Later researches by Mrs Ayrton (Electrician, 1898, 41, p. 720),
however, showed that for a direct-current arc of given size with solid
carbons, the observed values of V can be better represented as a
function both of A and of L of the form
c + dL
V = a + bL + ------.
A
In the case of direct-current arcs formed with solid carbons, Edlund
and other observers agree that the arc resistance R may be expressed
by a simple straight line law, R = e + fL. If the arc is formed with
cored carbons, Mrs Ayrton demonstrated that the lines expressing
resistance as a function of arc length are no longer straight, but
that there is a rather sudden dip down when the length of the arc is
less than 3 mm.
The constants in the above equation for the potential difference of
the carbons were determined by Mrs Ayrton in the case of solid carbons
to be--
11.7 + 10.5L
V = 38.9 + 2.07L + ------------.
A
There has been much debate as to the meaning to be given to the
constant a in the above equation, which has a value apparently not far
from forty volts for a direct-current arc with solid carbons. The
suggestion made in 1867 by Edlund (_Phil. Mag._, 1868, 36, p. 358),
that it implied the existence of a counter-electromotive force in the
arc, was opposed by Luggin in 1889 (_Wien. Ber._ 98, p. 1198), Ernst
Lecher in 1888 (_Wied. Ann._, 1888, 33, p. 609), and by Franz Stenger
in 1892 (_Id._ 45, p. 33); whereas Victor von Lang and L. M. Arons in
1896 (_Id._ 30, p. 95), concluded that experiment indicated the
presence of a counter-electromotive force of 20 volts. A. E. Blondel
concludes, from experiments made by him in 1897 (_The Electrician_,
1897, 39, p. 615), that there is no counter-electromotive force in the
arc greater than a fraction of a volt. Subsequently W. Duddëll (_Proc.
Roy. Soc._, 1901, 68, p. 512) described experiments tending to prove
the real existence of a counter-electromotive force in the arc,
probably having a thermo-electric origin, residing near the positive
electrode, and of an associated lesser adjuvant _e.m.f._ near the
negative carbon.
This fall in voltage between the carbons and the arc is not uniformly
distributed. In 1898 Mrs Ayrton described the results of experiments
showing that if V1 is the potential difference between the positive
carbon and the arc, then
9 + 3.1L
V1 = 31.28 + --------;
A
and if V2 is the potential difference between the arc and the negative
carbon, then
13.6
V2 = 7.6 + ----,
A
where A is the current through the arc in amperes and L is the length
of the arc in millimetres.
The total potential difference between the carbons, minus the fall in
potential down the arc, is therefore equal to the sum of V1 + V2 = V3.
22.6 + 3.1L
Hence V3 = 38.88 + -----------.
A
The difference between this value and the value of V, the total
potential difference between the carbons, gives the loss in potential
due to the true arc. These laws are simple consequences of
straight-line laws connecting the work spent in the arc at the two
electrodes with the other quantities. If W be the work spent in the
arc on either carbon, measured by the product of the current and the
potential drop in passing from the carbon to the arc, or vice versa,
then for the positive carbon W = a + bA, if the length of arc is
constant, W = c + dL, if the current through the arc is constant, and
for the negative carbon W = e + fA.
In the above experiments the potential difference between the carbons
and the arc was measured by using a third exploring carbon as an
electrode immersed in the arc. This method, adopted by Lecher, F.
Uppenborn, S. P. Thompson, and J. A. Fleming, is open to the objection
that the introduction of the third carbon may to a considerable extent
disturb the distribution of potential.
The total work spent in the continuous-current arc with solid carbons
may, according to Mrs Ayrton, be expressed by the equation