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The Law of Electrodynamicsby H. ASPDENIBM United KingdomWinchester, England

Laboratories, Ltd.

ABSTRACT: Th e in determ in ate state of the emp ir ical la w of electrodynamic action betweentwo curr ent elements is reviewed. It is shown that a form of the law ful ly consistent wi thexper im ent can be deduced from sim ple N ewtonian dynami cs. Th is law point s to anomaliesto be expected in electr odynam ic in teracti ons betw een electr ic par ti cles of un equa l cha r ge-mass ratio. Th is resul t ma y account for cer tai n hi therto unexplai ned anomal ies in electr icdischarge phenomena.

i n t r o d u c t i o nA summary of the early development of the law of electrodynamic action

between current elements is given by Tricker (1). He presents the basic paperof Ampere on this subject, and the criticisms sustained by Amperes law andalternative formulations by Biot and Savart and by Grassmann. Alsomentioned is the general empirical formulation by Whittaker (2) in which heproposes a simplified new law based upon new assumptions. The commonproblem is that none of these laws is fully consistent with Newtons ThirdLaw when applied to interactions between individual elements of current.Yet, all the formulations appear to give the correct answers when used inintegrated form to apply to interactions involving a closed circuit. Trickerthen reaches the seemingly inevitable conclusion that an isolated element ofsteady current is a contradiction in terms, and thus leaves open the problemof how two electrons in motion really react owing to their electrodynamicinteraction. Here, it is proposed to apply directly Newtonian principles to theproblem of two interacting particles. From very simple considerations aformulation of the law of interaction is deduced which is fully consistent withthe empirically derived general formula of electrodynamics but, oddlyenough, is different from the laws deduced, by assumption, by Ampere, Biotand Savart, Grassmann, and Whittaker.Emp i r i c a l A n a l y s i s

Four basic empirical facts were relied upon by Ampere in deriving his law :(a) The effect of a current is reversed when the direction of the current isreversed.(b) The effect of a current flowing in a circuit twisted into small sinuositiesis the same as if the circuit were smoothed out.

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Brief Communications(c) The force exerted by a closed circuit on an element of another circuit

is at right angles to the latter.(d) The force between two elements of circuits is unaffected when all linear

dimensions are increased proportionately, the current-strengthsremaining unaltered.Ampere combined with the above the assumption that the force between

two current elements acts along the line joining them, and thus obtainedhis law:

p = Eii,(3(z.r)(& .F) 2(& X) r

r5 - p3 1 * (1)

Here, F denotes the force acting upon an element z of a circuit of currentstrength i and due to a current i in an element z. The line from d7Eo & isthe vector distance F; Ic is arbitrary and depends upon the units chosen,although its polarity may be determined by using the law to verify theobservation :

(e) Two extended parallel circuit elements in close proximity mutuallyrepel one another when carrying current in opposite directions, orattract when carrying current in the same direction.

Prom the analysis by Whittaker (2), disregarding Amperes assumption,the general formulation consistent with obs&vations (a) toto be:

F = /& (3(&F)(Z.i)F 2(Z.Z7I;+4&.F)Z~

r5 - r3 r3- B(& .@ B(& z)Z+3B(& F)(i& )F\_ - I .

(d) is found

(2)Here, A and B denote arbitrary constants. Whittaker then assumed linearforce balance as represented by symmetry in & and &. This involvesequating A and -B. In its simplest form, with k and A both equal to unity,the law becomes

P = ~{(~.r)~+(~.~)~-(d7;.~)r}. (3)Inspection shows that this formulation satisfies observation (e). However,Whittaker made no mention of the all-important experimental discovery ofTrouton and Noble (3). Their experiment demonstrated that separatedcharges in a capacitor resulted in no tendency for the capacitor to turn whenin linear motion transverse to its suspension. Put another way:

(f) There is no interaction torque out of balance between anti-parallelcurrent elements.

This balance of torque action is not assured by the simple formulation ofWhittaker in Eq. 3. To satisfy observation (f) terms other than those in Imust cancel when & is equal to - &. This applies to the general formulationwhen A = B.

