An introduction to the mathematical theory of attraction ...

Examples. 271
Again, it appears as in Art. 131, that
qll q12 * * *. qlm, q22 q235 * * * q2m, * * * qm
are independent of the state of the field outside Am+,.:Hence
the charges on the interior conductors are functions of the
differences between the values of the potential on them and
its value on the surrounding conductor, and are independent
of the state of the field external to the latter.
EXAIMPLES.
1. If there be a system of conductors and a new conductor be brought into
the field, the coefficient of potential of any one of the others on itself is
diminished.
The final result here is the same as if the portion of space originally unoccupied, and subsequently occupied by the new conductor, were rendered
capable of conducting electricity. The consequence of this change would be a
new distribution of electricity brought about by the electric forces, and accompanied, therefore, by a diminution of the electric energy of the system. If we
now suppose all the conductors uncharged except Ai, the original energy is
plneli2, and after the introduction of the new conductor the energy becomes
p'liel2, but, as this is less than the original, we have p' i <pll.
2. If two conductors occupying the field be placed in electric communication
so as to form a single one, determine the capacity of the new conductor in terms
of the coefficients of capacity and induction of the original system.
If we suppose the two conductors originally at the same potential Z, this is
also the value of the potential on the single conductor formed by their union,
and if Q be the capacity of this conductor, we have
el = (ql + q2) L, e2 = (q2 + q22) L, e + e2 = QL;
whence
Q = qiî + q22 + 2q12.
3. Show that the capacity of any conductor is less than that of another conductor geometrically capable of surrounding the former.
If 42, supposed non-conducting, were made to surround A1, no change would
take place in the electric condition of A1. If then electric communication
between A1 and A2 were established, and A2 rendered conducting, the electric
charge on A4 would be transferred to the external surface of A2, and in effecting
this transference the electric forces would do work, and therefore the electric
energy would be diminished. Hence if e be the charge originally on A1, we have
e2 e2
q2 qi
and therefore, q2 > ql, where ql and q2 are the capacities of A4 and A2.