where d and d' are the distances of the fixed and movable plates from
one another in the two cases, and A is some constant. We thus find V
in terms of the constant and the difference of the two screw readings.
[Illustration: FIG. 4.--Kelvin's Absolute Electrometer.]
Lord Kelvin's absolute electrometer (fig. 4) involves the same
principle. There is a certain fixed guard disk B having a hole in it
which is loosely occupied by an aluminium trap door plate, shielded by
D and suspended on springs, so that its surface is parallel with that
of the guard plate. Parallel to this is a second movable plate A, the
distances between the two being measurable by means of a screw. The
movable plate can be drawn down into a definite sighted position when
a difference of potential is made between the two plates. This
sighted position is such that the surface of the trap door plate is
level with that of the guard plate, and is determined by observations
made with the lenses H and L. The movable plate can be thus depressed
by placing on it a certain standard weight W grammes.
Suppose it is required to measure the difference of potentials V and
V' of two conductors. First one and then the other conductor is
connected with the electrode of the lower or movable plate, which is
moved by the screw until the index attached to the attracted disk
shows it to be in the sighted position. Let the screw readings in the
two cases be d and d'. If W is the weight required to depress the
attracted disk into the same sighted position when the plates are
unelectrified and g is the acceleration of gravity, then the
difference of potentials of the conductors tested is expressed by the
formula
_______
/8[pi]gW
V - V' = (d - d') / -------,
\/ S
where S denotes the area of the attracted disk.
The difference of potentials is thus determined in terms of a weight,
an area and a distance, in absolute C.G.S. measure or electrostatic
units.
[Illustration: FIG. 5.]
_Symmetrical Electrometers_ include the dry pile electrometer and
Kelvin's quadrant electrometer. The principle underlying these
instruments is that we can measure differences of potential by means of
the motion of an electrified body in a symmetrical field of electric
force. In the dry pile electrometer a single gold-leaf is hung up
between two plates which are connected to the opposite terminals of a
dry pile so that a certain constant difference of potential exists
between these plates. The original inventor of this instrument was
T.G.B. Behrens (_Gilb. Ann._, 1806, 23), but it generally bears the name
of J.G.F. von Bohnenberger, who slightly modified its form. G.T. Fechner
introduced the important improvement of using only one pile, which he
removed from the immediate neighbourhood of the suspended leaf. W.G.
Hankel still further improved the dry pile electrometer by giving a slow
motion movement to the two plates, and substituted a galvanic battery
with a large number of cells for the dry pile, and also employed a
divided scale to measure the movements of the gold-leaf (_Pogg. Ann._,
1858, 103). If the gold-leaf is unelectrified, it is not acted upon by
the two plates placed at equal distances on either side of it, but if
its potential is raised or lowered it is attracted by one disk and
repelled by the other, and the displacement becomes a measure of its
potential.
[Illustration: FIG. 6.--Kelvin's Quadrant Electrometer.]
A vast improvement in this instrument was made by the invention of the
quadrant electrometer by Lord Kelvin, which is the most sensitive form
of electrometer yet devised. In this instrument (see fig. 5) a flat
paddle-shaped needle of aluminium foil U is supported by a bifilar
suspension consisting of two cocoon fibres. This needle is suspended in
the interior of a glass vessel partly coated with tin-foil on the
outside and inside, forming therefore a Leyden jar (see fig. 6). In the
bottom of the vessel is placed some sulphuric acid, and a platinum wire
attached to the suspended needle dips into this acid. By giving a charge
to this Leyden jar the needle can thus be maintained at a certain
constant high potential. The needle is enclosed by a sort of flat box
divided into four insulated quadrants A, B, C, D (fig. 5), whence the
name. The opposite quadrants are connected together by thin platinum
wires. These quadrants are insulated from the needle and from the case,
and the two pairs are connected to two electrodes. When the instrument
is to be used to determine the potential difference between two
conductors, they are connected to the two opposite pairs of quadrants.
The needle in its normal position is symmetrically placed with regard to
the quadrants, and carries a mirror by means of which its displacement
can be observed in the usual manner by reflecting the ray of light from
it. If the two quadrants are at different potentials, the needle moves
from one quadrant towards the other, and the image of a spot of light on
the scale is therefore displaced. Lord Kelvin provided the instrument
with two necessary adjuncts, viz. a replenisher or rotating
electrophorus (q.v.), by means of which the charge of the Leyden jar
which forms the enclosing vessel can be increased or diminished, and
also a small aluminium balance plate or gauge, which is in principle the
same as the attracted disk portable electrometer by means of which the
potential of the inner coating of the Leyden jar is preserved at a known
value.
