This formula is important in connexion with the capacity of electric
cables, which consist of a cylindrical conductor (a wire) enclosed in
a conducting sheath. If the dielectric or separating insulator has a
constant K, then the capacity becomes K times as great.
Capacity of two parallel planes.
"Edge effect."
The capacity of two parallel planes can be calculated at once if we
neglect the distribution of the lines of force near the edges of the
plates, and assume that the only field is the uniform field between
the plates. Let V1 and V2 be the potentials of the plates, and let a
charge Q be given to one of them. If S is the surface of each plate,
and d their distance, then the electric force E in the space between
them is E = (V1-V2)/d. But if [sigma] is the surface density, E =
4[pi][sigma], and [sigma] = Q/S. Hence we have
(V1 - V2) d = 4[pi]Q/S or C = Q/(V1 - V2) = S/4[pi]d (13).
In this calculation we neglect altogether the fact that electric force
distributed on curved lines exists outside the interspace between the
plates, and these lines in fact extend from the back of one plate to
that of the other. G.R. Kirchhoff (_Gesammelte Abhandl._ p. 112) has
given a full expression for the capacity C of two circular plates of
thickness t and radius r placed at any distance d apart in air from
which the edge effect can be calculated. Kirchhoff's expression is as
follows:--
[pi]r^2 r / 16[pi]r(d+t) d + t\
C = ------- + ------ ( d log_[epsilon] ------------ + t log_[epsilon] ----- ) (14).
4[pi]d 4[pi]d \ [epsilon]d^2 t /
In the above formula [epsilon] is the base of the Napierian
logarithms. The first term on the right-hand side of the equation is
the expression for the capacity, neglecting the curved edge
distribution of electric force, and the other terms take into account,
not only the uniform field between the plates, but also the
non-uniform field round the edges and beyond the plates.
Guard plates.
In practice we can avoid the difficulty due to irregular distribution
of electric force at the edges of the plate by the use of a guard
plate as first suggested by Lord Kelvin.[8] If a large plate has a
circular hole cut in it, and this is nearly filled up by a circular
plate lying in the same plane, and if we place another large plate
parallel to the first, then the electric field between this second
plate and the small circular plate is nearly uniform; and if S is the
area of the small plate and d its distance from the opposed plate, its
capacity may be calculated by the simple formula C = S/4[pi]d. The
outer larger plate in which the hole is cut is called the "guard
plate," and must be kept at the same potential as the smaller inner or
"trap-door plate." The same arrangement can be supplied to a pair of
coaxial cylinders. By placing metal plates on either side of a larger
sheet of dielectric or insulator we can construct a condenser of
relatively large capacity. The instrument known as a Leyden jar (q.v.)
consists of a glass bottle coated within and without for three parts
of the way up with tinfoil.
Systems of condensers.
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 capacities, then
[Sigma](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.