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distance between A and B increases, the two charges are not so
strongly bound mutually and tend to spread, or that the effect of
the negative charge on A upon the potential of B becomes less
and the potential of B increases.

If A and B be pushed closer together the pith balls will again
drop down. The conclusion is that other things being equal the
capacity of a condenser varies inversely as the distance apart of
the conducting surfaces.

87. Location of Charge of a Condenser. In the course of
some experiments with a Ley den jar which contained water
instead of an inner coating of tin-foil, Franklin, having charged
the jar, poured out the water into another vessel and expected





Fig. 39.

thus to obtain the liquid highly charged. His tests however giving
no marked results, he thought to repeat the experiment and poured
fresh water into the jar when, to his surprise, he found the jar to be
almost as highly charged as in the beginning. He concluded that
the charge, since it remained behind, could not have been dis-
tributed in the liquid and must have been spread over the surface
of the glass. To demonstrate this he constructed a jar with
movable coatings (Fig. 39). After this jar has been charged, the



STATIC ELECTRICITY. 67

inner coating C may be lifted out by inserting a glass rod in the
hook and then the glass B may be taken out of the outer coating
A. C and A may now be shown to have no appreciable charge
either separately or together, but if the jar be reassembled it will
give almost as large a spark as it would have given just after
charging. The coatings therefore serve merely as paths by which
the charge is conducted about over the surface of the glass and
the surface of this glass is the seat of the charge. This may also
be shown with the condenser represented in Fig. 38 if a sheet of
some non-conducting material be inserted between the plates but
not if the medium between be air or gas.

We saw in Par. 60 that every electric field consists of non-
conductors and is bounded by conductors, and elsewhere (Par. 57)
it was stated that the medium within the limits of a field is not
passive or inert but takes part in the transmission of the electrical
effects and is subjected to certain mechanical strains. Of this
there are many proofs. For example, if a beam of polarized light
be passed through a piece of glass not under mechanical strain no
effect is produced but should the glass be strained, then the beam
on emergence will, if allowed to fall upon a white surface, produce
certain color effects. Such a beam passed through a piece of glass
placed in an electrical field will reveal the presence of strains*
Again, if shortly after a Leyden jar has been discharged, the dis-
charger be again applied, an additional spark may be obtained and
sometimes even a third. The production of this residual charge
may be hastened by tapping the jar. This is sometimes explained
by saying that a portion of the charge has soaked into the glass
but it is perhaps better to say that the material has been strained
so near its elastic limit that, like an overloaded spring when the
load is removed, it does not return instantly to its primary posi-
tion. There is no residual charge in an air condenser. Also when
a Leyden jar is charged and discharged rapidly a number of times
the glass grows warm just as does a spring when rapidly com-
pressed and extended. Finally, the discharge of a jar, while
apparently a simple phenomenon, is in reality complex and by the
application of instantaneous photography to the image of the
spark in a rapidly rotating mirror (Par. 688) it can be shown to be
in the nature of an oscillation, sparks of decreasing intensity passing
back and forth just as a released spring vibrates with decreasing
amplitude back and forth across its neutral position. This, with



68



ELEMENTS OF ELECTRICITY.



the proof that the charge of a Leyden jar lies on the surface of the
glass, would seem to justify us in saying that the real seat of the
charge is along the bounding surfaces of the non-conductor en-
closed within the limits of the field and that the energy of the charge
is due to the stresses set up in this medium; the conductor there-
fore plays a minor part.

88. Capacity of a Spherical Condenser. The capacity of a
condenser is measured by the quantity of electricity that must be
imparted to one plate (the other plate being
connected to the earth or "grounded" and
hence at zero potential) to raise its potential
unity. For many condensers the capacity
must be measured, for others it may be calcu-
lated. For example, let it be required to deter-
mine the capacity of a spherical condenser.
Let A (Fig. 40) be a metallic- sphere surrounded
by the metallic sphere B and separated from B
by a thickness of air t. If R be the radius of A,
that of B is R' = R+t. Let B be connected to
earth. The potential of B is therefore zero. If
a charge Q be imparted to A, a charge Q will




B



Fig. 40.



be induced upon the inner surface of B. The potential of A due
to its own charge is Q/R (Par. 80) ; the potential of A due to the
charge on B is

Q

"irn

The resultant potential of A is (Par. 74)



=
R

And since (Par. 79)



R + t R(R +
' = Q/V

QRR' RR'



RR'



Qt ~F~

or the capacity varies directly as the area of the conducting sur-
faces and, as was shown in Par. 86, inversely as the thickness of
the layer of air separating these surfaces.

