Font size

than the same mass of copper in a more compact form. As the

zinc and copper plates are flattened out and increased in size the

glass cell must keep pace, but as it gets larger it increases rapidly

in cost. Reflection will show that two cells in parallel are elec-

trically equal to a single cell with plates twice as large. Therefore,

the usual method of increasing the cross-sectional area of a

battery is to join cells in parallel.

295. Wire Tables. As the practical electrician has to deal

largely with wires, it is important that he should possess infor-

mation as to the different sizes, their dimension, weight, resistance,

etc. Such data is embodied in wire tables which are issued by the

wire manufacturers and are also found in the various electrical

hand-books. The sizes of wire are designated by numbers corre-

sponding to certain wire gauges. It is unfortunate that there are

in existence four or five of these gauges and that their numbers

do not correspond nor do their sizes of wire vary in accordance

with any fixed rule. In this country the gauge in most common

use is the American wire gauge of the Brown and Sharpe Company.

The Birmingham wire gauge is also in use. The No. 1 wire on the

Brown and Sharpe gauge is very nearly .3 of an inch in diameter,

and the smallest wire, or No. 40, is about .003 of an inch. There

are four sizes larger than No. 1 and they are designated single 0,

VOLTAIC ELECTRICITY.

221

double 0, treble 0, etc. The No. 10 wire on the B. & S. gauge is

just about .1 of an inch in diameter and if of copper its resistance

is about one ohm per 1000 feet. As a rule of thumb, by subtracting

three from the gauge number of any wire we get the number of

the wire whose cross-sectional area is twice as great. The cross-

sectional area of No. 7 is twice that of No. 10.

COPPER WIRE TABLE, BROWN AND SHARPE GAUGE.

Resistance at 20 C.

Size of

wire

Diameter,

inches

Ohma per

foot

Feet per

ohm

Pounds per

foot

0000

0.460

0.00004893

20,440

0.6405

000

0.4096

0.00006170

16,210

0.5080

00

0.3648

0.00007780

12,850

0.4028

0.3249

0.00009811

10,190

0.3195

1

0.2893

0.0001237

8,083

0.2533

2

0.2576

0.0001560

6,410

0.2009

3

0.2294

0.0001967

5,084

0.1593

4

0.2043

0.0002480

4,031

0.1264

5

0.1819

0.0003128

3,197

0.1002

6

0.1620

0.0003944

2,535

0.07946

7

0.1443

0.0004973

2,011

0.06302

8

0.1285

0.0006271

1,595

0.04998

9

0.1144

0.0007908

1,265

0.03963

10

0.1019

0.0009972

1,003

0.03143

11

0.09074

0.001257

795.3

0.02493

12

0.08081

0.001586

630.7

0.01977

13

0.07196

0.001999

500.1

0.01568

14

0.06408

0.002521

396.6

0.01243

15

0.05707

0.003179

314.5

0.009858

16

0.05082

0.004009

249.4

0.007818

17

0.04526

0.005055

197.8

0.006200

18

0.04030

0.006374

156.9

0.004917

19

0.03589

0.008038

124.4

0.003899

20

0.03196

0.01014

98.66

0.003092

21

0.02846

0.01278

78.24

0.002452

22

0.02535

0.01612

62.05

0.001945

23

0.02257

0.02032

49.21

0.001542

24

0.02010

0.02563

39.02

0.001223

25

0.01790

0.03231

30.95

0.0009699

26

0.01594

0.04075

24.54

0.0007692

27

0.01420

0.05138

19.46

0.0006100

28

0.01264

0.06479

15.43

0.0004837

29

0.01126

0.08170

12.24

0.0003836

30

0.01003

0.1030

9.71

0.0003042

222 ELEMENTS OF ELECTRICITY.

296. Circular Measure of Wires. Owing to the errors likely

to occur from lack of agreement in the sizes of the various wire

gauges, it is becoming more and more the custom among elec-

tricians to designate wires by their diameters expressed in thous-

andths of an inch or mils, indeed, by recent orders of the War

Department this has been made mandatory for our army. If we

compare the area of cross-section of a wire whose diameter is one

mil with that of one whose diameter is n mils we see, since the

areas of circles are to each other as the squares of their diameters,

that the cross-section of the larger wire is n 2 times greater than

that of the smaller. Because of this very simple relation, the area

of cross-section of a wire of one mil diameter is taken as the unit

of area and called a circular mil To find the area in circular mils

of the cross-section of any other wire we simply square its diameter

expressed in thousandths of an inch. This method of comparison

is very much simpler than expressing the cross-sections in square

inches. A piece of wire one foot long and one mil in diameter is

called a mil foot. The resistance of a mil foot of annealed copper

is 9.59 ohms at 32 F and 10.505 ohms at 75 F. With this data

we may, by applying the law given in Par. 287, determine the

resistance of a copper wire of any size and length.

-\

VOLTAIC ELECTRICITY. 223

CHAPTER 25.

OHM'S LAW.

297. Ohm's Law. As a result of his investigations, Ohm

announced in 1827 the law which bears his name and which is to

the effect that in any electric circuit the current varies directly

as the electro-motive force and inversely as the resistance of the

circuit. Expressed in symbols this becomes

I- E

~ R

in which, if E be the E. M. F. in

volts and R the resistance in ohms, I is the current in amperes.

