forces exerted by this conductor on a point charged with unit
quantity, that is, double the quantity at each point, or what is the
same thing, double the total charge. Plainly, equilibrium still
subsists. But carry, now, the positive electrostatic unit towards
the conductor. Everywhere we shall have to overcome double the force
of repulsion we did before, everywhere we shall have to expend
double the work. By doubling the charge of the conductor a double
potential has been produced. Charge and potential go hand in hand,
are proportional. Consequently, calling the total quantity of
electricity of a conductor _Q_ and its potential _V_, we can write:
_Q = CV_, where _C_ stands for a constant, the import of which will
be understood simply from noting that _C = Q/V_.[32] But the
division of a number representing the units of quantity of a
conductor by the number representing its units of potential tells us
the quantity which falls to the share of the unit of potential. Now
the number _C_ here we call the capacity of a conductor, and have
substituted, thus, in the place of the old relative determination of
capacity, an absolute determination.[33]
In simple cases the connexion between charge, potential, and
capacity is easily ascertained. Our conductor, let us say, is a
sphere of radius _r_, suspended free in a large body of air. There
being no other conductors in the vicinity, the charge _q_ will then
distribute itself uniformly upon the surface of the sphere, and
simple geometrical considerations yield for its potential the
expression _V = q/r_. Hence, _q/V = r_; that is, the capacity of a
sphere is measured by its radius, and in the C. G. S. system in
centimetres.[34] It is clear also, since a potential is a quantity
divided by a length, that a quantity divided by a potential must be
a length.
Imagine (Fig. 36) a jar composed of two concentric conductive
spherical shells of the radii _r_ and _r₁_, having only air between
them. Connecting the outside sphere with the earth, and charging the
inside sphere by means of a thin, insulated wire passing through the
first, with the quantity _Q_, we shall have _V = (r₁-r)/(r₁r)Q_, and
for the capacity in this case _(r₁r)/(r₁-r)_, or, to take a specific
example, if _r = 16_ and _r₁ = 19_, a capacity of about 100
centimetres.
[Illustration: Fig. 36.]
We shall now use these simple cases for illustrating the principle
by which capacity and potential are determined. First, it is clear
that we can use the jar composed of concentric spheres with its
known capacity as our unit jar and by means of this ascertain, in
the manner above laid down, the capacity of any given jar _F_. We
find, for example, that 37 discharges of this unit jar of the
capacity 100, just charges the jar investigated at the same
striking distance, that is, at the same potential. Hence, the
capacity of the jar investigated is 3700 centimetres. The large
battery of the Prague physical laboratory, which consists of sixteen
such jars, all of nearly equal size, has a capacity, therefore, of
something like 50,000 centimetres, or the capacity of a sphere, a
kilometre in diameter, freely suspended in atmospheric space. This
remark distinctly shows us the great superiority which Leyden jars
possess for the storage of electricity as compared with common
conductors. In fact, as Faraday pointed out, jars differ from simple
conductors mainly by their great capacity.
[Illustration: Fig. 37.]
For determining potential, imagine the inner coating of a jar _F_,
the outer coating of which communicates with the ground, connected
by a long, thin wire with a conductive sphere _K_ placed free in a
large atmospheric space, compared with whose dimensions the radius
of the sphere vanishes. (Fig. 37.) The jar and the sphere assume at
once the same potential. But on the surface of the sphere, if that
be sufficiently far removed from all other conductors, a uniform
layer of electricity will be found. If the sphere, having the radius
_r_, contains the charge _q_, its potential is _V = q/r_. If the
upper half of the sphere be severed from the lower half and
equilibrated on a balance with one of whose beams it is connected by
silk threads, the upper half will be repelled from the lower half
with the force _P = q²/8r² = 1/8V²_. This repulsion _P_ may be
counter-balanced by additional weights placed on the beam-end, and
so ascertained. The potential is then _V = [sqrt](8P)_.[35]
That the potential is proportional to the square root of the force
is not difficult to see. A doubling or trebling of the potential
means that the charge of all the parts is doubled or trebled; hence
their combined power of repulsion quadrupled or nonupled.
