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force only one-half as great, or, if separated by an equal thickness

of mica will repel each other with a force only one-sixth as great.

The force between two charged bodies in air is not varied by com-

pressing or by rarifying the air and for this reason and on account

of the absence of any absolute measure we use air as our standard.

The ratio of the force exerted between two charged bodies in a

certain dielectric to the force exerted between the same bodies

with the same charges at the same distance apart in air may be

called the dielectric coefficient of repulsion and is the coefficient

by which the force exerted between two charged bodies in air

would be multiplied in order to obtain the force between the same

two bodies under the same conditions in the medium to which the

coefficient pertained. For oil this would therefore be 1/2, for

mica 1/6, etc.

For gases and liquids this coefficient might be determined by

the use of Coulomb's balance as explained above but it is obvious

that this method could not be applied to solids. However, we shall

see later on (Par. 91) how it may be otherwise determined and at

the same time it will be shown why it is written in the form of a

fraction or as I/ k.

In problems involving forces exerted between charged bodies

in other media than air, the appropriate value of 1/fc should be

used and when in discussions in the following pages this coefficient

does not appear it is to be understood that the dielectric is air.

56. Unit Quantity of Electricity. Representing by / the force

of attraction or of repulsion, by q and q' the two charges and by

d their distance apart we may combine the three laws discussed

above and express them mathematically thus

Since, as was explained in Par. 10, electricians have agreed to

follow the C. G. S. system of units, / in this expression must be

measured in dynes and d in centimeters. In the torsion balance

42 ELEMENTS OF ELECTRICITY.

where the two gilded pith balls were of equal size, q and q' are

equal, and since the dielectric is air, l/k = 1, hence the above ex-

pression becomes

If we assume / to be one dyne and d to be one centimeter we

obtain q = l, whence follows at once the definition: An electro-

static unit of electricity is that quantity which when placed at a centi-

meter's distance in air from a similar and equal quantity repels it

with a force of one dyne.

The expression "in air" is essential to this definition as is also

the term "electrostatic" for, as we shall see later (Par. 228), there

is another and different unit of quantity, the coulomb, which is

based upon current relations. The coulomb is three billion

<3xl0 9 ) times as large as the electrostatic unit.

STATIC ELECTRICITY. 43

CHAPTER 8.

ELECTRIC FIELD.

57. Electric Field. We have seen that a charged body attracts

non-electrified bodies and others with opposite charge and repels

those with similar charge, therefore, in the space around an elec-

trified body all bodies experience a force either of attraction or of

repulsion and this space is called the field of the charge. As we

recede from the charged body the force falls off rapidly and to fix

its limits with more definiteness we define the electric field as that

space surrounding a charged body in which the force of attraction or

of repulsion is perceptible. If there be more than one charged body

involved each produces a certain effect and they have a resultant

field. 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. Between two

oppositely charged bodies there is a tension as if they were being

pulled together by invisible rubber bands and at the same time a

stress at right angles to the bands pushing the bands apart.

58. Intensity of Field. It may aid the beginner in his concep-

tion if he consider a field as analogous to a current of water. In

the electric field there is no matter in actual movement but in a

sense there is a flow of force and -light charged bodies, such as pith

balls, if their charges are all of one kind, will be swept along in one

direction just as corks are carried by a river. In order that a

charged body be acted upon it must be in the field, just as the corks

to be carried along must be in the stream. Finally, as we may

measure the strength of a stream by the push it exerts upon a

board of unit area inserted in it, so we measure the strength or

intensity of a field by the push it exerts upon a unit charge placed

in it. We therefore define a unit field as that field which acts with

a force of one dyne upon a unit charge placed in it. It follows from

the deduction of Par. 56, that the field produced at a distance d

in air from a charge q must be q/d 2 . In any other medium than

air the field must be q/kd 2 . If we say that a field has a strength of

44

ELEMENTS OF ELECTRICITY.

three we mean that it will pull or push such a unit charge with a

force of three dynes. If the charge itself be not unity, the force

with which it is acted upon is equal to the product of the charge

by the strength of the field.

59. Direction of Field. Suppose that we have a horizontal sheet

of glass in whose center there is a charged metal sphere (Fig. 23).

Fig. 23.

If a small pith ball be released upon the glass anywhere near the

sphere, it will first be attracted to the sphere, will become charged

and will then be shot away in a radial line. The force acts along

these lines and they therefore indicate the direction of the field.

Since opposite charges would move in opposite directions we, by

convention, define the positive direction of a field as that direction in

which a free positive charge would move.

I

Fig. 24.

If we continue this experiment, substituting for the single sphere

two placed some distance apart and charged, one positively, the

other negatively (Fig. 24), the pith ball will no longer follow

STATIC ELECTRICITY.

45

rectilinear paths but curves emerging from one sphere and enter-

ing the other. These curves indicate the direction of the resultant

field at the successive points through which they pass. A posi-

tively-charged ball at C is repelled by A along the line CD and

attracted by B along the line CE. A being the nearer, the force

CD is greater than CE and the ball moves along the resultant CF

which indicates therefore the direction of the field at the point C.