In its determination Ohm employed the rather crude appliances

which he extemporized for the purpose (Par. 284). Since his time^

the delicacy and accuracy of electrical apparatus have been

immensely increased, yet the most careful and refined observations

serve merely to afford stronger confirmation of his conclusions.

The importance of this law can not be over-estimated. In the

study and application of electricity it is fundamental and in one

form or another it is met at every turn. On account of its very

simplicity there is sometimes a failure to recognize that it is

unique, and occasionally it is spoken of as "self evident." Such

is far from being the case. There is no material substance which

follows such a law. Pressure causes liquids and gases to flow

through pipes, yet if this pressure be doubled the flow is by no

means doubled.

When applying the law to a more or less complex circuit, E

represents the total E. M. F. and R the total resistance. Thus there

may be several cells or batteries or electrical machines contrib-

uting to the E. M. F., in which case the sum of the E. M. F.s

must be taken. Again, through error or by design a cell or battery

may be reversed so as to oppose the remaining E. M. F. Such

opposing E. M. F. is spoken of as counter E. M. F. or back E. M. F.

Back E. M. F. is also produced by polarization (Par. 198) and,

as we shall see later, by the operation of motors in the circuit. In

224

ELEMENTS OF ELECTRICITY.

summing up the total E. M. F. of the circuit, back E. M. F. is to

be considered as negative. The resistance R includes not only

the resistance of the line but also that of the contacts, joints and

connections and of the electrolyte and elements of the cells. The

law can therefore be given

T E' + E" + E'" + E"" + (fee.

R' 4- R" + E'" + R"" + &c.

or the current

in the circuit is equal to the algebraic sum of the separate

E. M. F.s divided by the sum of the separate resistances.

298. Drop of Potential. The three quantities, current, electro-

motive force and resistance are bound together by Ohm's law so

that any two being given, the third may be determined. It may

at first sight appear unnecessary to state such a self-evident truth

but it is desirable to lay especial emphasis upon the fact for, until

the student has become familiar with the law, the tendency is

rather to restrict its use to the determination of current only.

The law may be put in the form

E = IR

and it is helpful to the

beginner if he will accustom himself to interpret this as meaning

that E is the electro-motive force necessary to drive a current

of strength / through a resistance R.

>

V ]

D" F"

r~<~

I

D 1

t ^^^

1

\^ i

1 F ^ B

Fig. 121.

Suppose AB (Fig. 121) to represent a portion of an electric

circuit, the point A being of higher potential than B, and suppose

that by means of one of the instruments to be described later

(Chapter 34) we measure the difference in potential between

A and B. Lay off on some convenient scale A A' proportional

to this difference of potential. If we move along A B to some

point D and measure the difference of potential between D and B

VOLTAIC ELECTRICITY. 225

we will find it to be less than that at A, or represented by DD'~

Likewise, at F this difference of potential is still smaller and is

proportional to FF', that is, as we move from A towards B the

difference of potential between the successive points and B

steadily grows less, or there is a falling off from the difference of

potential represented by A A'. At D, for example, this drop of

potential is D"D' and at F it is F"F'.

The drop of potential between any two points is always equal

to the product of the current into the resistance between the points.

Certain elementary applications of this principle will be shown

in the following paragraphs.

299. Ohm's Law Applies to Any Portion of the Circuit. In Par.

297 we saw that Ohm's law was applicable to the entire circuit

even though this be made complex by including heterogeneous

resistances and sources of E. M. F. It also applies to any portion

of a circuit, that is, the current flowing between any two points

in a circuit is equal to the difference of potential between these

two points divided by the resistance between them. We have seen

(Par. 229) that the current at every cross-section of a circuit is the

same; if, therefore, we determine it at one point we have it for

any other point. Knowing the current, if we have the resistance

between two points we can, by what we have shown in the pre-

ceding paragraph, determine the difference of potential, or drop,

between the two points. These principles enable us to solve a

variety of problems. For example, let A BCD, Fig. 122, repre-

sent part of an electric circuit. The resistance of the portion AB is

12 ohms, that of the incandescent lamp EC is 220 ohms, that of

CD is 8 ohms. The difference of potential between A and B is 6

volts. What current is flowing in the circuit and what is the poten-

tial of the points A, B and C if that of D be taken as zero?

A 12, OHMS B

-" *

.^^ 220 OHMS

8 OHMS

Fig. 122.'

V fi 1

The current between A and ^ = ^ = To = o am P ere > which is

also the current for the rest of the circuit. The drop from B to C =

226 ELEMENTS OF ELECTRICITY.

1^ = ^X220 = 110 volts; that from C to D = ^X8 = 4 volts. The

potential of C is therefore 4 volts, that of B is 114, and that of A

is 120.

300. Division of Current in Divided Circuit. This principle of

drop of potential furnishes a simple determination of the division

of an electric current in a divided circuit.

I' R'

A j I"

R"

D

r

R 1 "

Fig. 123.

Let Fig. 123 represent a divided circuit of three branches whose

resistances are respectively R', R" and R'". Call the correspond-

ing currents /', I" and /"' '. The current in the main branch upon

arriving at A divides into these three portions which reunite at B.