Let us consider a special case. I wish to produce the potential 40
on the sphere. What additional weight must I give to the half sphere
in grammes that the force of repulsion shall maintain the balance in
exact equilibrium? As a gramme weight is approximately equivalent
to 1000 units of force, we have only the following simple example to
work out: _40×40 = 8× 1000.x_, where _x_ stands for the number of
grammes. In round numbers we get _x_ = 0.2 gramme. I charge the jar.
The balance is deflected; I have reached, or rather passed, the
potential 40, and you see when I discharge the jar the associated
spark.[36]
The striking distance between the knobs of a machine increases with
the difference of the potential, although not proportionately to
that difference. The striking distance increases faster than the
potential difference. For a distance between the knobs of one
centimetre on this machine the difference of potential is 110. It
can easily be increased tenfold. Of the tremendous differences of
potential which occur in nature some idea may be obtained from the
fact that the striking distances of lightning in thunder-storms is
counted by miles. The differences of potential in galvanic batteries
are considerably smaller than those of our machine, for it takes
fully one hundred elements to give a spark of microscopic striking
distance.
* * * * *
We shall now employ the ideas reached to shed some light upon
another important relation between electrical and mechanical
phenomena. We shall investigate what is the potential _energy_, or
the _store of work_, contained in a charged conductor, for example,
in a jar.
If we bring a quantity of electricity up to a conductor, or, to
speak less pictorially, if we generate by work electrical force in a
conductor, this force is able to produce anew the work by which it
was generated. How great, now, is the energy or capacity for work of
a conductor of known charge _Q_ and known potential _V_?
Imagine the given charge _Q_ divided into very small parts _q_,
_q₁_, _q₂_ ..., and these little parts successively carried up to
the conductor. The first very small quantity _q_ is brought up
without any appreciable work and produces by its presence a small
potential _V__{'}. To bring up the second quantity, accordingly, we
must do the work _q__{'}_V__{'}, and similarly for the quantities
which follow the work _q__{''}_V__{''}, _q__{'''}_V__{'''}, and so
forth. Now, as the potential rises proportionately to the quantities
added until the value _V_ is reached, we have, agreeably to the
graphical representation of Fig. 38, for the total work performed,
_W = 1/2QV_,
which corresponds to the total energy of the charged conductor.
Using the equation _Q_ = _CV_, where _C_ stands for capacity, we
also have,
_W = 1/2CV²_, or _W = Q²/2C_.
It will be helpful, perhaps, to elucidate this idea by an analogy
from the province of mechanics. If we pump a quantity of liquid,
_Q_, gradually into a cylindrical vessel (Fig. 39), the level of the
liquid in the vessel will gradually rise. The more we have pumped
in, the greater the pressure we must overcome, or the higher the
level to which we must lift the liquid. The stored-up work is
rendered again available when the heavy liquid _Q_, which reaches up
to the level _h_, flows out. This work _W_ corresponds to the fall
of the whole liquid weight _Q_, through the distance _h_/2 or
through the altitude of its centre of gravity. We have
_W = 1/2Qh_.
Further, since _Q_ = _Kh_, or since the weight of the liquid and the
height _h_ are proportional, we get also
_W = 1/2Kh²_ and _W = Q²/2K_.
[Illustration: Fig. 38.]
[Illustration: Fig. 39.]
As a special case let us consider our jar. Its capacity is _C_ =
3700, its potential _V_ = 110; accordingly, its quantity _Q = CV_ =
407,000 electrostatic units and its energy _W = 1/2QV_ = 22,385,000
C. G. S. units of work.
The unit of work of the C. G. S. system is not readily appreciable
by the senses, nor does it well admit of representation, as we are
accustomed to work with weights. Let us adopt, therefore, as our
unit of work the gramme-centimetre, or the gravitational pressure of
a gramme-weight through the distance of a centimetre, which in round
numbers is 1000 times greater than the unit assumed above; in this
case, our numerical result will be approximately 1000 times smaller.
Again, if we pass, as more familiar in practice, to the
kilogramme-metre as our unit of work, our unit, the distance being
increased a hundred fold, and the weight a thousand fold, will be
100,000 times larger. The numerical result expressing the work done
is in this case 100,000 times less, being in round numbers 0.22
kilogramme-metre. We can obtain a clear idea of the work done here
by letting a kilogramme-weight fall 22 centimetres.