The space about the two spheres may be regarded as permeated

with similar lines symmetrically distributed around the line join-

ing the two centers.

Fig. 25.

Had the two spheres contained like charges, the paths would

have been as represented in Fig. 25.

60. Lines of Force. These lines indicating the direction of the

resultant electric force at the points in the field through which

they pass are called lines of force. They start from a positively-

charged surface and terminate upon a negatively-charged surface.

They therefore have opposite charges at their two ends and never

extend between bodies with like charges. They never penetrate

below the surface or pass through a conductor. They are always

perpendicular to the terminal surfaces at the points of origin and

termination, otherwise there would, be a component parallel to

the surface and a movement of electricity along this surface would

result. They never intersect, for in that case two tangents could

be drawn at one point, that is, there could be two resultants at

one point which is an absurdity. It follows from the foregoing

that every electric field consists of non-conductors and is bounded

by conductors.

61. Graphic Representation of Intensity of Field. In mechan-

ics, in order to treat graphically, to discuss mathematically and

46 ELEMENTS OF ELECTRICITY.

to interpret geometrically problems involving parallel forces dis-

tributed over a surface or among the particles of a mass we, by

convention, represent the direction and the intensity of the forces

by the direction and length respectively of a right line and for the

entire system may substitute a single resultant whose length is

the sum of the lengths of the separate components and whose

point of application is the center of gravity of the surface or of

the mass. In the case of electric fields however, the intensity

varies from point to point and in general the lines of force are not

parallel, therefore, instead of representing this intensity by the

length of a resultant, we agree to represent it by the number

of lines of force per unit of area, the area being taken perpen-

dicular to the lines. By convention, therefore, a unit field is that

field which contains one line of force per square centimeter of cross-

section.

It is not meant by this convention that in moving about in a

unit field the force is experienced at intervals of one centimeter

only and that the intervening space is blank, any more than by

representing the attraction of gravity upon a body by a single line

we imply that there is no gravity in the adjacent space. In a

similar manner we might consider beams of light as made up of a

number of parallel lines or rays and might agree to measure the

intensity of the beam by the number of rays per centimeter of

cross-section. Two beams of light passing through circles of the

same size may differ in intensity and therefore include a different

number of rays, yet on a cross-section of each the illumination is

uniformly and continuously distributed, so two fields may differ

in intensity yet in each the force exists at every point.

In representing lines of force graphically the positive direction,

or direction in which a free positive charge would move, should

always be indicated by arrow-heads.

We conclude by saying that lines of force are those imaginary

lines in a field which by their direction indicate the direction of the

resultant field and by their number indicate its intensity.

62. Tubes of Force. Another convention which avoids this

apparent intermittent distribution of the lines of force and which

is much used in mathematical discussions of electric fields is that

of tubes of force. There are supposed to originate from the surface

of a positively-charged body certain tubular surfaces various in

cross-section and frequently curved but lying side by side like the

STATIC ELECTRICITY. 47

cells of a honeycomb and including within themselves all the

space about the body. Their walls are parallel to the lines of force

of the field and therefore at every cross-section of one of these

tubes the same number of lines would be cut. They terminate

upon a negatively-charged surface. If the portion of the surface

of the charged body included in the base of the tube contains one

unit of electricity, the tube is called a unit tube. It follows from

this conception (and also from Par. 31) that the quantities of

electricity upon the terminal surfaces of a tube are equal and

opposite and a further consequence is that in the case of two

parallel planes near together, one of which is charged, the tubes

at a distance from the edges are parallel and the surface density

upon the central portions of the two planes equal and of opposite

signs. This principle is utilized in the attracted disc electrometer

described later (Par. 101).

It is difficult to represent these tubes graphically and we gener-

ally do so by drawing a single line supposed to be the axis of the

tube, so that after all we resort to lines of force.

63. Lines of Force from Unit Charge. If in Fig. 26 A repre-

sents a unit charge and B a similar and equal charge at a distance

of one centimeter from A, B will be re- ^~ ^

pelled with a force of one dyne. A unit /' ^^\

field is that field which acts with a force of / \

one dyne upon a unit charge placed in it. / ^ \

A is surrounded by its own field and B is J vx (p ^

in it, therefore at B there is a unit field. V % /

The same is true for every point at a dis- \ /

tance of one centimeter from A, that is, Sx "- ^

the surface of a sphere with A as a center Fi s- 26 -

and a radius of one centimeter is a unit field. From Par. 61 there

is in a unit field one line of force per square centimeter of cross-

section. The surface of this sphere is 4 ?r square centimeters and

since each contains one line of force, 4 TT lines of force radiate from

a unit charge.

64. Gauss' Theorem. If around one or more charged bodies

a closed surface be drawn, the number of lines of force which

pierce this surface is equal to 4 ?r times the total charge included

inside the surface. This follows at once from the preceding para-

graph. Each unit charge has 4 -K lines of force radiating from it,

48

ELEMENTS OF ELECTRICITY.

therefore from a charge q there would radiate 4 irq lines. This is

one way of expressing Gauss' Theorem, a principle of frequent

employment in mathematical discussions of electrostatic problems.

