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electric potential by a course of reasoning parallel to the foregoing.
Suppose A (Fig. 32) to be a positively-charged insulated sphere
and B a small sphere with a unit positive charge. B will be



54 . ELEMENTS OF ELECTRICITY.

repelled by A, the force varying inversely as the square of the
distance. At an infinite distance the force would be zero; at a
great distance it would be very small but as the distance be-



G-



B

-o



Fig. 32.



comes small it would increase rapidly. Should we begin with B
at an infinite distance and push it up towards A the work done at
first would be very, very small but would increase as we ap-
proached A and at any point as P the potential would be exactly
measured by the work expended in bringing B up to that point.
We therefore say that the electric potential at any point is measured
by the amount of work that must be spent in bringing up to that
point from an infinite distance a unit of positive electricity. Since
we use the C. G. S. system, electric potential as thus explained is
measured in ergs.

Had the unit charge upon B been negative, its potential at P
would have been negative and measured by the work expended in
pushing it back to an infinite distance.

From the above it follows that the difference of potential between
any two points is measured by the work expended in moving a unit
of positive electricity from one point to the other. Hence also, a unit
difference of potential exists between two surfaces when it requires
the expenditure of one erg to move a unit positive charge from one
to the other.

Parallel to the case of mechanical potential, a surface every
point of which is at the same potential is an equipotential surface.
Such a surface is that of any conductor in which no electricity is
in movement.

73. Zero Potential. Electricity not being matter, we recognize
it and measure it and its dynamical properties only by its effects.
If all bodies about us were at the same potential there could be no
movement of electricity among them and hence, with the exception
of mutual repulsion, there would be none of the manifestations
which we use in measurements. Repulsion of like charges depends
solely upon the quantity of the charges, their distance apart and
the medium in which they are situated and would be the same no
matter how high or how low their common potential, therefore,



STATIC ELECTRICITY. 55

there is no means of determining absolute potential but only rela-
tive potential, or, as it is usually expressed, "difference of potential."
Fortunately, there is no need of knowing the absolute potential,
just as in utilizing water power it is not necessary to know the
height above the sea but it is essential to know the difference of
level. A point at an infinite distance from all charged bodies
would be at zero potential but for convenience the potential of the
earth is taken as an arbitrary zero. This no more means that the
absolute potential of the earth is zero than that taking the melting
point of ice as zero implies that a lower temperature does not
exist or the taking of the sea level as zero means that we could not
go to greater depths.

74. Potential at a Point due to a Charge. If in Fig. 33 the
charge at P be unity, that at A be Q and the distance between



o



>



Fig. 33.

A and P be x centimeters, the force at P will be Q/x 2 dynes,
the work performed by the unit charge in moving from P to P',
a distance dx, will be

I'd* ergs

and the total work performed in moving from P to an infinite dis-
tance will be

'**-!

x = x

Hence the work expended in the opposite direction in moving
the unit charge from infinity up to P will also be Q/x ergs. But
from Par. 72, this measures the potential at the point P. There-
fore, the potential at any point due to a charge is equal to the
charge divided by the distance between the charge and the
point.

An important corollary of the foregoing is that the potential
at any point due to more than one charged body is equal to
the sum of the potentials at that point due to the bodies taken
separately.



56 ELEMENTS OF ELECTRICITY.

75. Expression for Electric Force. An expression for the elec-
tric force acting upon charged bodies may be deduced as follows:
Work is equal to force X path, hence

. work

force = TT-

path

The work performed in pushing a unit of positive electricity
from one point to another is equal to the product of the electric
force by the distance between the points. But from what we
have seen above (Par. 72) this work measures the difference of
potential between the two points, therefore the

, . - difference of potential

~ distance between the points

This is correct only on the assumption that the force has been
constant throughout the path but it is the exception when such
is the case. However, the nearer we take the two points together
the nearer we get to the true value of the force, hence, designating
the difference of potential between the two points by V and the
distance between them by x we have at the limit

, . , dV

electric force = -j

ax

or the electric force at any point is equal to the rate of change at that
point of potential per unit of length.

