ELECTRICITY
[ELECTEIC DISTRIBUTION.
The law of electric force between two quantities q and q
now becomes
Force =- .
a-
The unit of quantity which we have just defined is
called the electrostatic unit, in contradistinction to the
electromagnetic unit which we shall define hereafter.
Since the dimension of unit of force is [LMT~ 2 ], where
L,M,T symbolize units of length, mass, and time, we have
for the dimension of unit of electrical quantity [Q]
[Q] = [LF HL M*!- 1 ]. *
Quantitative Results concerning Distribution.
Tt has already been indicated that electricity in equili
brium resides on the surface of conducting bodies. We must
now review shortly the experimental method by which this
surface distribution has been more closely investigated. We
shall state here some of the general principles arrived at,
and one or two of the results, reserving others for quota
tion when we come to the mathematical theory of electrical
distribution.
The most important experiments are due to Coulomb.
He used the proof-plane and the torsion balance. Riess,
who afterwards made similar experiments, used methods
similar to those of Coulomb.
Allusion has already been made to the use of the proof-
plane, and it has been stated that whsn applied to any part
of the surface of an electrified body, it brings away just as
much electricity as originally occupied the part of the sur
face which it covers. If, therefore, we electrify the mov
able ball of the torsion balance in the same sense as the
body we are to examine, and note the repulsion caused by
the proof-plane when introduced in place of the fixed ball
after having touched in succession two parts of the surface
of the body, we can, from the indications of the balance,
calculate the ratio of the quantities of electricity on the
plane in the two cases, and hence the ratio of the electrical
densities at the two points of the surface. We suppose, of
course, that the proof-plane is small enough to allow us to
assume that the electrical density is sensibly uniform over the
small area covered by it. In some of his experiments Rie^s
used a small sphere (about two lines in diameter) instead
of the small disc of the proof-plane as Coulomb used it.
The sphere in such cases ought to be very small, and even
then, except in the case of plane surfaces, its use is objec
tionable, unless the object be merely to determine, by twice
touching the same point of the same conductor, the ratio
of the whole charges on the conductor at two different
limes. The fundamental requisite is that the testing body
shall, when applied, alter the form of the testing body as
little as possible, 1 and this requisite is best satisfied by a
Email disc, and the better the smaller the disc is. The
theoretically correct procedure would be to have a small
portion of the actual surface of the body movable. If we
could remove such a piece so as to break contact with all
neighbouring portions simultaneously, then we should,
by testing the electrification of this in the balance, get a
perfect measure of the mean electric surface density on the
removed portion. We shall see that Coulomb did employ
a method like this.
1 It is evident from what we have advanced here that the use of the
proof-plane to determine the electric density at points of a surface
where the curvature is very great, e.g., at edges or conical points is
inadmissible. If we attempt to determine the electrical density at the
vertex of a cone by applying a proof-sphere there, as Riess did, we
shall very evidently get a result much under the mark, owing to the
blunting of the point when the sphere is in situ. We should, on the
Dther hand, for an opposite reason, get too large a result by apply
ing a proof-plane edgewise to a point of a surface where the curvature
is continuous.
There are various ways of using the torsion balance in
researches on distribution. We may either electrify the
movable ball independently (as above described), or we
may electrify it each time by contact with the proof-plane
when it is inserted into the balance. It must be noticed
that the repulsion of the movable ball is in the first case
proportional to the charge on the proof-plane, but in the
second to the square of the charge, so that the indications
must be reduced differently.
In measuring we may either bring the movable ball to a
fixed position, in which case the whole torsion required to
keep it in this position is proportional to the charge on the
proof-plane (or to its square, if the second of the above
modes of operation be adopted), or we may simply observe
the angle of equilibrium and calculate the quantity from
that. It is supposed, for simplicity of explanation in all
that follows, that the former of the two alternatives is
adopted, and that the movable ball is always independently
charged.
The gradual loss of electricity experienced more or less
by every insulated conductor has already been alluded to.
This loss forms one of the greatest difficulties to be encoun
tered in such experiments as we are now describing. If
we apply the proof-plane to a part of a conductor and take
the balance reading, giving a torsion r l say, and repeat the
observation, after time t, we shall get a different torsion
T 2 , owing to the loss of electricity in the interval. TLis loss,
partly if not mainly due to the insulating supports, depends
on a great many circumstances, some of which are entirely
beyond even the observation of the experimenter. We may
admit, however, what experiment confirms within certain
small limits, that the rate of loss of electricity is propor
tional to the charge, and we shall call tUz (the loss per
t
unit of time on hypothesis of uniformity) the coefficient of
dissipation (8), This coefficient, although, as we have im
plied, tolerably constant for one experiment, will vary very
much from experiment to experiment, and from day to
day ; it depends above all on the weather.
Supposing we have determined this coefficient by such
an observation as the above, then we can calculate the
torsion T , which we should have observed had we touched
the body at any interval t after the first experiment ; for
we have, provided t be small,
T = TJ - & = T 2 + S(t - ( )
In particular, if t = t, we have
r -ifo + Tj).
Coulomb used this principle in comparing the electric
densities at two points A and A of the same conductor.
He touched the two points a number of times in succession,
first A, then A , then A again, and so on, observing the cor
responding torsions TU T/, T. 2 ,T 2 ,&c., the intervals between
the operations being very nearly equal. He thus got for
the ratio of the densities at A and A the values T - - J>
- -- , &c. These values ought to be all
T! +T 2 , ZTg
equal: the mean of them was taken as the best result.
In certain cases, where the rapidity of the electric dissi
pation was too great to allow the above method to be
applied, Riess used the method of paired proof-planes. For
a description of this, and for some elaborate calculations
on the subject of electrical dissipation, the reader is referred
to Riess s work.
The cage method is well adapted for experiments on
distribution. The proof-plane, proof-sphere, or paired
proof-planes may all be used in conjunction with it. If
the cage be fairly well insualted, and a tolerably deli
Page:Encyclopædia Britannica, Ninth Edition, v. 8.djvu/32
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