This amount of work, accordingly, is performed on the charging of
the jar, and on its discharge appears again, according to the
circumstances, partly as sound, partly as a mechanical disruption of
insulators, partly as light and heat, and so forth.
The large battery of the Prague physical laboratory, with its
sixteen jars charged to equal potentials, furnishes, although the
effect of the discharge is imposing, a total amount of work of only
three kilogramme-metres.
In the development of the ideas above laid down we are not
restricted to the method there pursued; in fact, that method was
selected only as one especially fitted to familiarise us with the
phenomena. On the contrary, the connexion of the physical processes
is so multifarious that we can come at the same event from very
different directions. Particularly are electrical phenomena
connected with all other physical events; and so intimate is this
connexion that we might justly call the study of electricity the
theory of the general connexion of physical processes.
With respect to the principle of the conservation of energy which
unites electrical with mechanical phenomena, I should like to point
out briefly two ways of following up the study of this connexion.
A few years ago Professor Rosetti, taking an influence-machine,
which he set in motion by means of weights alternately in the
electrical and non-electrical condition with the same velocities,
determined the mechanical work expended in the two cases and was
thus enabled, after deducting the work of friction, to ascertain the
mechanical work consumed in the development of the electricity.
I myself have made this experiment in a modified, and, as I think,
more advantageous form. Instead of determining the work of friction
by special trial, I arranged my apparatus so that it was eliminated
of itself in the measurement and could consequently be neglected.
The so-called fixed disk of the machine, the axis of which is
placed vertically, is suspended somewhat like a chandelier by three
vertical threads of equal lengths _l_ at a distance _r_ from the
axis. Only when the machine is excited does this fixed disk, which
represents a Prony's brake, receive, through its reciprocal action
with the rotating disk, a deflexion _[alpha]_ and a moment of
torsion which is expressed by _D = (Pr²/l)[alpha]_, where _P_ is the
weight of the disk.[37] The angle _[alpha]_ is determined by a
mirror set in the disk. The work expended in _n_ rotations is given
by _2n[pi]D_.
If we close the machine, as Rosetti did, we obtain a continuous
current which has all the properties of a very weak galvanic
current; for example, it produces a deflexion in a multiplier which
we interpose, and so forth. We can directly ascertain, now, the
mechanical work expended in the maintenance of this current.
If we charge a jar by means of a machine, the energy of the jar
employed in the production of sparks, in the disruption of the
insulators, etc., corresponds to a part only of the mechanical work
expended, a second part of it being consumed in the arc which forms
the circuit.[38] This machine, with the interposed jar, affords in
miniature a picture of the transference of force, or more properly
of work. And in fact nearly the same laws hold here for the
economical coefficient as obtain for large dynamo-machines.
Another means of investigating electrical energy is by its
transformation into heat. A long time ago (1838), before the
mechanical theory of heat had attained its present popularity, Riess
performed experiments in this field with the help of his electrical
air-thermometer or thermo-electrometer.
[Illustration: Fig. 40.]
If the discharge be conducted through a fine wire passing through
the globe of the air-thermometer, a development of heat is observed
proportional to the expression above-discussed _W = 1/2QV_. Although
the total energy has not yet been transformed into measurable heat
by this means, in as much as a portion is left behind in the spark
in the air outside the thermometer, still everything tends to show
that the total heat developed in all parts of the conductor and
along all the paths of discharge is the equivalent of the work
1/2_QV_.
It is not important here whether the electrical energy is
transformed all at once or partly, by degrees. For example, if of
two equal jars one is charged with the quantity _Q_ at the potential
_V_ the energy present is 1/2_QV_. If the first jar be discharged
into the second, _V_, since the capacity is now doubled, falls to
_V_/2. Accordingly, the energy 1/4_QV_ remains, while 1/4_QV_ is
transformed in the spark of discharge into heat. The remainder,
however, is equally distributed between the two jars so that each on
discharge is still able to transform 1/8_QV_ into heat.