76. Electro-Motive Force. In the example of mechanical
potential in Par. 71 above, if the cord be only partly paid out the
weight will fall a corresponding distance, the tendency always
being for the body to move from a point of high potential to one
of lower. In the case of electricity there is a like tendency, and the
insulation of a charged body may be regarded as analogous to the
cord since it restrains the charge from flowing from the body to
another of lower potential. If the charged body be connected to a
body of lower potential it is analogous to paying out the cord, and
if it be connected through a conductor to the earth the effect is
analogous to cutting the cord.

In the illustration of electric potential, if the little sphere
pushed up from an infinite distance and containing the unit posi-
tive charge be released it will be pushed back, the charge and the
sphere both moving. If instead of releasing the sphere, it be con-
nected, say through a conducting wire, with the earth, the charge
alone will be pushed back along the wire to a point of zero poten-



STATIC ELECTRICITY. 57

tial. In this case no movement of matter is involved but only of
the charge. If new charges be supplied to the little sphere as fast
as the previous charges flow away, it will be kept at a constant
potential and the successive charges following along the wire will
constitute a continuous stream. This is what is known as current
electricity and is discussed later.

Mechanical force is defined as that which moves or tends to
move or tends to produce a change of motion in matter. In the
case of the movement of electricity however no matter is involved.
The first force might therefore be named "matter-motive force/'
the second in centra-distinction, is named "electro-motive force,"
and can be defined as that force which moves or tends to move
electricity. It is represented in symbols as E. M. F.

77. Practical Unit of Electro -Motive Force. Reverting to
our comparison of potential to water level, the flow of water is
produced by a force and this force is the pressure due to the "head"
or difference of level between the surface of the water and the out-
let. So the flow of electricity is produced by the electro-motive
force which in turn is caused by the difference of potential between
the two ends of the path. The difference of level in the case of
water is measured in feet, the corresponding pressure is measured
in pounds per square inch, and for any given difference in level
the pressure in pounds per square inch may be obtained by multi-
plying this difference expressed in feet by the factor .434. In the
case of water the cause and effect are so closely connected that we
often hear such expressions as "a pressure of 30 feet."

The practical unit of electric pressure or of electro-motive force
is called the volt and will be defined later. Difference in potential
expressed in ergs is, for reasons given later, converted into the
corresponding electro-motive force in volts by multiplying by 300.
Similar to the case of water it has become usual to confound cause
and effect and it is customary to speak of a difference of potential
of so many volts. Some writers even go to the extent of stating
that difference of potential and electro-motive force are two names
for one and the same thing. In view of this, insistence upon the
distinction becomes academic and of no practical importance and
hereafter will not be dwelt upon.

78. Summary. The gist of the preceding discussion upon
potential is that whenever a charge of electricity is produced, it



58 ELEMENTS OF ELECTRICITY.

may be regarded as brought up from infinity or from a point of
zero potential and whenever a difference of potential is developed,
the charge must either have been pushed against a repulsion or
pulled against an attraction. In either case, just like a spiral
spring which has been compressed or extended, it has a tendency
to fly back and can be retained in its position only by a continua-
tion of the push or pull or by the interposition of an insulator.
The more the mechanical or chemical energy expended in bringing
up the charge, the greater its potential energy or the greater its
tendency to fly back when released. The potential of the charge
is measured by the work in ergs spent in bringing up a portion of
it equal to one positive unit. The force with which the unit
charge when released would be pushed back, or the electro-motive
force, is measured in units called volts whose number is obtained
by multiplying the ergs by 300.



STATIC ELECTRICITY.



59



CHAPTER 10.

ELECTROSTATIC CAPACITY.