A conducting sphere of one centimeter radius has unit capacity,
that is, one electrostatic unit raises its potential unity. If it be
surrounded by a concentric conducting sphere connected to the



STATIC ELECTRICITY.



69



earth and leaving an air space of one millimeter (1/25 of an inch)
between the two, its capacity becomes 11, that is, eleven units of
electricity must now be imparted to it to raise its potential unity.
The appropriateness of the term "condenser" is hence apparent.

89. Capacity of a Plate Condenser. Let AB (Fig. 41) be a
plate of glass of thickness t upon the opposite sides of which are
pasted equal circular discs, E and F, of tin-foil, one of which, as
F, is connected to the earth. Let the radius of these discs be R.
If now a positive charge be imparted to E it will induce and bind
an equal opposite charge upon F and repel into the A

earth an equal positive charge. If the surface
density of E be 5, that of F (as shown in Par. 62)
will be 8. A unit positive charge placed between
E and F will be repelled from E with a force of
j . 2 7r5 dynes (Pars. 66 and 55) and attracted to F
with an equal force, the total force being ^Awd.
The work done in moving this unit charge from
F to E, a distance t, is ^ . Airdt. According to what
was shown in Par. 72, this measures the difference of

potential between F and E, hence



W//7/////,
Fig. 41.

F being connected to earth, its potential V" is zero, hence the
potential of E is



But (Par. 79) the capacity K = Q/V, hence
K-k Q

J\. /v



The face of the disc E is nR 2 , the charge upon it is wR 2 d.
stituting this for Q in the above expression we get

R 2



Sub-



that is, the capacity of a condenser is different with different
dielectrics and, as has already been shown, varies directly with
the area of the conducting surfaces and inversely as their distance
apart.



70 ELEMENTS OF ELECTRICITY.

90. Dielectric Capacity. The fact that the capacity of a con-
denser varies with the medium between the plates may be shown
by a simple experiment. If the air condenser (Fig. 38) be charged
to a certain potential and then, without altering the charge or the
distance apart of the plates, a slab of paraffine be inserted between
them, the potential will immediately drop. If mica be used the
drop will be even greater. Since the potential is reduced the
condenser will require a greater charge to bring it to its original
potential, that is, by substituting for air these other media its
capacity is increased.

Since without changing the geometrical arrangement of a con-
denser but by substituting one dielectric for another we alter its
capacity, and since we have seen that the charge resides on this
dielectric and not on the conducting plates, we naturally associate
the idea of capacity with the dielectric itself and therefore speak
of dielectric capacity. We use air as the standard of comparison
and when we say that the dielectric capacity of mica is six we mean
that a condenser in which mica is the dielectric has six times the
capacity of one with air as the dielectric but otherwise precisely
similar. The dielectric capacity of a substance is therefore meas-
ured by the ratio of the capacity of a condenser in which the sub-
stance is employed as the dielectric to that of the same condenser
in which air has been substituted for the substance. This ratio is
represented by k in the last expression in the preceding paragraph.
This factor k is sometimes called the dielectric coefficient since it
is the coefficient by which the capacity of an air condenser must
be multiplied to obtain the capacity of the same condenser in
which the corresponding dielectric has been substituted for air.
Reference to Par. 55 will show that this is the reciprocal of what
was there called the "dielectric coefficient of repulsion," whence
it follows that in a medium whose dielectric coefficient is k, the
force exerted between charged bodies is |th as much as the force
exerted between these bodies in air.

