These rules provide means for calculating the resultant capacity when
any number of condensers are joined up in any way.
If one condenser is charged, and then joined in parallel with another
uncharged condenser, the charge is divided between them in the ratio
of their capacities. For if C1 and C2 are the capacities and Q1 and Q2
are the charges after contact, then Q1/C1 and Q2/C2 are the potential
differences of the coatings and must be equal. Hence Q1/C1 = Q2/C2 or
Q1/Q2 = C1/C2. It is worth noting that if we have a charged sphere we
can perfectly discharge it by introducing it into the interior of
another hollow insulated conductor and making contact. The small
sphere then becomes part of the interior of the other and loses all
charge.
_Measurement of Capacity._--Numerous methods have been devised for the
measurement of the electrical capacity of conductors in those cases in
which it cannot be determined by calculation. Such a measurement may
be an _absolute_ determination or a _relative_ one. The dimensions of
a capacity in electrostatic measure is a length (see UNITS, PHYSICAL).
Thus the capacity of a sphere in electrostatic units (E.S.U.) is the
same as the number denoting its radius in centimetres. The unit of
electrostatic capacity is therefore that of a sphere of 1 cm.
radius.[9] This unit is too small for practical purposes, and hence a
unit of capacity 900,000 greater, called a microfarad, is generally
employed. Thus for instance the capacity in free space of a sphere 2
metres in diameter would be 100/900,000 = 1/9000 of a microfarad. The
electrical capacity of the whole earth considered as a sphere is about
800 microfarads. An absolute measurement of capacity means, therefore,
a determination in E.S. units made directly without reference to any
other condenser. On the other hand there are numerous methods by which
the capacities of condensers may be compared and a relative
measurement made in terms of some standard.
Relative determinations.
One well-known comparison method is that of C.V. de Sauty. The two
condensers to be compared are connected in the branches of a
Wheatstone's Bridge (q.v.) and the other two arms completed with
variable resistance boxes. These arms are then altered until on
raising or depressing the battery key there is no sudden deflection
either way of the galvanometer. If R1 and R2 are the arms' resistances
and C1 and C2 the condenser capacities, then when the bridge is
balanced we have R1 : R2 = C1 : C2.
Another comparison method much used in submarine cable work is the
method of mixtures, originally due to Lord Kelvin and usually called
Thomson and Gott's method. It depends on the principle that if two
condensers of capacity C1 and C2 are respectively charged to
potentials V1 and V2, and then joined in parallel with terminals of
opposite charge together, the resulting potential difference of the
two condensers will be V, such that
(C1V1 - C2V2)
V = ------------- (16);
(C + C)
and hence if V is zero we have C1 : C2 = V2 : V1.
The method is carried out by charging the two condensers to be
compared at the two sections of a high resistance joining the ends of
a battery which is divided into two parts by a movable contact.[10]
This contact is shifted until such a point is found by trial that the
two condensers charged at the different sections and then joined as
above described and tested on a galvanometer show no charge. Various
special keys have been invented for performing the electrical
operations expeditiously.
A simple method for condenser comparison is to charge the two
condensers to the same voltage by a battery and then discharge them
successively through a ballistic galvanometer (q.v.) and observe the
respective "throws" or deflections of the coil or needle. These are
proportional to the capacities. For the various precautions necessary
in conducting the above tests special treatises on electrical testing
must be consulted.
Absolute determinations.
In the absolute determination of capacity we have to measure the ratio
of the charge of a condenser to its plate potential difference. One of
the best methods for doing this is to charge the condenser by the
known voltage of a battery, and then discharge it through a
galvanometer and repeat this process rapidly and successively. If a
condenser of capacity C is charged to potential V, and discharged n
times per second through a galvanometer, this series of intermittent
discharges is equivalent to a current nCV. Hence if the galvanometer
is calibrated by a potentiometer (q.v.) we can determine the value of
this current in amperes, and knowing the value of n and V thus
determine C. Various forms of commutator have been devised for
effecting this charge and discharge rapidly by J.J. Thomson, R.T.
Glazebrook, J.A. Fleming and W.C. Clinton and others.[11] One form
consists of a tuning-fork electrically maintained in vibration of
known period, which closes an electric contact at every vibration and
sets another electromagnet in operation, which reverses a switch and
moves over one terminal of the condenser from a battery to a
galvanometer contact. In another form, a revolving contact is used
driven by an electric motor, which consists of an insulating disk
having on its surface slips of metal and three wire brushes a, b, c
(see fig. 2) pressing against them. The metal slips are so placed
that, as the disk revolves, the middle brush, connected to one
terminal of the condenser C, is alternately put in conductive
connexion with first one and then the other outside brush, which are
joined respectively to the battery B and galvanometer G terminals.
From the speed of this motor the number of commutations per second can
be determined. The above method is especially useful for the
determinations of very small capacities of the order of 100
electrostatic units or so and upwards.
[Illustration: FIG. 2.]
_Dielectric constant._--Since all electric charge consists in a state of
strain or polarization of the dielectric, it is evident that the
physical state and chemical composition of the insulator must be of
great importance in determining electrical phenomena. Cavendish and
subsequently Faraday discovered this fact, and the latter gave the name
"specific inductive capacity," or "dielectric constant," to that quality
of an insulator which determines the charge taken by a conductor
embedded in it when charged to a given potential. The simplest method of
determining it numerically is, therefore, that adopted by Faraday.[12]
He constructed two equal condensers, each consisting of a metal ball
enclosed in a hollow metal sphere, and he provided also certain
hemispherical shells of shellac, sulphur, glass, resin, &c., which he
could so place in one condenser between the ball and enclosing sphere
that it formed a condenser with solid dielectric. He then determined the
ratio of the capacities of the two condensers, one with air and the
other with the solid dielectric. This gave the dielectric constant K of
the material. Taking the dielectric constant of air as unity he obtained
the following values, for shellac K = 2.0, glass K = 1.76, and sulphur K
= 2.24.