79. Electrostatic Capacity. At several points in the preceding
pages reference has been made to electric capacity. The word
"capacity" in its application to electricity is used in a sense quite
different from its ordinary acceptance and necessarily so, as the
following will show. The capacity of a vessel is the volume which
it will contain and is fixed once for all. If by the capacity of a
conductor we meant the amount of electricity which could be im-
parted to it, the term would be indefinite for as conditions vary
the same conductor could contain very different amounts. Re-
sorting to analogy, conductors can be compared to vertical



Fig. 34.

cylindrical vessels differing in cross-section and of indeterminate
height (Fig. 34). The amount of water which could be placed in
any of these vessels would depend upon the cross-section, upon
the height to which the inflowing liquid could be raised by the
supply pump and also upon the strength of the material, that is,
the height to which the cylinder could be filled before the pressure
caused the bottom or sides to yield. Hence, keeping the cross-
section constant, the capacity might be varied by using a more
powerful pump or a stronger material for the vessel. The only
basis of comparison in terms of contents would therefore be the
amount of water that would cause the pressure per square inch
on the bottom to increase a definite amount, or, since this pressure
varies directly as the head, the amount of water that would raise
the level in the vessel a certain distance, say one foot. If one



60 ELEMENTS OF ELECTRICITY.

gallon raises the level one foot in a certain cylinder and it requires
two gallons to do the same in another, the second cylinder may be
said to have twice the capacity of the first. The same amount of
water will raise the level more quickly in a small cylinder than in
a large one.

If various insulated conductors be connected to a charged body
of higher potential, electricity will flow from the source into them
until a common potential is reached. The small conductors will
receive least for less is required to raise their potential to the com-
mon level. Upon reaching the common potential the flow will
cease but should the potential of the source be increased the flow
will again begin and continue until a new common potential is
reached. This can be continued, the potential of the conductors
steadily rising, but finally the strain on the medium surrounding
the conductors becomes so great as to overcome its dielectric
strength (see Par. 93), there is a breakdown and a discharge
occurs. Hence the total quantity of electricity which can be
transferred to a conductor, besides varying with the size of the
conductor also depends upon the difference of potential between
the conductor and the source of electricity and upon the dielectric
strength of the surrounding medium and is therefore indefinite.
On the other hand, the capacity of a conductor is measured by the
quantity of electricity which must be imparted to it to raise its
potential one unit, and is perfectly fixed and definite.

If a charge Q imparted to a body raises its potential V units,
then a charge Q/V would raise its potential one unit. But, by
the preceding definition, this is the measure of the capacity of the
body, and representing the capacity by K, we have the relation
between these three quantities given by the expression

K $
= V

80. Capacity of a Sphere. The capacity of most bodies must
be determined by actual measurement but for a few of simple
geometrical form it may be calculated. The capacity of a sphere
may be determined as follows. In Par. 74 it was shown that the
potential at a point due to a charge Q at a distance x from the
point is Q/x. If the charge Q be upon the surface of a sphere it
acts as if concentrated at the center of the sphere (Par. 65), and
hence the distance between the charge and the point must be



STATIC ELECTRICITY. 61

measured from the center of the sphere. Therefore, the potential
of a point infinitely near the surface of the sphere (that is, the
potential of the sphere itself) is V = Q/R. In other words, the
potential of a sphere varies directly as the charge and inversely as
the radius. In the above expression if V = 1, Q must be equal to
R, that is, to maintain unit potential as R varies, Q must vary in
the same ratio and preserve numerical equality with R. We also
see that the capacity of a sphere varies directly as its radius. This
may be shown directly by substituting in the expression for
capacity, K= Q/V, the above value V= Q/R, whence we obtain

K = R

or the number of units of electricity required to raise the potential
of a sphere by unity is equal to the number of centimeters in the
radius of the sphere. A unit charge would therefore raise by unity
the potential of a sphere of one centimeter radius and such a
sphere is said to have unit capacity.

Certain interesting consequences follow from the foregoing, two
of which we shall now notice.