The drop from A to B is the same over each of the three routes,

therefore

I'R'=I"R"=T"R'"

which may be written

f - f" 1'"

R' ' R" ' R"'

that is, the current

in the branches of a divided circuit are to each other inversely as the

resistances of the respective branches.

In making an actual calculation, if the fractions in the second

member of this proportion be brought to a common denominator,

their numerators indicate at once the relation between the several

currents.

If there be but two branches, the above becomes

i - . -*-

R' ' R

which may be written

/' \l" = R" \R'

a somewhat simple

form for calculations.

301. Shunts. In practical electricity it frequently becomes

necessary to employ a divided circuit of two branches which must

jt . ji

' "

VOLTAIC ELECTRICITY. 227

be so proportioned that the main current divides between them

in accordance with some desired ratio. For example, suppose that

we wish to measure a current which is much larger than can be

measured directly by the instruments at our disposal. If we can

arrange a divided circuit so that exactly one-hundredth of the

total current flows through one branch, we can measure this

small current and always know that the entire current is one

hundred times greater. This division is brought about by shunts

(Par. 293).

In Fig. 124 we desire to measure the current flowing in AD.

G is our measuring instrument which with its connecting wires,

BG and GC has a resistance of R ohms. EC is the shunt. What

must be the resistance of the shunt so that one-hundredth of the

total current will flow through G ?

Fig. 124.

Call the current through the instrument /; that through BC

will be 997. If x be the resistance of BC, then, as shown in the

preceding paragraph

/ X R = 997 X x

whence _ R

~99

or the resistance of the shunt

must be one-ninety-ninth of the resistance of G and its connecting

wires and leads. In a similar manner we can determine the

resistance of shunts to bring about division of the total current in

any desired ratio. It is to be noted that these shunts are con-

structed for use with a particular instrument and cannot be used

with another of different resistance.

302. Rheostats. A consideration of Ohm's law, I = E/R, will

show that by varying R we can vary the current inversely and

suggests that by introducing or removing resistance from a cir-

cuit we may regulate the current at. will. Instruments for this

purpose are called rheostats. The principle of their use will be

228 ELEMENTS OF ELECTRICITY.

understood from the diagram (Fig. 125). A series of metal con-

tacts are arranged upon the arc of a circle DE and connected

between these contacts are resistance coils. Pivoted at the center

B

t>f the arc is a metal arm CD which can be moved about over the

contacts. Suppose the current to come in by A. As represented

in the figure, it must now traverse all the coils from D to E before

it can leave by B, and it is therefore cut down. Had the arm CD

been still farther to the left, the circuit would have been broken

entirely, R would have been infinite and the current zero. As the

arm is slid around to E the coils are successively cut out, the

resistance correspondingly reduced, and the current correspond-

ingly increased, reaching its maximum when the arm reaches E.

The controller by which the motorman starts and stops a trolley

car is similar in principle.

It will be shown later that regulation of current by rheostat is

a wasteful method and except for temporary purposes, such as for

starting and stopping motors, should not be employed.

303. KirchoflPs Laws. Where an electric circuit is composed

of interlacing branches and especially where there are in it several

seats of electro-motive force, confusion and uncertainty may

arise as to the correct way of applying Ohm's law in the determina-

tion of the separate currents and pgtentials. To obviate this,

Kirchoff has formulated a set of rules which render this applica-

tion almost mechanical. These are:

I. // several conductors meet at a common point, the algebraic sum

of the currents in these conductors is zero.

This is but a statement of the fact that electricity does not

accumulate at a point and that therefore as much flows away as

flows to the point. If currents flowing to the point be considered

positive, those flowing away must be regarded as negative.

II. // two or more conductors form a closed figure, or a mesh in a

network of conductors, the sum of the products of each current of this

VOLTAIC ELECTRICITY.

229

mesh into the resistance through which it passes is equal to the algebraic

sum of the electro-motive forces acting around this same mesh.

This is another statement of the fact that the total drop of

potential in going around a closed circuit is equal to the sum of

the partial drops. The convention must be adopted that in going

around a closed circuit, if the E. M. F. acting in a clockwise direc-

tion be considered positive, that acting in the opposite direction

is negative.

304. Example of Application of Kirchoff's Laws. By combin-

ing these laws it is always possible to obtain as many independent

equations as there are unknown quantities and hence these

unknown quantities may be determined. The following concrete

example will make the matter clear. Fig. 126 represents a net-

work of conductors in two of the branches of which there are bat-

teries E and F, sources of E. M. F. The currents in the separate

branches are designated ii, i 2 , i z , etc., and their assumed direction

is indicated by the arrows. In the final solution, a negative value

of a current indicates that the actual direction is opposite to that

assumed. The E. M. F. of the batteries and the resistances of the

branches are indicated on the diagram. We are required to deter-

mine the currents in the separate branches.

From Kirchoff's first law we obtain the following "point

equations" :

point a

point 6

point c

point d

ii i s =

-f i- i\ =

i* is =

-f it ib =

230 ELEMENTS OF ELECTRICITY.

p

From the second law we obtain the following "voltage

equations":

mesh iii&t 5ii + 2iz 4- 3i 4 = 3

mesh fciaie 3i 3 4t 6 2^ =

mesh iii&s 4i 6 + 4^ 5 3i 4 = 2

mesh w 3 ^5 5ii -h 3i 3 -f- 44 = 3 2

We now have eight independent equations from which to deter-

mine six unknown quantities, and the remainder of the process is

but a matter of combination and elimination.

