The following is a Letter to the Editor of the IEE journal 'Electronics and Power' published in the April, 1965 issue at p. 137.
Dear Sir - Sir Edmund Whittaker, in his historical writings about the theory of electricity, [WHITTAKER, E: 'Aether and Electricity (Classical Theories)', (Nelson, 1951), pp. 84-87] reports that Amptre based his analysis of the mutual action of currents upon the following experimental observations:
(a) the effect of a current is reversed when the direction of the current is reversed,
(b) the effect of a current flowing in a circuit twisted into small sinuosities is the same as if the circuit were smoothed out,
(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-strengths remaining unaltered.
Although these data are adequate to allow the formulation of the laws of forces between a closed current circuit and an individual
current element, they do not allow one to obtain a conclusive result for the law of force between two individual current elements.
There has been much speculation on this subject. It assumes importance when considering effects between individual charged
particles in motion and is therefore of some significance in plasma physics.
With this in mind, the article by Dr. A. A. Ware (January 1965, Electronics and Power, p. 12) relating to controlled thermonoculear power assumes a particular interest. On p. 14 of that article there is a reproduction of a photograph which shows, very clearly, that a column of mercury carrying current develops instabilities by extending itself to form sinuosities. This experimental discovery, unknown in Ampere's time, might well provide the additional fact needed to solve the problem of the true law of electrodynamic force. In the research application described by Ware there is no doubt that every effort will have been taken to ensure that the mercury column is well screened from the magnetic effects of the current in its return loop. Therefore, the following experimental observation can seemingly be added to the four stated above:
(e) an element of a circuit carrying a constant current has an intrinsic tendency to increase in length.
It is a curious result that the combination of the observations (b) and (c) is consistent with the increase in length of the mercury column causing the column to assume its sinuous form.
Using the observations (a)-(d), Whittaker has shown that the force F on a circuit element ds' due to a current i in a circuit element ds is given by:
F = (ii'/r3)[(ds.r)ds' + (ds'.r)ds - (ds.ds)r] ...... (1)
where r is the line from ds to ds' and i' is the current in ds'. In this expression the currents and the term r3 are scalar quantities, whereas ds, ds' and r are vectors. There is an assumption in the derivation that there is no out-of-balance linear force between the elements, though there is normally out-of-balance couple.
From Whittaker's analysis, one other equation for the force is equally likely:
F = (ii'/r3)[(ds'.r)ds - (ds.r)ds' - (ds.ds')r] ...... (2)
This is based on the supporting assumption that there is no out-of-balance couple on the elements, though there may normally be
out-of-balance linear force.
Both equation (1) and equation (2) satisfy the conditions (a)-(d).
Considering the tendency to form the sinuosities in a straight column of mercury, it is seen that ds and ds' as well as r are parallel.
In this case, equations (1) and (2) both reduce to:
F = (ii'/r2)[(ds.ds']
where all quantities are scalar. As derived from equation (1) the axial force is repulsive, whereas as derived from equation (2) the force is attractive. The electromagnetic component of the energy of the mercury column may be shown from this to be proportional to
(ii'/L)(L)2, where L is the length of the mercury column, but is positive or negative according to whether the mutual force between its elements is repulsive or attractive. If the force is repulsive and equation (1) applies, a decrease in L reduces overall energy. If the force is attractive and equation (2) applies, the larger L then the less the overall energy. Therefore, from the experimental observation (e), it is clear that the tendency for L to increase corresponding to equation (2) is applicable.
It is concluded that the expression given by equation (2) is the basic law of electrodynamic force between two current elements.
Unlike equation (1), the law given by equation (2) is particularly interesting because it includes in its range of application a state in which, for any direction of r, there is no out-of-balance force or torque acting between the elements. Thus,
it can, for this particular state, satisfy fully the law that action balances reaction. The state is that in which the elements as charged particles in motion move mutually parallel. Under these conditions two like charges having like motions experience electrodynamic forces of attraction satisfying the inverse square law, a statement which hitherto has not been supported by experimental evidence or theory. This might well further attempts to account for the nature of gravitational force in terms of electromagnetic action.
I am hopeful that by these comments the very important significance of the photograph published in your January issue will not pass unnoticed.
