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
(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
(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.
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
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.
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:
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.
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
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
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
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.
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
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].