An example of the application of this theorem is given in the follow-

ing paragraph.

65. Field about a Uniformly- Charged Sphere. Let (Fig. 27)

be the center of a uniformly-charged sphere, its surface density

,*~ ~^ N being 5, and let P be an external point at

/' \ a distance D from this center. Through

/ \ P pass a sphere with as a center. If the

charge on the original sphere be q, then

according to Gauss' Theorem 4 irq lines of

/ force pierce the sphere P. The area of the

\^ / sphere P is 4?rD 2 and the distribution of

x ^~ <-'' the lines of force is uniform, therefore the

Fig. 27. number of lines per square centimeter is

4 7rg/4 TrZ) 2 , or q/D 2 . But (Par. 61) this measures the intensity of

the field at P or, in other words, measures the force with which

a unit charge at P is acted upon, whence we see that the charge

upon a uniformly-charged sphere acts upon external points as if it

were concentrated at the center.

If the external point be indefinitely near the surface of the sphere

the force exerted will be q/R 2 . Substituting for q its value 4 wR 2 5,

this becomes 4?r5, that is, the field very near the surface of a

charged sphere is equal to 4?r

times the surface density of the

charge. Coulomb extended this

theorem to include charged

bodies of any shape.

66. Field near a Uniformly-

Charged Plane. Let AB (Fig.

28) be a uniformly-charged

plane, its surface density being

5; to find the force exerted upon

a unit positive charge at P at a

distance D from the plane. Let

PC be the perpendicular from

Fig. 28.

the point to the plane.

y -f dy describe a zone.

With C as a center and radii y and

The area of the zone is 2 iry.dy. The

STATIC ELECTRICITY. 49

charge upon this zone is 2 iry.dy.5. The force exerted at P by

this charge is

2<n-y.d ,

dy

z

The normal component in the direction PF is

2ir.d.y.COSa -,

- . dy

z 2

The integral of the components from all the zones will give the

total force. To prepare the above expression for integration cos a

and z 2 must be expressed in terms of y.

From the figure z 2 = D 2 -f- y 2 and cos a = = , ==

,- 2irD.d.y ,

Hence df = , = . dy

v(D 2 +# 2 ) 3

And integrating / = - 2 / + C

VD 2 + y 2

Taking this between the limits y = oo and y =

/ = 2 w5 dynes

In this expression D does not appear, so that the force is inde-

pendent of the position of P with respect to the plane. If the

charged plane be not of indefinite extent, the expression is still

approximately correct if P be so near the plane that the dimen-

sions of the plane as compared to this distance are very great.

67. Force Exerted upon an Internal Point by a Uniformly-

Charged Sphere. Consider a uniformly-charged insulated sphere

remote from other bodies. Let P

(Fig. 29) be any point within such

a sphere and let AB be any line

' drawn through this point. Let the

tangents at A and B represent the

traces of the tangent planes at those

points. The line AB makes equal

angles with these planes. With P

as a vertex describe about PB as Fig729.

an axis a slender cone EPF. Pro-

long its elements beyond P thus describing a second cone GPH.

Suppose the bases of these cones to be charged, the surface density

being the same as that of the sphere. They are similar since they

50 ELEMENTS OF ELECTRICITY.

have equal solid angles at the vertices and their axes make equal

angles a with their bases. Let PB = R, PA=r, the area of the

base EF = S, that of GH = s, the charge on EF = Q, that on GH = q.

The force exerted by Q upon a unit charge at P is Q . sin a /R 2 , that

exerted by q is q . sin a/r 2 . The charges on the bases of these cones

being of the same surface density

Q : q = S :s

hence the above expressions are proportional to S/R 2 and s/r 2 ,

respectively. The cones being similar

S :s = R 2 :r 2

whence S/R 2 = s/r 2 , or the forces exerted upon P are equal and

opposite.

As the cones are made smaller their bases approach coincidence

with the surface of the sphere. The whole surface of the sphere

can be thus divided up by pairs of cones, the effects of the charges

upon whose bases exactly neutralize one another, therefore, the

charge upon the surface exerts no force at an internal point.

This is true whatever the shape of the conductor or surface

distribution of the charge but in only a few cases can the conditions

be given a sufficiently simple mathematical expression to permit

of ready proof.

This fact was shown experimentally by Faraday. He con-

structed of tin-foil and wire-netting an insulated cubical chamber

into which he entered with his most delicate electroscopes. The

chamber was then charged so highly that great sparks and brush

discharges were escaping from the corners, yet his instruments

gave no indications at all.