Yours faithfully, H. ASPDEN
IBM Research Laboratories
Hursley Park, Winchester, Hants.
25th January 1965
[Dr. Ware writes: Dr. Aspden raises the subject of the force between two current elements which, although interesting, is I
think academic. In practice, an element of conductor always experiences the force due to an entire circuit. Elements of circuits can never exist in isolation. Even in the case of a
single moving charged particle the circuit is closed by the displacement current. The mercury-column experiment, to which I
referred in my article, is no exception. An element of the column is acted upon by the whole of the circuit of which it is part. Any screening will either modify the return part of the circuit or introduce new closed circuits, but the resultant system is always made up of a series of closed circuits.
As stated by Sir Edmund Whittaker in the book referred to by Dr. Aspden, the different formulas all yield the same result for the
force on a current element due to a complete circuit. Where Dr. Aspden goes wrong is in integrating the formulas for only the length of the mercury column and not the whole circuit. The rest of the circuit is also acting on an element of the column. Hence the experiment does not distinguish between the various formulas.]
Commentary: The topic I raised in the above Letter to the Editor of that IEE journal is very important and it cannot just be brushed aside by the above response by Dr. Ware. I well knew that the integrated effect of the closed current circuit so far as its action on a segment of itself amounts to zero. This, in theory, requires that middle term in equation (2) above to cancel to zero for such a closed circuit situation. That eliminates forces acting axially along the current path and reduces the force given by equation (2) to a scalar product version of the familiar Lorentz force law which we usually see expressed in vector product notation.
However, here was an experiment involving a falling column of mercury carrying a high current and developing as a result a sinuous motion during its fall. That had to be produced by the self-action of the electrodynamic effects of that current, meaning the whole closed circuit flow of the current. So we can see how the lateral deflection of the column from the vertical arises, there being scope for producing forces on the column acting at right angles to its current, that is in a horizontal direction. What was apparent from the photograph illustrating those sinuosities was not just the increase of that lateral deflection as the mercury was falling but also the fact that at the bottom of its fall when it joined the pool of mercury at its base that column had come back to its central axial position. Now lateral forces alone could not account for that. I have therefore to insist that the evidence points to forces holding the column together and able to pull it back to its central axis at the bottom of its fall.
Nor, indeed, can one just declare that every charge in motion is really part of a closed loop circuit, thanks to displacement currents in the field environment. Think what that means if we consider two electrons travelling along a common line. If each has its own current loop then the current loop of one electron acts on the other electron to apply force to it that can only be at right angles to its motion. There would be no electrodynamic force acting between the two electrons, as I say there is according to the force law of equation (2) above. The idea that gravity can be an electrodynamic interaction force is then washed away and along with it all hope of finding the ultimate Unified Field Theory. Surely common sense says that there must be scope for electrodynamic forces acting on those electrons along the axis of their motion. How else can one expect energy to be fed to and from electrons in their interplay with a magnetic field as part of the process of magnetic induction. Do remember the need to explain how energy goes from a solenoidal into the 'field' and returns to the solenoid as the current is switched off.
Forces asserted by displacement currents are forces exerted by the aether. Yet physicists tell us the aether is a figment of 19th century imagination. Then if, as I have done in my Letter above, I say that two current circuit elements acting on one another develop a force according to equation (2) above, then I am told by Dr. Ware that I am ignoring the effect of displacement currents and these are part of the whole circuit. My concern about the connection with the force of gravity and the electrodynamic forces internal to that mercury column which somehow hold it together yet extend it in length is not heeded.
Now, of course, in raising this issue at all in the professional forum to which I belonged I was only laying the foundations for a stronger attack on the problem. I was mindful of anomalies that existed in the forces exerted on cathodes where electrical discharges involved heavy ions and not just electrons. These were forces acting along the discharge axis. This was the territory known as the 'cold cathode discharge'. I knew that those forces were electrodynamic in character because they varied as the square of the discharge current and that the anomalous forces could be 100 and more times what might be expected from self-pinch pressure in the discharge as estimated using the standard Lorentz Law. Accordingly, I pursued the above matter one step further by submitting another Letter to the Editor of Electronics and Power. It was published in June 1965 as can be seen by pressing [1965b].