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Brief CommunicationsD y n am i c A n a l y s i s

Consider now two particles of mass m and m. They are separated by thedistance vector 7. The centre of inertia of this two-particle system is takento be distant a and b respectively from m and m. Then,

mb = ma. (4)Let V, the velocity of m, tend to change, decreasing by %. This must arise

from a force -m(%/ d t ) acting on m in the direction V.Let V, the velocity of m, tend to change, decreasing by I%. This must

arise from a force -m (& / d t ) acting on m in the direction V.These two forces on m and m will, in the general case, produce a turning

moment in the system. Since there is no evidence that any system can beginto turn merely by its own internal interactions, the forces in the system mustprevent an out-of-balance couple from asserting itself. Therefore, on m weadd the force :

iii b a37-m-- =-m-d t a atfrom Eq. 4 . On m we add the force:

%a Z_m-- = -_m__.d t b d tThe total force on m now becomes

,d;-m dt in V direction,

-mc in V direction,dt-i?f in F direction,

where --F is the force we now assume to act directly on m as a result of itselectrodynamic interaction with m. This force is a fully balanced interactionforce. I n summary, although we now have three force components, two havearisen by action and reaction and one is really fictitious. The prime directelectrodynamic force induces acceleration in a particle and therefore theinertial reaction. For generality, the inertial reaction force component whichcould correspond to a turning action has been introduced in the inertialformulation but since it must be zero it has been offset by a balancing term.

Consider the rate of energy change at m, i.e.

We then remove the kinetic energy term and equate the remainder to zero.

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Brief CommunicationsThus,

Similarly for m,(6)

d;mdt= F (F.21)210c- -.r (w.v) (7)We can now evaluate the resultant force acting on each particle. For theparticle of mass m, Eqs. 6 and 7 may be used to derive the general forceexpression

p= ~-(q.i;.)v~(F.s)o~-(,.;)?).(w.v )rI f we now assume the particles to be electrons, the masses m and mbecome equal and since the effective current elements e5, eZ may be written

iz, i% where e denotes the electron charge in electromagnetic units, Eq. 8becomes,

(9)A comparison of Eq. 9 with Eq. 2 shows it to be of the general form

developed empirically by Whittaker. The comparison is satisfied if A is - 1and B is - 1. Further, this satisfies observation (f), as expected from thebasis of the analysis. We then develop the identity,

F, = kii(& .Z)r2

and, from observation (e), with & = & realize that k is + 1. Of interest thenis the fact that this force is exactly that deduced theoretically by evaluatingthe interaction component of the integrated magnetic field energy clue to thetwo current elements and assuming that it tends to increase. This result fullysupports the belief that the isolated current element as a steady currentelement is contradictory. The inertial effects of the charge carriers areimportant to the understanding of the full law of electrodynamics. However,it has been shown that a unique law of electrodynamics can be formulated as

The odd thing about this is that the law thus deduced differs from that ofAmpBre, Whittaker, Biot and Savart, and Grassmann. Further, we havededuced that this law is only valid for interactions between charge elementsof equal mass. If the masses of charge carriers in different parts of the samecircuit are different the middle term in the above formulation has a coefficientequal to the mass ratio.

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Brief CommunicationsSince this middle term represents a forceomponent along the direction of

current flow, we may predict that in a discharge circuit where electrons carrycurrent in a cathode but positive ions contribute to the current to the cathodethere will be an electrodynamic force manifested along the discharge.Similarly, some manifestation of the predicted anomalous forces shouldappear in plasma work.

Many authors have found anomalous cathode reaction forces in dischargestudies. For example, Kobel (4) found an anomalous cathode reaction forceof 250 dyn at 16 A and 1400 dyn at 35 A. This is of the order of 100i2, wherei is the current in absolute units. This quadrature current phenomenon hasdefied explanation. Mere reaction momentum considerations lead to arelatively small cathode reaction force which is linearly dependent uponcurrent. Even using Eq. 1, for example, any element of current in a con-tinuous filament is subjected to balancing forces from the filament currenton either side. There is no force action along the filament. This also appliesto the classical formulations of the electrodynamic law. However, bearing inmind that in a discharge at least some of the current at the electrodessuddenly is transported by ions and not electrons, the mlm factor to be usedwith the middle term of Eq. 10 assumes importance. On one side of thiscurrent junction at the cathode electrons act upon ions in the discharge andon the other side ions act on ions. It works out that there is an out-of-balanceforce productive of a cathode reaction by impact from the ions. This force isthe product of the constituent ion current component squared and the ratioof the ion mass to the electron mass. Forces of the order of lOOi as ound byKobel are therefore readily explained.

It may be concluded that the resolution of this long-standing problem ofthe true nature of this basic electrodynamic law is not a mere academic topic.Some deeper understanding of the law will have practical consequences indischarge and plasma control.

References(1) R. A. R. Tricker, Early E lectrodynamics, Oxford, New York, Pergamon Press,

1965.(2) Sir E. Whittaker, Aether and Electricity, London, Nelson, 1951.(3) F. T. Trouton and H. R. Noble, The Mechanical Forces Acting on a Charged

Condenser Moving Through Space, Ph i l . Trans. R. Sot. L ondon, Vol. 202A,pp. 165-181, 1903.

(4) E. Kobel, Phys. Rev., Vol. 36, p. 1636, 1930.

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