68. The Charge Resides on the Surface. The proof of the

statement in Par. 38 that the charge resides on the surface of an

insulated conductor follows from the foregoing. Suppose that we

might have an insulated sphere with a charge distributed uniformly

throughout its substance. This charge will induce on surrounding

objects a charge of the opposite kind. The attraction between

these charges will cause the charge in the sphere to move out to

the surface. The portions of the charge upon the surface mutually

repel each other and thus spread over the entire exterior. No

part of the charge could be crowded off into the interior of the

sphere for we have just seen that the charge on the surface exerts

no force in the interior.

STATIC ELECTRICITY. 51

CHAPTER 9.

POTENTIAL.

69. Cause of Movement of Electric Charges. If a charged

conductor be connected to the earth it will be instantly discharged.

If two equal insulated spheres containing unequal charges be

brought into contact there will be a flow from the greater charge

to the lesser until the two charges are equalized and equilibrium

established. If the spheres be of unequal size yet contain equal

charges a portion of the charge of the smaller sphere will flow to

the larger. Finally, if these unequal spheres have charges of the

same surface density a portion of the charge of the larger will flow

to the smaller. The movement is therefore not entirely deter-

mined by difference in the size of the conductors or by inequality

either of the charges or of surface density and we naturally ask

why does it take place. It is produced by what is designated a

difference of electric potential.

70. Physical Analogues of Electric Potential. It has been

remarked that one of the reasons why the study of electricity is

difficult for the beginner is that although we have a sense of weight,

of force, of direction, of velocity, etc., we are devoid of an electric

sense and therefore such expressions as intensity of current,

quantity of electricity, electric pressure, electric potential, etc., are

pure abstractions. To convey a physical conception of these and

to aid in our explanations we are compelled to resort to analogies.

In explaining electric potential it is frequently compared to

temperature and to water level.

Making the first comparison, it is not size or shape of the bodies

or quantity of heat contained but difference of temperature which

determines whether heat shall pass from one body to another, the

flow taking place from the body whose temperature is the higher.

Thus a red-hot nail loses heat when dipped into a bucket of hot

water, although the water may contain several hundred times

more units of heat than the nail; and no matter how they differ in

size there is no net transfer of heat between two bodies at the same

52

ELEMENTS OF ELECTRICITY.

temperature. So with electricity, there is always a flow when

conductors at different potentials are brought into contact, the

flow (of positive electricity) taking place from the conductor of

higher potential, and there is no flow if their potentials are the

same.

Again, if two vessels containing water (Fig. 30) are connected

by a pipe there will be a flow

from the vessel in which the

water stands at the higher level

and this is irrespective of the

actual amounts of water in the

two. There will be no flow if

the level in the two is the

same.

These analogies can not be

carried too far. For example,

Flg - 30 - it will be noted that a change

in temperature is accompanied by a change in volume and often

by a change in state, but conductors show no such changes when

their charges are varied.

71. Mechanical Potential. Consider a cord (Fig. 31) attached

to a weight W and running over a pulley. By the expenditure of

a certain amount of work on the free end of

the cord the weight can be raised against

the force of gravity through a vertical dis-

tance to a new position W. In this new

position it has a certain amount of stored up

energy, or ability to do work, or potentiality,

for if the free end of the cord be released the

weight will drop back to W and in doing so

will, if proper mechanical arrangements be

made and losses by friction be not con-

sidered, do as many foot-pounds of work as

were expended originally in raising it to the

position W.

To raise it to W", higher than W, would

require a greater expenditure of energy but

also its potential energy at W" would be Fi s- 3L

correspondingly greater than that at W. We see then that its

potential energy, or for brevity its potential, varies with its posi-

w

STATIC ELECTRICITY. 53

tion with respect to the surface of the earth (more strictly with

respect to the center of gravity of the earth) and at every different

level we reach it has a corresponding different potential, always

exactly measured by the amount of work expended in moving the

weight from the surface of the earth to that level.

Potential as thus explained is not an inherent property of the

weight for in its various changes of position the weight in itself

does not change. It sometimes becomes desirable to compare the

potential which a body has at one point with that which it would

have at another point and we therefore speak of the potential of

this body at the point, or simply of the potential of the point, but

points in space have no potential and we mean the potential which

the body when moved to the point acquires due to the work ex-

pended in the movement.

In ordinary mechanical problems the force of gravity at any

one spot is considered constant and the potential of a body varies

directly with the vertical distance through which it is raised,

therefore it suffices to give the height and this height in feet multi-

plied by the weight of the body in pounds gives the foot-pounds

by which the potential of the body is measured. However, should

we take this force as following the law of inverse squares, the

amount of work done in raising a body one foot from a certain

level would differ from the amount done in raising this same body

one foot from some other level. The relative potentials of the two

points would not in this case be given directly by their heights but

by the work expended in raising the same weight to the respective

points. Logically therefore we would compare the potentials of

points by comparing the work expended in moving a unit weight

against the force of gravity and from the surface of the earth to

the respective points.

Theoretically, since it is neither raised nor lowered, no work is

expended in moving a body about on a level. Every point on such

a surface has the same potential and it could therefore be called

an equipotential surface. A unit difference of potential exists

between two levels when a unit of work must be expended in

moving a unit weight from one to the other.

