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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

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

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