305. Lost Volts and Useful Volts. Should there be connected

up to a circuit of resistance R a cell whose E. M. F. is E and in-

ternal resistance r, the resulting current would be given by the

expression

R + r

which may be written

E = IR + Ir

Interpreting this as explained in Par. 298, we see that a part of

the E. M. F. of the cell is spent in driving the current through the

external resistance R and the remainder in driving this current

through the internal resistance of the cell. The volts, Ir, con-

sumed on the interior of the cell are called the lost volts and we

profit only by those upon the external circuit, or IR, which are

therefore called the available or useful volts. Since the less the lost

volts, the more the useful volts, it is of importance to keep the

former at a minimum. Ir may be reduced in two ways; by reduc-

ing the current or by decreasing the internal resistance. If there

be no current, there is of course no wastage. The internal resist-

ance may be reduced by selecting an electrolyte of low resistance

(though usually choice is restricted), by bringing the plates closer

together, and by increasing the size of the plates (Par. 294). Lost

volts have also to be considered in the operation of electrical

machinery.

306. Short Circuit. The commonest source of injury to elec-

trical machinery is a short circuit, which may be defined as the

removal, usually accidental, of the greater part of the resistance

from a "live" circuit.

VOLTAIC ELECTRICITY. 231

v

Suppose B (Fig. 127) to represent a battery supplying current

for the incandescent lamp L. The internal resistance of the bat-

tery is almost negligible, the resistance of the wires should be very

4.

Fig. 127.

small. Suppose the E. M. F. of the battery to be 111 volts, the

resistance of the lamp to be 220 ohms and that of the battery

and wires to be 2 ohms. The current is

1 = 220 + 2 = 2 am P ere

If by some accident the wire should sag, as shown by the dotted

line, and touch the lower wire at P, at that instant the current

would be short circuited through the point P, the resistance of the

lamp and of the wire beyond P being eliminated. The current is

now

I = -= = HI amperes

or it has suddenly in-

creased over two hundred times. If the wires had been designed

to carry only ten or fifteen amperes they will be fused, apparatus

in the circuit will be "burned out," insulation will be charred and

possibly fires started. To avoid the injury resulting from such

accidents, use is made of fuses, pieces of soft, easily-fusible wire

inserted in the circuit which is to be protected. If the current

exceeds that which the fuses are intended to carry, they melt

before damage is done to the rest of the circuit. This same pro-

tection is also afforded by certain automatic apparatus called

overload switches (Par. 414).

307. Definitions Based Upon Ohm's Law. Since the three

quantities 7, E and R are bound together by Ohm's law, any one

may be defined in terms of the other two. Thus the ampere is

sometimes defined as the current produced by an E. M. F. of one

volt applied to a conductor whose resistance is one ohm. So also

the volt is defined as that E. M. F. which applied to a resistance of

one ohm will produce in it a current of one ampere. The ohm may

232 ELEMENTS OF ELECTRICITY.

be similarly defined but such definition adds but little to our

knowledge.

Since Ohm's law may be written

and since E and / fluctuate

together so that R remains always constant, the resistance of a

conductor is defined by some writers as the ratio of the difference

of potential of the ends of the conductor to the current produced

in it. To define a property as a ratio is not altogether satisfactory.

It is perhaps better to say that this ratio affords a measure of the

resistance.

VOLTAIC ELECTRICITY. 233

CHAPTER 26.

MEASUREMENT OF RESISTANCE.

308. Measurement of Resistance. One of the most important

classes of measurements with which the electrician has to deal is

that of resistance. Logically, this subject should have been taken

up in connection with that of resistance in Chapter 24, but the

methods employed could not be clearly presented until after the

consideration of Ohm's law and the explanation, as given in Chap-

ter 25, of the principles of the drop of potential and the division

of current in a divided circuit. Even now we shall have to antici-

pate certain principles which can not be fully developed until

later.

In these measurements, the methods to be employed vary with

the amount and character of the resistance. Thus, very high and

very low resistances are measured in a different way from those

covering a moderate range. Again, the measurement of the in-

ternal resistance of cells and of the resistance of electrolytes must

be undertaken in an entirely different manner from that of a

metallic conductor. These facts will be brought out in the follow-

ing pages.

309. Drop of Potential Proportional to Resistance Passed

Over. If there exists between AB (Fig. 128), two points of a cir-

cuit, a difference of potential E, there will be a flow of electricity

from the point of higher potential to that of lower. The value of

this current as given by Ohm's law is I = E/R, whence E=IR,

which last expression, as we have already seen (Par. 298), may be

interpreted as expressing the fact that E is the electro-motive

force required to drive a current of strength / through the con-

ductor of resistance R.

234 ELEMENTS OF ELECTRICITY.

To drive the same current through a resistance only one-half as

great requires only one-half as much E. M. F., or, if the resistance

of AM be one-half of the total resistance between A and B, then

one-half of the total E. M. F. will be expended in driving the cur-

rent from A to M, and the difference of potential between M and

B is only one-half of that between A and B. In more general

terms, for a constant current, the expenditure of E. M. F., or the

drop of potential, is directly proportional to the resistance passed

over.