72. Electric Potential. We arrive at a definite conception of

of mica will repel each other with a force only one-sixth as great.

The force between two charged bodies in air is not varied by com-

pressing or by rarifying the air and for this reason and on account

of the absence of any absolute measure we use air as our standard.

The ratio of the force exerted between two charged bodies in a

certain dielectric to the force exerted between the same bodies

with the same charges at the same distance apart in air may be

called the dielectric coefficient of repulsion and is the coefficient

by which the force exerted between two charged bodies in air

would be multiplied in order to obtain the force between the same

two bodies under the same conditions in the medium to which the

coefficient pertained. For oil this would therefore be 1/2, for

mica 1/6, etc.

For gases and liquids this coefficient might be determined by

the use of Coulomb's balance as explained above but it is obvious

that this method could not be applied to solids. However, we shall

see later on (Par. 91) how it may be otherwise determined and at

the same time it will be shown why it is written in the form of a

fraction or as I/ k.

In problems involving forces exerted between charged bodies

in other media than air, the appropriate value of 1/fc should be

used and when in discussions in the following pages this coefficient

does not appear it is to be understood that the dielectric is air.

56. Unit Quantity of Electricity. Representing by / the force

of attraction or of repulsion, by q and q' the two charges and by

d their distance apart we may combine the three laws discussed

above and express them mathematically thus

Since, as was explained in Par. 10, electricians have agreed to

follow the C. G. S. system of units, / in this expression must be

measured in dynes and d in centimeters. In the torsion balance

42 ELEMENTS OF ELECTRICITY.

where the two gilded pith balls were of equal size, q and q' are

equal, and since the dielectric is air, l/k = 1, hence the above ex-

pression becomes

If we assume / to be one dyne and d to be one centimeter we

obtain q = l, whence follows at once the definition: An electro-

static unit of electricity is that quantity which when placed at a centi-

meter's distance in air from a similar and equal quantity repels it

with a force of one dyne.

The expression "in air" is essential to this definition as is also

the term "electrostatic" for, as we shall see later (Par. 228), there

is another and different unit of quantity, the coulomb, which is

based upon current relations. The coulomb is three billion

<3xl0 9 ) times as large as the electrostatic unit.

STATIC ELECTRICITY. 43

CHAPTER 8.

ELECTRIC FIELD.

57. Electric Field. We have seen that a charged body attracts

non-electrified bodies and others with opposite charge and repels

those with similar charge, therefore, in the space around an elec-

trified body all bodies experience a force either of attraction or of

repulsion and this space is called the field of the charge. As we

recede from the charged body the force falls off rapidly and to fix

its limits with more definiteness we define the electric field as that

space surrounding a charged body in which the force of attraction or

of repulsion is perceptible. If there be more than one charged body

involved each produces a certain effect and they have a resultant

field. 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. Between two

oppositely charged bodies there is a tension as if they were being

pulled together by invisible rubber bands and at the same time a

stress at right angles to the bands pushing the bands apart.

58. Intensity of Field. It may aid the beginner in his concep-

tion if he consider a field as analogous to a current of water. In

the electric field there is no matter in actual movement but in a

sense there is a flow of force and -light charged bodies, such as pith

balls, if their charges are all of one kind, will be swept along in one

direction just as corks are carried by a river. In order that a

charged body be acted upon it must be in the field, just as the corks

to be carried along must be in the stream. Finally, as we may

measure the strength of a stream by the push it exerts upon a

board of unit area inserted in it, so we measure the strength or

intensity of a field by the push it exerts upon a unit charge placed

in it. We therefore define a unit field as that field which acts with

a force of one dyne upon a unit charge placed in it. It follows from

the deduction of Par. 56, that the field produced at a distance d

in air from a charge q must be q/d 2 . In any other medium than

air the field must be q/kd 2 . If we say that a field has a strength of

44

ELEMENTS OF ELECTRICITY.

three we mean that it will pull or push such a unit charge with a

force of three dynes. If the charge itself be not unity, the force

with which it is acted upon is equal to the product of the charge

by the strength of the field.

59. Direction of Field. Suppose that we have a horizontal sheet

of glass in whose center there is a charged metal sphere (Fig. 23).

Fig. 23.

If a small pith ball be released upon the glass anywhere near the

sphere, it will first be attracted to the sphere, will become charged

and will then be shot away in a radial line. The force acts along

these lines and they therefore indicate the direction of the field.

Since opposite charges would move in opposite directions we, by

convention, define the positive direction of a field as that direction in

which a free positive charge would move.

I

Fig. 24.

If we continue this experiment, substituting for the single sphere

two placed some distance apart and charged, one positively, the

other negatively (Fig. 24), the pith ball will no longer follow

STATIC ELECTRICITY.

45

rectilinear paths but curves emerging from one sphere and enter-

ing the other. These curves indicate the direction of the resultant

field at the successive points through which they pass. A posi-

tively-charged ball at C is repelled by A along the line CD and

attracted by B along the line CE. A being the nearer, the force

CD is greater than CE and the ball moves along the resultant CF

which indicates therefore the direction of the field at the point C.