81. Case of Two United Spheres. If two unequal spheres be
placed in contact or be joined by a conductor and a charge be im-
parted to either they will come to a common potential, or will
share the charge in proportion to their capaci-
ties, which, from the preceding paragraph, is

also in proportion to their radii. Suppose the
radius of A (Fig. 35) to be twice that of B, the
charge upon A will be twice the charge upon B. Flg * 3o>

The surfaces of these spheres being to each other as the squares of
their radii, the surface of A is four times that of B. The surface
density of the charge on A is therefore as 2/4, that of the charge
on B is as 1/1, or the surface density on B, although B has the
smaller charge, is twice that on A. If B is very small as compared
to A, its surface density will become very great and we have seen
(Par. 41) that if the surface density exceeds 20 units per square
centimeter a discharge will take place. This is the explanation
of the action of points already described (Par. 42).

82. Case of Two Coalescing Spheres. Suppose two equal
charged spheres, A and B (Fig. 36), should coalesce produc-
ing a resultant sphere C. If the radius of A be r and that




62 ELEMENTS OF ELECTRICITY.

of C be R, since the volume of a sphere = firr 3 we have




Hence # 3 = 2r 3 or R = ^/2.r = 1.26r.

Hence, since the capacity of a sphere varies
directly with its radius, it will require 1.26 times
Fig. 36. as large a charge to raise the potential of the
sphere C one unit as is required to raise that of A or of B one unit.
But by the coalescing of the spheres C receives twice as great a
charge as A or B, or .74 times more than necessary to bring it to
the same potential, and hence its potential is greater than that
of A or B.

It is known that evaporation is accompanied by the production
of electricity, the vapor being charged. As the vapor begins to
condense, the molecules unite into globules, these microscopic
globules into larger ones and these into still larger ones until drops
of rain result. By this coalescing the potential is enormously in-
creased until a final point is reached when a disruptive discharge,
a flash of lightning, takes place. This is an explanation which has
been advanced to account for thunder storms.

83. Condensers. In the discussion of capacity in Pars. 79 and
80 above, the conductors were supposed to be remote from all
other bodies. Should the conductor to which the charge is given
be near to a second, this last being connected to the earth, a very
different state of affairs will result. A charge imparted to the first
will repel from the second into the earth a similar and almost equal
charge and induce and attract into it an opposite and almost equal
charge. In Par. 74 it was stated that the potential at a given
point due to more than one charged body is equal to the sum of
the potentials at that point due to the bodies taken separately.
The potential of the first body is therefore the sum of the potentials
due to its own charge and to the induced charge and these being of
opposite signs the resultant potential is much less. The potential
being less, a greater charge is required to raise the potential of the
first body a certain amount than was required when this body was
remote from all others, in other words, its capacity is increased.
We see then that the capacity of a conductor is increased by the
proximity of another which is earth connected, and since a greater
charge can now be given to it before a given change of potential is
produced, such an arrangement is called a condenser. The earliest



STATIC ELECTRICITY. 63

form of a condenser was the Leyden Jar which we shall now con-
sider.

84. Invention of the Leyden Jar. The invention of the Leyden
jar is in dispute, the merit having been claimed for three or more
persons. Priestly, noted as the discoverer of oxygen, has left a
contemporaneous account of the event which is in substance as
follows: Dr. Muschenbroek of Leyden in experimenting with
static electricity was much troubled by the rapidity with which
his conductors lost their charge and ascribed this loss to some
"effluvium" in the surrounding air. He therefore thought to pro-
tect his charged body by surrounding it by a non-conducting
vessel which would shield it from the atmosphere. To test this,
he poured some water into a glass jar and holding the jar in his
left hand he led a charge into the water by a wire attached to the
prime conductor of the crude machine he was using. After giving
the handle of the machine a few turns he attempted to disengage
with his right hand the wire from the prime conductor but as he
touched it there was a flash and he was subjected to a strong con-
vulsive shock. In a letter describing this experience he states that
he felt himself struck in his arms, shoulders and breast so that he
lost his breath and was two days before he recovered from the
effects of the blow and the terror. He added that he would not
take a second shock for the whole Kingdom of France.