91. Determination of Dielectric Capacity. In Faraday's deter-
mination of dielectric capacity he used spherical condensers
similar to the one represented in Fig. 40 but with the opening in
the outer sphere closed by an insulating stopper through which
the stem of the inner sphere passed. The outer sphere was sup-
plied with a stop cock by which the air between the spheres could
be drawn off and liquids or gases introduced, also the outer sphere



STATIC ELECTRICITY. 71

could be separated into halves when it became necessary in in-
serting or removing other materials. Two of these condensers of
equal size were taken. In one air was retained as the dielectric;
into the other was introduced the substance whose dielectric
capacity was to be determined. Suppose the space in the second
one to be filled with oil. The air condenser was now charged to a
certain potential which was carefully measured by the torsion
balance. The outer coatings of the two condensers were next
placed in contact, either directly or through a third body, and were
thus brought to a common potential. Finally, the inner coatings
were brought into contact. The air condenser, being at a higher
potential, gave up a portion of its charge to the oil condenser until
equality of potential was reached. If the capacities of the two
condensers were equal, the charge would be divided equally
between them and the resultant potential would be one-half that
of the original potential. If the capacity of the oil condenser were
greater than that of the air condenser, the oil condenser would take
more than half the charge and the resultant potential would be
less than half the original potential. If the capacity of the oil
condenser were less than that of the air condenser, the resultant
potential would be greater than one-half of the original. In either
case, the resultant potential having been measured, the dielectric
capacity is calculated as follows. Let Q be the original charge of
the air condenser, V its original potential, V the potential of both
condensers after division of the charge, K their capacity when
used as air condensers and k the dielectric capacity of the oil.
From Par. 79 we have

K=Q, whence Q = VK

The charge in the air condenser after contact is

Q' = V'K
The charge in the oil condenser is

Q" = k(V'K}

The sum of the separate charges must be equal to the original

charge, hence

VK = V'K+k(V'K)
whence



72 ELEMENTS OF ELECTRICITY.

92. Dielectric Capacity of Various Substances. The dielectric
capacity of many insulating materials has been measured and some
of the accepted determinations are given in the table below.
There is wide variation in the results obtained by different inves-
tigators and this is due to the fact that the capacity of a condenser
is greater if the charge be slowly imparted than if it be suddenly
applied and as suddenly withdrawn. In the first case the medium
yields and accommodates itself to the stress put upon it. By the
so-called, instantaneous method of determining dielectric capacity,
the condenser is charged and discharged several hundred times per
second and the determinations are less than those obtained by the
slow methods. The dielectric capacity of a vacuum is about .94;
that of the various gases differs from that of air in the third or
fourth decimal place only.

Table of Dielectric Capacities.

Paper 1.5 Mica 4.0 to 8

Beeswax 1.8 Porcelain 4.4

Paraffine 2.0 to 2.3 Glycerine 16.5

Petroleum 2.0 to 2.4 Ethyl Alcohol 22.0

Ebonite 2.0 to 3.2 Methyl Alcohol 32.5

Rubber 2.2 to 2.5 Formic Acid 57.0

Shellac 2.7 to 3.6 Water 80.0

Glass 3.0 to 10. Hydrocyanic Acid 95.0

93. Dielectric Strength. The quantity of electricity which
must be imparted to a condenser to raise its potential unity de-
pends upon the capacity of the condenser. If the plates of a con-
denser be connected to two objects between which unit difference
of potential is maintained, the condenser will receive the charge
which is the measure of its capacity. If the difference of potential
between the two objects be doubled, the condenser will receive a
charge twice as great and so on. In other words, as has been
stated in Par. 79, the quantity of electricity which can be trans-
ferred to a condenser depends upon its capacity and also upon the
difference of potential maintained between the two plates. By
increasing this difference of potential, a greater and greater charge
can be given to the condenser but this can not go on indefinitely
for as the potential increases, the strain upon the dielectric in-
creases until finally it is pierced by a spark and the condenser is
discharged. The resistance which a medium offers to piercing by



STATIC ELECTRICITY. 73

the spark is called its dielectric strength and is measured by the
maximum difference in potential in volts which a given thickness
(one centimeter) of the medium will stand before piercing occurs.
It is difficult of accurate determination since it is affected by
temperature and pressure, by the size and shape of the bodies
between which the sparks pass and by the manner in which the
electric pressure is applied, that is whether constantly in one
direction or alternately in opposite directions.