The space about the two spheres may be regarded as permeated

with similar lines symmetrically distributed around the line join-

ing the two centers.

Fig. 25.

Had the two spheres contained like charges, the paths would

have been as represented in Fig. 25.

60. Lines of Force. These lines indicating the direction of the

resultant electric force at the points in the field through which

they pass are called lines of force. They start from a positively-

charged surface and terminate upon a negatively-charged surface.

They therefore have opposite charges at their two ends and never

extend between bodies with like charges. They never penetrate

below the surface or pass through a conductor. They are always

perpendicular to the terminal surfaces at the points of origin and

termination, otherwise there would, be a component parallel to

the surface and a movement of electricity along this surface would

result. They never intersect, for in that case two tangents could

be drawn at one point, that is, there could be two resultants at

one point which is an absurdity. It follows from the foregoing

that every electric field consists of non-conductors and is bounded

by conductors.

61. Graphic Representation of Intensity of Field. In mechan-

ics, in order to treat graphically, to discuss mathematically and

46 ELEMENTS OF ELECTRICITY.

to interpret geometrically problems involving parallel forces dis-

tributed over a surface or among the particles of a mass we, by

convention, represent the direction and the intensity of the forces

by the direction and length respectively of a right line and for the

entire system may substitute a single resultant whose length is

the sum of the lengths of the separate components and whose

point of application is the center of gravity of the surface or of

the mass. In the case of electric fields however, the intensity

varies from point to point and in general the lines of force are not

parallel, therefore, instead of representing this intensity by the

length of a resultant, we agree to represent it by the number

of lines of force per unit of area, the area being taken perpen-

dicular to the lines. By convention, therefore, a unit field is that

field which contains one line of force per square centimeter of cross-

section.

It is not meant by this convention that in moving about in a

unit field the force is experienced at intervals of one centimeter

only and that the intervening space is blank, any more than by

representing the attraction of gravity upon a body by a single line

we imply that there is no gravity in the adjacent space. In a

similar manner we might consider beams of light as made up of a

number of parallel lines or rays and might agree to measure the

intensity of the beam by the number of rays per centimeter of

cross-section. Two beams of light passing through circles of the

same size may differ in intensity and therefore include a different

number of rays, yet on a cross-section of each the illumination is

uniformly and continuously distributed, so two fields may differ

in intensity yet in each the force exists at every point.

In representing lines of force graphically the positive direction,

or direction in which a free positive charge would move, should

always be indicated by arrow-heads.

We conclude by saying that lines of force are those imaginary

lines in a field which by their direction indicate the direction of the

resultant field and by their number indicate its intensity.

62. Tubes of Force. Another convention which avoids this

apparent intermittent distribution of the lines of force and which

is much used in mathematical discussions of electric fields is that

of tubes of force. There are supposed to originate from the surface

of a positively-charged body certain tubular surfaces various in

cross-section and frequently curved but lying side by side like the

STATIC ELECTRICITY. 47

cells of a honeycomb and including within themselves all the

space about the body. Their walls are parallel to the lines of force

of the field and therefore at every cross-section of one of these

tubes the same number of lines would be cut. They terminate

upon a negatively-charged surface. If the portion of the surface

of the charged body included in the base of the tube contains one

unit of electricity, the tube is called a unit tube. It follows from

this conception (and also from Par. 31) that the quantities of

electricity upon the terminal surfaces of a tube are equal and

opposite and a further consequence is that in the case of two

parallel planes near together, one of which is charged, the tubes

at a distance from the edges are parallel and the surface density

upon the central portions of the two planes equal and of opposite

signs. This principle is utilized in the attracted disc electrometer

described later (Par. 101).

It is difficult to represent these tubes graphically and we gener-

ally do so by drawing a single line supposed to be the axis of the

tube, so that after all we resort to lines of force.

63. Lines of Force from Unit Charge. If in Fig. 26 A repre-

sents a unit charge and B a similar and equal charge at a distance

of one centimeter from A, B will be re- ^~ ^

pelled with a force of one dyne. A unit /' ^^\

field is that field which acts with a force of / \

one dyne upon a unit charge placed in it. / ^ \

A is surrounded by its own field and B is J vx (p ^

in it, therefore at B there is a unit field. V % /

The same is true for every point at a dis- \ /

tance of one centimeter from A, that is, Sx "- ^

the surface of a sphere with A as a center Fi s- 26 -

and a radius of one centimeter is a unit field. From Par. 61 there

is in a unit field one line of force per square centimeter of cross-

section. The surface of this sphere is 4 ?r square centimeters and

since each contains one line of force, 4 TT lines of force radiate from

a unit charge.

64. Gauss' Theorem. If around one or more charged bodies

a closed surface be drawn, the number of lines of force which

pierce this surface is equal to 4 ?r times the total charge included

inside the surface. This follows at once from the preceding para-

graph. Each unit charge has 4 -K lines of force radiating from it,

48

ELEMENTS OF ELECTRICITY.

therefore from a charge q there would radiate 4 irq lines. This is

one way of expressing Gauss' Theorem, a principle of frequent

employment in mathematical discussions of electrostatic problems.