310. Measurement of Resistance by Drop of Potential.

Should we have at our disposal a known resistance and an instru-

ment for measuring difference of potential (Chapter 34), the fore-

going affords us a means of measuring the resistance between any

two points in a circuit. For example, suppose that the resistance

R between A and M, Fig. 128, be known and that we desire to

determine the resistance x between M and B. We have simply

to measure with our instrument the drop E' between A and M,

and the drop E" between M and B. From the preceding paragraph

E' :E" = R:x

E"

whence x = -^7 R

This method supposes the current to be constant during the two

zinc and copper plates are flattened out and increased in size the

glass cell must keep pace, but as it gets larger it increases rapidly

in cost. Reflection will show that two cells in parallel are elec-

trically equal to a single cell with plates twice as large. Therefore,

the usual method of increasing the cross-sectional area of a

battery is to join cells in parallel.

295. Wire Tables. As the practical electrician has to deal

largely with wires, it is important that he should possess infor-

mation as to the different sizes, their dimension, weight, resistance,

etc. Such data is embodied in wire tables which are issued by the

wire manufacturers and are also found in the various electrical

hand-books. The sizes of wire are designated by numbers corre-

sponding to certain wire gauges. It is unfortunate that there are

in existence four or five of these gauges and that their numbers

do not correspond nor do their sizes of wire vary in accordance

with any fixed rule. In this country the gauge in most common

use is the American wire gauge of the Brown and Sharpe Company.

The Birmingham wire gauge is also in use. The No. 1 wire on the

Brown and Sharpe gauge is very nearly .3 of an inch in diameter,

and the smallest wire, or No. 40, is about .003 of an inch. There

are four sizes larger than No. 1 and they are designated single 0,

VOLTAIC ELECTRICITY.

221

double 0, treble 0, etc. The No. 10 wire on the B. & S. gauge is

just about .1 of an inch in diameter and if of copper its resistance

is about one ohm per 1000 feet. As a rule of thumb, by subtracting

three from the gauge number of any wire we get the number of

the wire whose cross-sectional area is twice as great. The cross-

sectional area of No. 7 is twice that of No. 10.

COPPER WIRE TABLE, BROWN AND SHARPE GAUGE.

Resistance at 20 C.

Size of

wire

Diameter,

inches

Ohma per

foot

Feet per

ohm

Pounds per

foot

0000

0.460

0.00004893

20,440

0.6405

000

0.4096

0.00006170

16,210

0.5080

00

0.3648

0.00007780

12,850

0.4028

0.3249

0.00009811

10,190

0.3195

1

0.2893

0.0001237

8,083

0.2533

2

0.2576

0.0001560

6,410

0.2009

3

0.2294

0.0001967

5,084

0.1593

4

0.2043

0.0002480

4,031

0.1264

5

0.1819

0.0003128

3,197

0.1002

6

0.1620

0.0003944

2,535

0.07946

7

0.1443

0.0004973

2,011

0.06302

8

0.1285

0.0006271

1,595

0.04998

9

0.1144

0.0007908

1,265

0.03963

10

0.1019

0.0009972

1,003

0.03143

11

0.09074

0.001257

795.3

0.02493

12

0.08081

0.001586

630.7

0.01977

13

0.07196

0.001999

500.1

0.01568

14

0.06408

0.002521

396.6

0.01243

15

0.05707

0.003179

314.5

0.009858

16

0.05082

0.004009

249.4

0.007818

17

0.04526

0.005055

197.8

0.006200

18

0.04030

0.006374

156.9

0.004917

19

0.03589

0.008038

124.4

0.003899

20

0.03196

0.01014

98.66

0.003092

21

0.02846

0.01278

78.24

0.002452

22

0.02535

0.01612

62.05

0.001945

23

0.02257

0.02032

49.21

0.001542

24

0.02010

0.02563

39.02

0.001223

25

0.01790

0.03231

30.95

0.0009699

26

0.01594

0.04075

24.54

0.0007692

27

0.01420

0.05138

19.46

0.0006100

28

0.01264

0.06479

15.43

0.0004837

29

0.01126

0.08170

12.24

0.0003836

30

0.01003

0.1030

9.71

0.0003042

222 ELEMENTS OF ELECTRICITY.

296. Circular Measure of Wires. Owing to the errors likely

to occur from lack of agreement in the sizes of the various wire

gauges, it is becoming more and more the custom among elec-

tricians to designate wires by their diameters expressed in thous-

andths of an inch or mils, indeed, by recent orders of the War

Department this has been made mandatory for our army. If we

compare the area of cross-section of a wire whose diameter is one

mil with that of one whose diameter is n mils we see, since the

areas of circles are to each other as the squares of their diameters,

that the cross-section of the larger wire is n 2 times greater than

that of the smaller. Because of this very simple relation, the area

of cross-section of a wire of one mil diameter is taken as the unit

of area and called a circular mil To find the area in circular mils

of the cross-section of any other wire we simply square its diameter

expressed in thousandths of an inch. This method of comparison

is very much simpler than expressing the cross-sections in square

inches. A piece of wire one foot long and one mil in diameter is

called a mil foot. The resistance of a mil foot of annealed copper

is 9.59 ohms at 32 F and 10.505 ohms at 75 F. With this data

we may, by applying the law given in Par. 287, determine the

resistance of a copper wire of any size and length.