The dielectric strength of air has been investigated by a number
of observers. A minimum difference of potential of 300 volts is
required to produce a spark at all, even across a space of less than
.01 of an inch. Sparks pass more readily between points than
between bodies of other shapes. The strength increases with the
density of the air, whether produced by falling temperature or by
increasing barometric pressure. If air be under a pressure of 500
pounds per square inch, it can be hardly pierced at all. On the
other hand, a vacuum offers an equal resistance. To throw a
spark between two points an inch apart requires about 20,000
volts and to produce a 15-inch spark requires 145,000 volts. To
pierce one centimeter of paraffine requires 130,000 volts, one
centimeter of ebonite about 200,000 and one centimeter of mica
about 350,000.

94. Commercial Condensers. Condensers are used, as will be
explained later, in certain electrical measurements, in telegraphy




HBB5SB93B

Fig. 42.

and in the production of high potential electricity by means of
induction coils. They are usually constructed of alternate layers
of tin-foil and mica or of tin-foil and waxed paper pressed tightly
together and thus including a large surface in very small bulk.
The alternate sheets of foil are connected as shown diagrammat-
ically in Fig. 42 (in which the shaded spaces represent the paper



74 ELEMENTS OF ELECTRICITY.

and the heavy lines the foil) and the whole is contained in a rect-
angular or cylindrical case provided with the proper terminals.
The one represented in the figure is of invariable capacity but by
connecting the sheets of foil together in groups attached to separate
terminals it is possible to use at will different fractions of the entire
condenser.

95. Practical Unit of Capacity. The practical unit of capacity,
the farad , is denned as the capacity of that body whose potential
is raised one volt by one coulomb of electricity. The coulomb will
be defined later (Par. 228) but we have already seen (Par. 56) that
it is three billion (3X10 9 ) times as large as the electrostatic unit
of quantity. We have also seen (Par. 77) that the electrostatic
unit of potential is equal to 300 volts. Since one electrostatic unit
of quantity raises the potential of a sphere of one centimeter radius
300 volts, one coulomb would raise the potential of such a sphere
to 3X10 9 X300, or nine hundred billion (9X10 11 ) volts, and a
sphere of 9X10 11 centimeters radius would be raised one volt by
one coulomb and would therefore have a capacity of one farad.
The radius of such a sphere is about 5,600,000 miles or about
1,400 times as large as that of the earth. A farad is therefore so
great that in practice one-millionth of a farad (or a micro-farad)
is used. An isolated sphere of 9xl0 5 centimeters radius (about
5.6 miles) would have a capacity of one micro-farad. A mica-tin-
foil condenser containing about 25 square feet of tin-foil, has also
a capacity of about one micro-farad.

Since a sphere of 9X10 5 centimeters radius has a capacity of
one micro-farad, a sphere of one centimeter radius (or a sphere
of unit electrostatic capacity) has a capacity of

micr - farad



96. Work Expended in Charging a Condenser. In Par. 72 it
was shown that the potential at a point was measured by the work
done in bringing up to that point from an infinite distance, or from
a point of zero potential, a unit charge. If the potential be V, we
mean that the work done in bringing up the unit charge is V ergs.
The work done in bringing up a charge Q would [therefore be QV
ergs, although the potential of the point would still remain V, that
is, the assumption is that the charge brought up does not increase
the potential of the point. The potential in this case is analogous



STATIC ELECTRICITY. 75

to the head of a body of water which body is of such extent that
its level is not^appreciably altered by the pumping up of additional
quantities. However, the case is different if the charge is to be
brought up to a body of limited capacity. Suppose we have a
sphere of unit capacity and at zero potential. At first sight it
might seem that to transfer to this sphere from zero potential a
certain charge would not require the expenditure of any energy.
But suppose the charge to be brought up by successive portions.
The first portion could be brought up without the expenditure of
energy but would raise the potential of the sphere and would repel
the second portion as the latter approached. These two portions
would repel the third still more strongly and so on, the work re-
quired to bring up the successive portions increasing in regular
progression. The potential in this second case is analogous to the
head of water in a narrow vessel, each portion that is added raising
the level and thus increasing the work which must be expended to
bring up the succeeding portion.