An example of the application of this theorem is given in the follow-

ing paragraph.

65. Field about a Uniformly- Charged Sphere. Let (Fig. 27)

be the center of a uniformly-charged sphere, its surface density

,*~ ~^ N being 5, and let P be an external point at

/' \ a distance D from this center. Through

/ \ P pass a sphere with as a center. If the

charge on the original sphere be q, then

according to Gauss' Theorem 4 irq lines of

/ force pierce the sphere P. The area of the

\^ / sphere P is 4?rD 2 and the distribution of

x ^~ <-'' the lines of force is uniform, therefore the

Fig. 27. number of lines per square centimeter is

4 7rg/4 TrZ) 2 , or q/D 2 . But (Par. 61) this measures the intensity of

the field at P or, in other words, measures the force with which

a unit charge at P is acted upon, whence we see that the charge

upon a uniformly-charged sphere acts upon external points as if it

were concentrated at the center.

If the external point be indefinitely near the surface of the sphere

the force exerted will be q/R 2 . Substituting for q its value 4 wR 2 5,

this becomes 4?r5, that is, the field very near the surface of a

charged sphere is equal to 4?r

times the surface density of the

charge. Coulomb extended this

theorem to include charged

bodies of any shape.

66. Field near a Uniformly-

Charged Plane. Let AB (Fig.

28) be a uniformly-charged

plane, its surface density being

5; to find the force exerted upon

a unit positive charge at P at a

distance D from the plane. Let

PC be the perpendicular from

Fig. 28.

the point to the plane.

y -f dy describe a zone.

With C as a center and radii y and

The area of the zone is 2 iry.dy. The

STATIC ELECTRICITY. 49

charge upon this zone is 2 iry.dy.5. The force exerted at P by

this charge is

2<n-y.d ,

dy

z

The normal component in the direction PF is

2ir.d.y.COSa -,

- . dy

z 2

The integral of the components from all the zones will give the

total force. To prepare the above expression for integration cos a

and z 2 must be expressed in terms of y.

From the figure z 2 = D 2 -f- y 2 and cos a = = , ==

,- 2irD.d.y ,

Hence df = , = . dy

v(D 2 +# 2 ) 3

And integrating / = - 2 / + C

VD 2 + y 2

Taking this between the limits y = oo and y =

/ = 2 w5 dynes

In this expression D does not appear, so that the force is inde-

pendent of the position of P with respect to the plane. If the

charged plane be not of indefinite extent, the expression is still

approximately correct if P be so near the plane that the dimen-

sions of the plane as compared to this distance are very great.

67. Force Exerted upon an Internal Point by a Uniformly-

Charged Sphere. Consider a uniformly-charged insulated sphere

remote from other bodies. Let P

(Fig. 29) be any point within such

a sphere and let AB be any line

' drawn through this point. Let the

tangents at A and B represent the

traces of the tangent planes at those

points. The line AB makes equal

angles with these planes. With P

as a vertex describe about PB as Fig729.

an axis a slender cone EPF. Pro-

long its elements beyond P thus describing a second cone GPH.

Suppose the bases of these cones to be charged, the surface density

being the same as that of the sphere. They are similar since they

50 ELEMENTS OF ELECTRICITY.

have equal solid angles at the vertices and their axes make equal

angles a with their bases. Let PB = R, PA=r, the area of the

base EF = S, that of GH = s, the charge on EF = Q, that on GH = q.

The force exerted by Q upon a unit charge at P is Q . sin a /R 2 , that

exerted by q is q . sin a/r 2 . The charges on the bases of these cones

being of the same surface density

Q : q = S :s

hence the above expressions are proportional to S/R 2 and s/r 2 ,

respectively. The cones being similar

S :s = R 2 :r 2

whence S/R 2 = s/r 2 , or the forces exerted upon P are equal and

opposite.

As the cones are made smaller their bases approach coincidence

with the surface of the sphere. The whole surface of the sphere

can be thus divided up by pairs of cones, the effects of the charges

upon whose bases exactly neutralize one another, therefore, the

charge upon the surface exerts no force at an internal point.

This is true whatever the shape of the conductor or surface

distribution of the charge but in only a few cases can the conditions

be given a sufficiently simple mathematical expression to permit

of ready proof.

This fact was shown experimentally by Faraday. He con-

structed of tin-foil and wire-netting an insulated cubical chamber

into which he entered with his most delicate electroscopes. The

chamber was then charged so highly that great sparks and brush

discharges were escaping from the corners, yet his instruments

gave no indications at all.

68. The Charge Resides on the Surface. The proof of the

statement in Par. 38 that the charge resides on the surface of an

insulated conductor follows from the foregoing. Suppose that we

might have an insulated sphere with a charge distributed uniformly

throughout its substance. This charge will induce on surrounding

objects a charge of the opposite kind. The attraction between

these charges will cause the charge in the sphere to move out to

the surface. The portions of the charge upon the surface mutually

repel each other and thus spread over the entire exterior. No

part of the charge could be crowded off into the interior of the

sphere for we have just seen that the charge on the surface exerts

no force in the interior.

