for the force between moving charges.

 

Fig. 2 The Classical and Gaussian equations for the force between moving charges.

Fig. 3. The Hering Furnace

Fig. 4 The welder’s circuit

Fig. 5 Unipolar induction

Dr. Domina Eberle Spencer,

Professor of Mathematics.

University of Connecticut,

Storrs, Conn. 06268