* * * * *
We have here discussed electricity in the limited phenomenal form in
which it was known to the inquirers before Volta, and which has been
called, perhaps not very felicitously, "statical electricity." It is
evident, however, that the nature of electricity is everywhere one
and the same; that a substantial difference between statical and
galvanic electricity does not exist. Only the quantitative
circumstances in the two provinces are so widely different that
totally new aspects of phenomena may appear in the second, for
example, magnetic effects, which in the first remained unnoticed,
whilst, _vice versa_, in the second field statical attractions and
repulsions are scarcely appreciable. As a fact, we can easily
show the magnetic effect of the current of discharge of an
influence-machine on the galvanoscope although we could hardly have
made the original discovery of the magnetic effects with this
current. The statical distant action of the wire poles of a galvanic
element also would hardly have been noticed had not the phenomenon
been known from a different quarter in a striking form.
If we wished to characterise the two fields in their chief and most
general features, we should say that in the first, high potentials
and small quantities come into play, in the second small potentials
and large quantities. A jar which is discharging and a galvanic
element deport themselves somewhat like an air-gun and the bellows
of an organ. The first gives forth suddenly under a very high
pressure a small quantity of air; the latter liberates gradually
under a very slight pressure a large quantity of air.
In point of principle, too, nothing prevents our retaining the
electrostatical units in the domain of galvanic electricity and in
measuring, for example, the strength of a current by the number of
electrostatic units which flow per second through its cross-section.
But this would be in a double aspect impractical. In the first
place, we should totally neglect the magnetic facilities for
measurement so conveniently offered by the current, and substitute
for this easy means a method which can be applied only with
difficulty and is not capable of great exactness. In the second
place our units would be much too small, and we should find
ourselves in the predicament of the astronomer who attempted to
measure celestial distances in metres instead of in radii of the
earth and the earth's orbit; for the current which by the magnetic
C. G. S. standard represents the unit, would require a flow of some
30,000,000,000 electrostatic units per second through its
cross-section. Accordingly, different units must be adopted here.
The development of this point, however, lies beyond my present
task.
FOOTNOTES:
[Footnote 26: A lecture delivered at the International Electrical
Exhibition, in Vienna, on September 4, 1883.]
[Footnote 27: If the two bodies were oppositely electrified they
would exert attractions upon each other.]
[Footnote 28: The quantity which flows off is in point of fact less
than _q_. It would be equal to the quantity _q_ only if the inner
coating of the jar were wholly encompassed by the outer coating.]
[Footnote 29: Rigorously, of course, this is not correct. First, it
is to be noted that the jar _L_ is discharged simultaneously with
the electrode of the machine. The jar _F_, on the other hand, is
always discharged simultaneously with the outer coating of the jar
_L_. Hence, if we call the capacity of the electrode of the machine
_E_, that of the unit jar _L_, that of the outer coating of _L_,
_A_, and that of the principal jar _F_, then this equation would
exist for the example in the text: _(F + A)/(L + E) = 5_. A cause of
further departure from absolute exactness is the residual charge.]
[Footnote 30: Making allowance for the corrections indicated in the
preceding footnote, I have obtained for the dielectric constant of
sulphur the number 3.2, which agrees practically with the results
obtained by more delicate methods. For the highest attainable
precision one should by rights immerse the two plates of the
condenser first wholly in air and then wholly in sulphur, if the
ratio of the capacities is to correspond to the dielectric constant.
In point of fact, however, the error which arises from inserting
simply a plate of sulphur that exactly fills the space between the
two plates, is of no consequence.]
[Footnote 31: As this definition in its simple form is apt to give
rise to misunderstandings, elucidations are usually added to it. It
is clear that we cannot lift a quantity of electricity to _K_,
without changing the distribution on _K_ and the potential on _K_.
Hence, the charges on _K_ must be conceived as fixed, and so small a
quantity raised that no appreciable change is produced by it. Taking
the work thus expended as many times as the small quantity in
question is contained in the unit of quantity, we shall obtain the
potential. The potential of a body _K_ may be briefly and precisely
defined as follows: If we expend the element of work _dW_ to raise
the element of positive quantity _dQ_ from the earth to the
conductor, the potential of a conductor _K_ will be given by _V =
dW/dQ_.]