-\

VOLTAIC ELECTRICITY. 223

CHAPTER 25.

OHM'S LAW.

297. Ohm's Law. As a result of his investigations, Ohm

announced in 1827 the law which bears his name and which is to

the effect that in any electric circuit the current varies directly

as the electro-motive force and inversely as the resistance of the

circuit. Expressed in symbols this becomes

I- E

~ R

in which, if E be the E. M. F. in

volts and R the resistance in ohms, I is the current in amperes.

In its determination Ohm employed the rather crude appliances

which he extemporized for the purpose (Par. 284). Since his time^

the delicacy and accuracy of electrical apparatus have been

immensely increased, yet the most careful and refined observations

serve merely to afford stronger confirmation of his conclusions.

The importance of this law can not be over-estimated. In the

study and application of electricity it is fundamental and in one

form or another it is met at every turn. On account of its very

simplicity there is sometimes a failure to recognize that it is

unique, and occasionally it is spoken of as "self evident." Such

is far from being the case. There is no material substance which

follows such a law. Pressure causes liquids and gases to flow

through pipes, yet if this pressure be doubled the flow is by no

means doubled.

When applying the law to a more or less complex circuit, E

represents the total E. M. F. and R the total resistance. Thus there

may be several cells or batteries or electrical machines contrib-

uting to the E. M. F., in which case the sum of the E. M. F.s

must be taken. Again, through error or by design a cell or battery

may be reversed so as to oppose the remaining E. M. F. Such

opposing E. M. F. is spoken of as counter E. M. F. or back E. M. F.

Back E. M. F. is also produced by polarization (Par. 198) and,

as we shall see later, by the operation of motors in the circuit. In

224

ELEMENTS OF ELECTRICITY.

summing up the total E. M. F. of the circuit, back E. M. F. is to

be considered as negative. The resistance R includes not only

the resistance of the line but also that of the contacts, joints and

connections and of the electrolyte and elements of the cells. The

law can therefore be given

T E' + E" + E'" + E"" + (fee.

R' 4- R" + E'" + R"" + &c.

or the current

in the circuit is equal to the algebraic sum of the separate

E. M. F.s divided by the sum of the separate resistances.

298. Drop of Potential. The three quantities, current, electro-

motive force and resistance are bound together by Ohm's law so

that any two being given, the third may be determined. It may

at first sight appear unnecessary to state such a self-evident truth

but it is desirable to lay especial emphasis upon the fact for, until

the student has become familiar with the law, the tendency is

rather to restrict its use to the determination of current only.

The law may be put in the form

E = IR

and it is helpful to the

beginner if he will accustom himself to interpret this as meaning

that E is the electro-motive force necessary to drive a current

of strength / through a resistance R.

>

V ]

D" F"

r~<~

I

D 1

t ^^^

1

\^ i

1 F ^ B

Fig. 121.

Suppose AB (Fig. 121) to represent a portion of an electric

circuit, the point A being of higher potential than B, and suppose

that by means of one of the instruments to be described later

(Chapter 34) we measure the difference in potential between

A and B. Lay off on some convenient scale A A' proportional

to this difference of potential. If we move along A B to some

point D and measure the difference of potential between D and B

VOLTAIC ELECTRICITY. 225

we will find it to be less than that at A, or represented by DD'~

Likewise, at F this difference of potential is still smaller and is

proportional to FF', that is, as we move from A towards B the

difference of potential between the successive points and B

steadily grows less, or there is a falling off from the difference of

potential represented by A A'. At D, for example, this drop of

potential is D"D' and at F it is F"F'.

The drop of potential between any two points is always equal

to the product of the current into the resistance between the points.

Certain elementary applications of this principle will be shown

in the following paragraphs.

299. Ohm's Law Applies to Any Portion of the Circuit. In Par.

297 we saw that Ohm's law was applicable to the entire circuit

even though this be made complex by including heterogeneous

resistances and sources of E. M. F. It also applies to any portion

of a circuit, that is, the current flowing between any two points

in a circuit is equal to the difference of potential between these

two points divided by the resistance between them. We have seen

(Par. 229) that the current at every cross-section of a circuit is the

same; if, therefore, we determine it at one point we have it for

any other point. Knowing the current, if we have the resistance

between two points we can, by what we have shown in the pre-

ceding paragraph, determine the difference of potential, or drop,

between the two points. These principles enable us to solve a

variety of problems. For example, let A BCD, Fig. 122, repre-

sent part of an electric circuit. The resistance of the portion AB is

12 ohms, that of the incandescent lamp EC is 220 ohms, that of

CD is 8 ohms. The difference of potential between A and B is 6

volts. What current is flowing in the circuit and what is the poten-

tial of the points A, B and C if that of D be taken as zero?

A 12, OHMS B

-" *

.^^ 220 OHMS

8 OHMS

Fig. 122.'