This experiment was quickly repeated by other investigators.
It was soon found that no appreciable charge could be given to
the jar unless it were held in the hand and that the amount of the
charge varied with the amount of the surface touched by the hand.
This led to the substitution of a metallic outer covering. It was
next discovered that the charge did not increase in proportion to
the amount of water in the jar but rather in proportion to the area
of the surface wetted and this led to the substitution of a lining of
tin-foil. Finally, it was found that, other conditions being the
same, the thinner the jar the greater the charge that it could be
given.

85. The Leyden Jar. The usual form consists of a wide-
mouthed glass jar (Fig. 37) coated inside and out for about two-
thirds of its height with tin-foil. It is closed with a stopper of
insulating material through which passes a brass rod terminating
above in a knob and below in a small chain which dangles long
enough to touch the tin-foil lining.



64



ELEMENTS OF ELECTRICITY.



To charge the jar, the outer coating must be connected to earth
either by being placed upon a wire or chain, one end of which is
grounded, or by being held in the hand and afforded a path through
the body. The knob is then held to the prime conductor of a
machine in operation and in a very short while the jar is charged.




Fig. 37.

The inner lining receives the same kind of charge as is generated
by the machine; the outer coating receives a charge of the opposite
kind. It can not be charged indefinitely. As we continue to turn
the handle of the machine, a point will be reached when the tension
between the two opposite charges becomes so great that either the
glass of the jar will be pierced or else a discharge will occur by the
charge creeping up the surface of the glass to the mouth of the jar
and thence down to the outer coating.

A jar once charged will remain so for some time. The inner and
outer charges are mutually bound and can not be removed by
touching the inner or the outer coatings separately, but if they be
touched simultaneously by any body which will afford a path
between the two charges, the jar is instantly discharged. Since
the effect of the discharge through the body is disagreeable and
may be dangerous, use is made of a discharger, a knobbed con-
ductor, hinged at the middle like a pair of tongs and furnished
with glass handles. It is held by the handles while, as shown in
Fig. 37, one knob is touched to the knob of the jar, the other to
the outer coating.



STATIC ELECTRICITY.



65



86. Explanation of Leyden Jar. Reflection and experiment
will show that the jar form of this apparatus is unimportant and
that the essential parts are two sheets of conducting material
separated by a thin non-conducting sheet. A window-pane set
on edge with a sheet of tin-foil pasted in the center on each side is
as efficient as a jar of equal area of glass and foil. Such an ar-
rangement was called by Franklin a "fulminating pane." If more
rigid metal sheets be substituted for the tin-foil and if they be
mounted upon an insulating support, the glass may be replaced
by a thin layer of air and the apparatus is then called an air
condenser.

The arrangement shown in Fig. 38 enables us to examine the
action of a condenser under various conditions. A and B are



- +

- +

- 4-

- +




B




Fig. 38.

vertical metallic plates mounted upon insulating stands by which
the distance between them may be varied. A is connected by a
chain or wire to the earth and B is connected to the prime con-
ductor C of a machine which we shall suppose is positively charged.
At first let A be remote from B. C being at a higher potential
than B, a charge will flow into B and B and C will reach a common
potential. If now A be moved up near B, the charge on B will
induce a negative charge on A and repel a positive charge into the
earth. The potential of B is the sum of that due to its own charge
and that due to the charge on A. This last being negative, the
potential of B is lowered and more charge will flow into B from C
until B and C are again brought to a common potential. Each
additional quantity that flows into B from C will induce a corre-



66



ELEMENTS OF ELECTRICITY.



spending quantity of negative electricity in A, the joint effect of
the two being to reduce the potential to which B, if remote from
A, would be raised and thus a much greater charge can be given
to B than would otherwise have been possible.

If the chains be now disconnected from A and from B and if A
and B be drawn apart, the pith balls attached to the supports will
be repelled more strongly, as if A and B had received greater
charges. This may be explained either by the fact that as the



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