[Footnote 32: In this article the solidus or slant stroke is used
for the usual fractional sign of division. Where plus or minus signs
occur in the numerator or denominator, brackets or a vinculum is
used.--_Tr._]
[Footnote 33: A sort of agreement exists between the notions of
thermal and electrical capacity, but the difference between the two
ideas also should be carefully borne in mind. The thermal capacity
of a body depends solely upon that body itself. The electrical
capacity of a body _K_ is influenced by all bodies in its vicinity,
inasmuch as the charge of these bodies is able to alter the
potential of _K_. To give, therefore, an unequivocal significance to
the notion of the capacity (_C_) of a body _K_, _C_ is defined as
the relation _Q_/_V_ for the body _K_ in a certain given position of
all neighboring bodies, and during connexion of all neighboring
conductors with the earth. In practice the situation is much
simpler. The capacity, for example, of a jar, the inner coating of
which is almost enveloped by its outer coating, communicating with
the ground, is not sensibly affected by charged or uncharged
adjacent conductors.]
[Footnote 34: These formulæ easily follow from Newton's theorem that
a homogeneous spherical shell, whose elements obey the law of the
inverse squares, exerts no force whatever on points within it but
acts on points without as if the whole mass were concentrated at its
centre. The formulæ next adduced also flow from this proposition.]
[Footnote 35: The energy of a sphere of radius _r_ charged with the
quantity _q_ is 1/2(_q_²/_r_). If the radius increase by the space
_dr_ a loss of energy occurs, and the work done is
1/2(_q_²/_r_²)_dr_. Letting _p_ denote the uniform electrical
pressure on unit of surface of the sphere, the work done is also
4_r_²[pi]_pdr_. Hence _p = (1/8r²[pi])(q²/r²)_. Subjected to the
same superficial pressure on all sides, say in a fluid, our half
sphere would be an equilibrium. Hence we must make the pressure _p_
act on the surface of the great circle to obtain the effect on the
balance, which is _r²[pi]p = 1/8(q²/r²) = 1/8V²_.]
[Footnote 36: The arrangement described is for several reasons not
fitted for the actual measurement of potential. Thomson's absolute
electrometer is based upon an ingenious modification of the
electrical balance of Harris and Volta. Of two large plane parallel
plates, one communicates with the earth, while the other is brought
to the potential to be measured. A small movable superficial portion
_f_ of this last hangs from the balance for the determination of the
attraction _P_. The distance of the plates from each other being _D_
we get _V = D[sqrt](8[pi]P/f)_.]
[Footnote 37: This moment of torsion needs a supplementary
correction, on account of the vertical electric attraction of the
excited disks. This is done by changing the weight of the disk by
means of additional weights and by making a second reading of the
angles of deflexion.]
[Footnote 38: The jar in our experiment acts like an accumulator,
being charged by a dynamo machine. The relation which obtains
between the expended and the available work may be gathered from the
following simple exposition. A Holtz machine _H_ (Fig. 40) is
charging a unit jar _L_, which after _n_ discharges of quantity _q_
and potential _v_, charges the jar _F_ with the quantity _Q_ at the
potential _V_. The energy of the unit-jar discharges is lost and
that of the jar _F_ alone is left. Hence the ratio of the available
work to the total work expended is
_½QV/[½QV + (n/2)qv]_ and as _Q = nq_, also _V/(V + v)_.
If, now, we interpose no unit jar, still the parts of the machine
and the wires of conduction are themselves virtually such unit jars
and the formula still subsists _V/(V + [sum]v)_, in which [sum]_v_
represents the sum of all the successively introduced differences of
potential in the circuit of connexion.]
ON THE PRINCIPLE OF THE CONSERVATION OF ENERGY.[39]
In a popular lecture, distinguished for its charming simplicity and
clearness, which Joule delivered in the year 1847,[40] that famous
physicist declares that the living force which a heavy body has acquired
by its descent through a certain height and which it carries with it in
the form of the velocity with which it is impressed, is the _equivalent_
of the attraction of gravity through the space fallen through, and that
it would be "absurd" to assume that this living force could be destroyed
without some restitution of that equivalent. He then adds: "You will
therefore be surprised to hear that until very _recently_ the universal
opinion has been that living force could be absolutely and irrevocably
destroyed at any one's option." Let us add that to-day, after
forty-seven years, the _law of the conservation of energy_, wherever
civilisation exists, is accepted as a fully established truth and
receives the widest applications in all domains of natural science.