According to the mathematical theory of the instrument,[3] if V and V'
are the potentials of the quadrants and v is the potential of the
needle, then the torque acting upon the needle to cause rotation is
given by the expression,
C(V - V') {v - 1/2(V + V')},
where C is some constant. If v is very large compared with the mean
value of the potentials of the two quadrants, as it usually is, then
the above expression indicates that the couple varies as the
difference of the potentials between the quadrants.
Dr J. Hopkinson found, however, before 1885, that the above formula
does not agree with observed facts (_Proc. Phys. Soc. Lond._, 1885, 7,
p. 7). The formula indicates that the sensibility of the instrument
should increase with the charge of the Leyden jar or needle, whereas
Hopkinson found that as the potential of the needle was increased by
working the replenisher of the jar, the deflection due to three volts
difference between the quadrants first increased and then diminished.
He found that when the potential of the needle exceeded a certain
value, of about 200 volts, for the particular instrument he was using
(made by White of Glasgow), the above formula did not hold good. W.E.
Ayrton, J. Perry and W.E. Sumpner, who in 1886 had noticed the same
fact as Hopkinson, investigated the matter in 1891 (_Proc. Roy. Soc._,
1891, 50, p. 52; _Phil. Trans._, 1891, 182, p. 519). Hopkinson had
been inclined to attribute the anomaly to an increase in the tension
of the bifilar threads, owing to a downward pull on the needle, but
they showed that this theory would not account for the discrepancy.
They found from observations that the particular quadrant electrometer
they used might be made to follow one or other of three distinct laws.
If the quadrants were near together there were certain limits between
which the potential of the needle might vary without producing more
than a small change in the deflection corresponding with the fixed
potential difference of the quadrants. For example, when the quadrants
were about 2.5 mm. apart and the suspended fibres near together at the
top, the deflection produced by a P.D. of 1.45 volts between the
quadrants only varied about 11% when the potential of the needle
varied from 896 to 3586 volts. When the fibres were far apart at the
top a similar flatness was obtained in the curve with the quadrants
about 1 mm. apart. In this case the deflection of the needle was
practically quite constant when its potential varied from 2152 to 3227
volts. When the quadrants were about 3.9 mm. apart, the deflection for
a given P.D. between the quadrants was almost directly proportional to
the potential of the needle. In other words, the electrometer nearly
obeyed the theoretical law. Lastly, when the quadrants were 4 mm. or
more apart, the deflection increased much more rapidly than the
potential, so that a maximum sensibility bordering on instability was
obtained. Finally, these observers traced the variation to the fact
that the wire supporting the aluminium needle as well as the wire
which connects the needle with the sulphuric acid in the Leyden jar in
the White pattern of Leyden jar is enclosed in a metallic guard tube
to screen the wire from external action. In order that the needle may
project outside the guard tube, openings are made in its two sides;
hence the moment the needle is deflected each half of it becomes
unsymmetrically placed relatively to the two metallic pieces which
join the upper and lower half of the guard tube. Guided by these
experiments, Ayrton, Perry and Sumpner constructed an improved
unifilar quadrant electrometer which was not only more sensitive than
the White pattern, but fulfilled the theoretical law of working. The
bifilar suspension was abandoned, and instead a new form of adjustable
magnetic control was adopted. All the working parts of the instrument
were supported on the base, so that on removing a glass shade which
serves as a Leyden jar they can be got at and adjusted in position.
The conclusion to which the above observers came was that any quadrant
electrometer made in any manner does not necessarily obey a law of
deflection making the deflections proportional to the potential
difference of the quadrants, but that an electrometer can be
constructed which does fulfil the above law.
The importance of this investigation resides in the fact that an
electrometer of the above pattern can be used as a wattmeter (q.v.),
provided that the deflection of the needle is proportional to the
potential difference of the quadrants. This use of the instrument was
proposed simultaneously in 1881 by Professors Ayrton and G.F.
Fitzgerald and M.A. Potier. Suppose we have an inductive and a
non-inductive circuit in series, which is traversed by a periodic
current, and that we desire to know the power being absorbed to the
inductive circuit. Let v1, v2, v3 be the instantaneous potentials of
the two ends and middle of the circuit; let a quadrant electrometer be
connected first with the quadrants to the two ends of the inductive
circuit and the needle to the far end of the non-inductive circuit,
and then secondly with the needle connected to one of the quadrants
(see fig. 5). Assuming the electrometer to obey the above-mentioned
theoretical law, the first reading is proportional to
/ v1 + v2\
v1 - v2 ( v3 - ------- )
\ 2 /
and the second to
/ v1 + v2\
v1 - v2 ( v2 - ------- ).
\ 2 /