If we have a number of such condensers we can combine them in “parallel” or in “series.” If all the plates on one side are connected together and also those on the other, the condensers are joined in parallel. If C1, C2, C3, &c., are the separate Systems of condensers. capacities, then Σ(C) = C1 + C2 + C3 + &c., is the total capacity in parallel. If the condensers are so joined that the inner coating of one is connected to the outer coating of the next, they are said to be in series. Since then they are all charged with the same quantity of electricity, and the total over all potential difference V is the sum of each of the individual potential differences V1, V2, V3, &c., we have
Q = C1V1 = C2V2 = C3V3 = &c., and V = V1 + V2 + V3 + &c.
The resultant capacity is C = Q/V, and
C = 1 / (1/C1 + 1/C2 + 1/C3 + &c.) = 1 / Σ(1/C)
(15).
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.
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 Relative determinations. 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