STATIC ELECTRICITY. 51

CHAPTER 9.

POTENTIAL.

69. Cause of Movement of Electric Charges. If a charged

conductor be connected to the earth it will be instantly discharged.

If two equal insulated spheres containing unequal charges be

brought into contact there will be a flow from the greater charge

to the lesser until the two charges are equalized and equilibrium

established. If the spheres be of unequal size yet contain equal

charges a portion of the charge of the smaller sphere will flow to

the larger. Finally, if these unequal spheres have charges of the

same surface density a portion of the charge of the larger will flow

to the smaller. The movement is therefore not entirely deter-

mined by difference in the size of the conductors or by inequality

either of the charges or of surface density and we naturally ask

why does it take place. It is produced by what is designated a

difference of electric potential.

70. Physical Analogues of Electric Potential. It has been

remarked that one of the reasons why the study of electricity is

difficult for the beginner is that although we have a sense of weight,

of force, of direction, of velocity, etc., we are devoid of an electric

sense and therefore such expressions as intensity of current,

quantity of electricity, electric pressure, electric potential, etc., are

pure abstractions. To convey a physical conception of these and

to aid in our explanations we are compelled to resort to analogies.

In explaining electric potential it is frequently compared to

temperature and to water level.

Making the first comparison, it is not size or shape of the bodies

or quantity of heat contained but difference of temperature which

determines whether heat shall pass from one body to another, the

flow taking place from the body whose temperature is the higher.

Thus a red-hot nail loses heat when dipped into a bucket of hot

water, although the water may contain several hundred times

more units of heat than the nail; and no matter how they differ in

size there is no net transfer of heat between two bodies at the same

52

ELEMENTS OF ELECTRICITY.

temperature. So with electricity, there is always a flow when

conductors at different potentials are brought into contact, the

flow (of positive electricity) taking place from the conductor of

higher potential, and there is no flow if their potentials are the

same.

Again, if two vessels containing water (Fig. 30) are connected

by a pipe there will be a flow

from the vessel in which the

water stands at the higher level

and this is irrespective of the

actual amounts of water in the

two. There will be no flow if

the level in the two is the

same.

These analogies can not be

carried too far. For example,

Flg - 30 - it will be noted that a change

in temperature is accompanied by a change in volume and often

by a change in state, but conductors show no such changes when

their charges are varied.

71. Mechanical Potential. Consider a cord (Fig. 31) attached

to a weight W and running over a pulley. By the expenditure of

a certain amount of work on the free end of

the cord the weight can be raised against

the force of gravity through a vertical dis-

tance to a new position W. In this new

position it has a certain amount of stored up

energy, or ability to do work, or potentiality,

for if the free end of the cord be released the

weight will drop back to W and in doing so

will, if proper mechanical arrangements be

made and losses by friction be not con-

sidered, do as many foot-pounds of work as

were expended originally in raising it to the

position W.

To raise it to W", higher than W, would

require a greater expenditure of energy but

also its potential energy at W" would be Fi s- 3L

correspondingly greater than that at W. We see then that its

potential energy, or for brevity its potential, varies with its posi-

w

STATIC ELECTRICITY. 53

tion with respect to the surface of the earth (more strictly with

respect to the center of gravity of the earth) and at every different

level we reach it has a corresponding different potential, always

exactly measured by the amount of work expended in moving the

weight from the surface of the earth to that level.

Potential as thus explained is not an inherent property of the

weight for in its various changes of position the weight in itself

does not change. It sometimes becomes desirable to compare the

potential which a body has at one point with that which it would

have at another point and we therefore speak of the potential of

this body at the point, or simply of the potential of the point, but

points in space have no potential and we mean the potential which

the body when moved to the point acquires due to the work ex-

pended in the movement.

In ordinary mechanical problems the force of gravity at any

one spot is considered constant and the potential of a body varies

directly with the vertical distance through which it is raised,

therefore it suffices to give the height and this height in feet multi-

plied by the weight of the body in pounds gives the foot-pounds

by which the potential of the body is measured. However, should

we take this force as following the law of inverse squares, the

amount of work done in raising a body one foot from a certain

level would differ from the amount done in raising this same body

one foot from some other level. The relative potentials of the two

points would not in this case be given directly by their heights but

by the work expended in raising the same weight to the respective

points. Logically therefore we would compare the potentials of

points by comparing the work expended in moving a unit weight

against the force of gravity and from the surface of the earth to

the respective points.

Theoretically, since it is neither raised nor lowered, no work is

expended in moving a body about on a level. Every point on such

a surface has the same potential and it could therefore be called

an equipotential surface. A unit difference of potential exists

between two levels when a unit of work must be expended in

moving a unit weight from one to the other.

72. Electric Potential. We arrive at a definite conception of

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