To determine the amount of work in bringing up in this manner
by n successive portions a charge Q. The first portion Q/n would
raise the potential of the unit sphere to Q/n. To bring a unit
charge to a point of such potential would, from the definition of
potential, require an expenditure of Q/n ergs, therefore to bring
up a charge of Q/n will require Q/n times as much or Q 2 /n 2 ergs.
The second portion would therefore require an expenditure of
Q 2 /n 2 ergs and the potential would become 2Q/n. Similarly, the
third portion would require 2Q 2 /n 2 ergs and the potential would
become 3Q/w and so on. To bring up n portions would require a
total expenditure of

IS! +*$+<$!+ + -



= {1 + 2 + 3+ . . . +(rc-l)
The sum of this series obtained by applying the formula



in which a is the first term, I the last, and n the number of terms
(in this case =n 1),

13 ^f ergs= ( i -9! ergs



76 ELEMENTS OF ELECTRICITY.

which when n increases indefinitely becomes

Q 2
^ergs

and the corresponding potential

n

which last also follows directly from the fact that the sphere is of
unit? capacity.

Since Q = V, the above expression for the work may be written

iQVergs

that is, the work done in bringing up from zero potential to a body
of limited capacity, likewise at zero potential, a charge Q by which
the potential of the body is raised to V is just one-half the work
done in bringing up the same charge from a point of zero potential
to a point whose potential is V.

97. Energy of a Condenser. If the body to which the charge is
brought is of capacity K instead of unity, the expression J QV,
since Q = V K, may be put in the form J Q 2 / K, that is, if a charge
Q be given to a condenser of capacity K, the work spent is propor-
tional to the square of the charge and inversely proportional to the
capacity of the condenser.

If the condenser be discharged it will give out as much energy
as was expended in charging it and therefore the expression
J Q 2 / K also represents the energy of discharge or the energy of
the condenser.

If for Q we substitute its value V K, the expression becomes

\V^K

that is, the energy of a condenser varies as the square of its
potential and as its capacity. This principle is utilized in the
quadrant electrometer to be described later (Par. 103).



STATIC ELECTRICITY. 77



CHAPTER 11.

ELECTROSTATIC MEASUREMENTS.

98. Electrostatic Measurements. The electrostatic quantities
which we most frequently desire to measure are quantity of charge
and difference of potential. Of these two, the latter is the more
important but if we may measure either one we may determine
the other indirectly. Thus, if an unknown charge raises the
potential of a certain conductor by a given amount, we have only
to find out how much its potential is raised by a unit charge and
can then determine at once the quantity of the unknown charge,
or similarly, can determine the potential to which a known charge
will raise a given conductor.

99. Unit Jars. At first, attempts were made to measure
charges directly by means of what were called "unit jars." These
were small Leyden jars, their outer coatings connected with a
knob which could be made to approach or recede from the knob
communicating with the inner lining. By adjusting the air gap
between these knobs a greater or a lesser charge could be given
to the jar before a discharge took place. They were used to
measure the charge imparted by a machine to a large Leyden jar
or to a battery of these jars. One was inserted between the
machine and the knob of the large jar. Obviously no charge could
pass to the large jar until the unit jar

filled up and discharged and the amount
was determined by counting the num-
ber of sparks. ( A

100. Principle of Electrometers.

Instruments for measuring differences
of electrostatic potential are called elec-
trometers. The principle upon which
they operate will be understood from
the following. Suppose A and B (Fig.

43) to be two bodies between which there exists a difference of
electrostatic potential which we desire to measure. For one reason




78 ELEMENTS OF ELECTRICITY.

or another it is generally impracticable to measure the difference
of potential between the bodies themselves and we therefore have
to transfer the potentials to the parts of our instrument. Let C and
D be two small spheres, D fixed and C attached to a spring which
can be extended or compressed and which has a scale from which the
force producing the extension or the compression can be read. If
A and C be connected by a wire they will at once attain the same
potential and the charge imparted to C will vary directly with the
potential of A. Likewise if B be connected with D, D will attain
the potential of B and acquire a charge proportional to this
potential. C and D will now attract or repel each other with a
ibrce which can be read from the scale and which is proportional
to the product of the charges which in turn are proportional to
the potentials. But C and D are of the same potentials as A and



Online LibraryWirt RobinsonThe elements of electricity → online text (page 7 of 46)