V fi 1

The current between A and ^ = ^ = To = o am P ere > which is

also the current for the rest of the circuit. The drop from B to C =

226 ELEMENTS OF ELECTRICITY.

1^ = ^X220 = 110 volts; that from C to D = ^X8 = 4 volts. The

potential of C is therefore 4 volts, that of B is 114, and that of A

is 120.

300. Division of Current in Divided Circuit. This principle of

drop of potential furnishes a simple determination of the division

of an electric current in a divided circuit.

I' R'

A j I"

R"

D

r

R 1 "

Fig. 123.

Let Fig. 123 represent a divided circuit of three branches whose

resistances are respectively R', R" and R'". Call the correspond-

ing currents /', I" and /"' '. The current in the main branch upon

arriving at A divides into these three portions which reunite at B.

The drop from A to B is the same over each of the three routes,

therefore

I'R'=I"R"=T"R'"

which may be written

f - f" 1'"

R' ' R" ' R"'

that is, the current

in the branches of a divided circuit are to each other inversely as the

resistances of the respective branches.

In making an actual calculation, if the fractions in the second

member of this proportion be brought to a common denominator,

their numerators indicate at once the relation between the several

currents.

If there be but two branches, the above becomes

i - . -*-

R' ' R

which may be written

/' \l" = R" \R'

a somewhat simple

form for calculations.

301. Shunts. In practical electricity it frequently becomes

necessary to employ a divided circuit of two branches which must

jt . ji

' "

VOLTAIC ELECTRICITY. 227

be so proportioned that the main current divides between them

in accordance with some desired ratio. For example, suppose that

we wish to measure a current which is much larger than can be

measured directly by the instruments at our disposal. If we can

arrange a divided circuit so that exactly one-hundredth of the

total current flows through one branch, we can measure this

small current and always know that the entire current is one

hundred times greater. This division is brought about by shunts

(Par. 293).

In Fig. 124 we desire to measure the current flowing in AD.

G is our measuring instrument which with its connecting wires,

BG and GC has a resistance of R ohms. EC is the shunt. What

must be the resistance of the shunt so that one-hundredth of the

total current will flow through G ?

Fig. 124.

Call the current through the instrument /; that through BC

will be 997. If x be the resistance of BC, then, as shown in the

preceding paragraph

/ X R = 997 X x

whence _ R

~99

or the resistance of the shunt

must be one-ninety-ninth of the resistance of G and its connecting

wires and leads. In a similar manner we can determine the

resistance of shunts to bring about division of the total current in

any desired ratio. It is to be noted that these shunts are con-

structed for use with a particular instrument and cannot be used

with another of different resistance.

302. Rheostats. A consideration of Ohm's law, I = E/R, will

show that by varying R we can vary the current inversely and

suggests that by introducing or removing resistance from a cir-

cuit we may regulate the current at. will. Instruments for this

purpose are called rheostats. The principle of their use will be

228 ELEMENTS OF ELECTRICITY.

understood from the diagram (Fig. 125). A series of metal con-

tacts are arranged upon the arc of a circle DE and connected

between these contacts are resistance coils. Pivoted at the center

B

t>f the arc is a metal arm CD which can be moved about over the

contacts. Suppose the current to come in by A. As represented

in the figure, it must now traverse all the coils from D to E before

it can leave by B, and it is therefore cut down. Had the arm CD

been still farther to the left, the circuit would have been broken

entirely, R would have been infinite and the current zero. As the

arm is slid around to E the coils are successively cut out, the

resistance correspondingly reduced, and the current correspond-

ingly increased, reaching its maximum when the arm reaches E.

The controller by which the motorman starts and stops a trolley

car is similar in principle.

It will be shown later that regulation of current by rheostat is

a wasteful method and except for temporary purposes, such as for

starting and stopping motors, should not be employed.

303. KirchoflPs Laws. Where an electric circuit is composed

of interlacing branches and especially where there are in it several

seats of electro-motive force, confusion and uncertainty may

arise as to the correct way of applying Ohm's law in the determina-

tion of the separate currents and pgtentials. To obviate this,

Kirchoff has formulated a set of rules which render this applica-

tion almost mechanical. These are:

I. // several conductors meet at a common point, the algebraic sum

of the currents in these conductors is zero.

This is but a statement of the fact that electricity does not

accumulate at a point and that therefore as much flows away as

flows to the point. If currents flowing to the point be considered

positive, those flowing away must be regarded as negative.

II. // two or more conductors form a closed figure, or a mesh in a

network of conductors, the sum of the products of each current of this

VOLTAIC ELECTRICITY.

229

mesh into the resistance through which it passes is equal to the algebraic

sum of the electro-motive forces acting around this same mesh.

This is another statement of the fact that the total drop of

potential in going around a closed circuit is equal to the sum of

the partial drops. The convention must be adopted that in going

around a closed circuit, if the E. M. F. acting in a clockwise direc-

tion be considered positive, that acting in the opposite direction

is negative.

304. Example of Application of Kirchoff's Laws. By combin-

ing these laws it is always possible to obtain as many independent

equations as there are unknown quantities and hence these

unknown quantities may be determined. The following concrete

example will make the matter clear. Fig. 126 represents a net-

work of conductors in two of the branches of which there are bat-

teries E and F, sources of E. M. F. The currents in the separate

branches are designated ii, i 2 , i z , etc., and their assumed direction

is indicated by the arrows. In the final solution, a negative value

of a current indicates that the actual direction is opposite to that

assumed. The E. M. F. of the batteries and the resistances of the

branches are indicated on the diagram. We are required to deter-

mine the currents in the separate branches.

