86.]
COEFFICIENTS OF POTENTIAL AND OF INDUCTION.
89
Similarly the work required to increase the charge of
is
, so that the whole work done in increasing the charge of the system is
|
.
|
|
If we suppose this process repeated an indefinitely great number of times, each charge being indefinitely small, till the total effect becomes sensible, the work done will be
|
;
|
|
where
means the sum of all the products of the potential of each element into the quantity of electricity in that element when
, and
is the initial and
the final value of
.
If we make
and
, we find for the work required to charge an unelectrified system so that the electricity is
and the potential
in each element,
|
.
|
|
General Theory of a System of Conductors.
86.] Let
be any number of conductors of any form. Let the charge or total quantity of electricity on each of these be
and let their potentials be
respectively.
Let us suppose the conductors to be all insulated and originally free of charge, and at potential zero.
Now let
be charged with unit of electricity, the other bodies being without charge. The effect of this charge on
will be to raise the potential of
to
, that of
to
, and that of
to
, where
, &c. are quantities depending on the form and relative position of the conductors. The quantity
may be called the Potential Coefficient of
on itself, and
may be called the Potential Coefficient of
on
, and so on.
If the charge upon
is now made
, then, by the principle of superposition, we shall have
|
.
|
|
Now let
be discharged, and
charged with unit of electricity, and let the potentials of
...
be
,
,...
potentials due to
on
will be
|
.
|
|
Similarly let us denote the potential of
due to a unit charge on
by
, and let us call
the Potential Coefficient of
on
,