From Kirchoff's first law we obtain the following "point

equations" :

point a

point 6

point c

point d

ii i s =

-f i- i\ =

i* is =

-f it ib =

230 ELEMENTS OF ELECTRICITY.

p

From the second law we obtain the following "voltage

equations":

mesh iii&t 5ii + 2iz 4- 3i 4 = 3

mesh fciaie 3i 3 4t 6 2^ =

mesh iii&s 4i 6 + 4^ 5 3i 4 = 2

mesh w 3 ^5 5ii -h 3i 3 -f- 44 = 3 2

We now have eight independent equations from which to deter-

mine six unknown quantities, and the remainder of the process is

but a matter of combination and elimination.

305. Lost Volts and Useful Volts. Should there be connected

up to a circuit of resistance R a cell whose E. M. F. is E and in-

ternal resistance r, the resulting current would be given by the

expression

R + r

which may be written

E = IR + Ir

Interpreting this as explained in Par. 298, we see that a part of

the E. M. F. of the cell is spent in driving the current through the

external resistance R and the remainder in driving this current

through the internal resistance of the cell. The volts, Ir, con-

sumed on the interior of the cell are called the lost volts and we

profit only by those upon the external circuit, or IR, which are

therefore called the available or useful volts. Since the less the lost

volts, the more the useful volts, it is of importance to keep the

former at a minimum. Ir may be reduced in two ways; by reduc-

ing the current or by decreasing the internal resistance. If there

be no current, there is of course no wastage. The internal resist-

ance may be reduced by selecting an electrolyte of low resistance

(though usually choice is restricted), by bringing the plates closer

together, and by increasing the size of the plates (Par. 294). Lost

volts have also to be considered in the operation of electrical

machinery.

306. Short Circuit. The commonest source of injury to elec-

trical machinery is a short circuit, which may be defined as the

removal, usually accidental, of the greater part of the resistance

from a "live" circuit.

VOLTAIC ELECTRICITY. 231

v

Suppose B (Fig. 127) to represent a battery supplying current

for the incandescent lamp L. The internal resistance of the bat-

tery is almost negligible, the resistance of the wires should be very

4.

Fig. 127.

small. Suppose the E. M. F. of the battery to be 111 volts, the

resistance of the lamp to be 220 ohms and that of the battery

and wires to be 2 ohms. The current is

1 = 220 + 2 = 2 am P ere

If by some accident the wire should sag, as shown by the dotted

line, and touch the lower wire at P, at that instant the current

would be short circuited through the point P, the resistance of the

lamp and of the wire beyond P being eliminated. The current is

now

I = -= = HI amperes

or it has suddenly in-

creased over two hundred times. If the wires had been designed

to carry only ten or fifteen amperes they will be fused, apparatus

in the circuit will be "burned out," insulation will be charred and

possibly fires started. To avoid the injury resulting from such

accidents, use is made of fuses, pieces of soft, easily-fusible wire

inserted in the circuit which is to be protected. If the current

exceeds that which the fuses are intended to carry, they melt

before damage is done to the rest of the circuit. This same pro-

tection is also afforded by certain automatic apparatus called

overload switches (Par. 414).

307. Definitions Based Upon Ohm's Law. Since the three

quantities 7, E and R are bound together by Ohm's law, any one

may be defined in terms of the other two. Thus the ampere is

sometimes defined as the current produced by an E. M. F. of one

volt applied to a conductor whose resistance is one ohm. So also

the volt is defined as that E. M. F. which applied to a resistance of

one ohm will produce in it a current of one ampere. The ohm may

232 ELEMENTS OF ELECTRICITY.

be similarly defined but such definition adds but little to our

knowledge.

Since Ohm's law may be written

and since E and / fluctuate

together so that R remains always constant, the resistance of a

conductor is defined by some writers as the ratio of the difference

of potential of the ends of the conductor to the current produced

in it. To define a property as a ratio is not altogether satisfactory.

It is perhaps better to say that this ratio affords a measure of the

resistance.

VOLTAIC ELECTRICITY. 233

CHAPTER 26.

MEASUREMENT OF RESISTANCE.

308. Measurement of Resistance. One of the most important

classes of measurements with which the electrician has to deal is

that of resistance. Logically, this subject should have been taken

up in connection with that of resistance in Chapter 24, but the

methods employed could not be clearly presented until after the

consideration of Ohm's law and the explanation, as given in Chap-

ter 25, of the principles of the drop of potential and the division

of current in a divided circuit. Even now we shall have to antici-

pate certain principles which can not be fully developed until

later.

In these measurements, the methods to be employed vary with

the amount and character of the resistance. Thus, very high and

very low resistances are measured in a different way from those

covering a moderate range. Again, the measurement of the in-

ternal resistance of cells and of the resistance of electrolytes must

be undertaken in an entirely different manner from that of a

metallic conductor. These facts will be brought out in the follow-

ing pages.

309. Drop of Potential Proportional to Resistance Passed

Over. If there exists between AB (Fig. 128), two points of a cir-

cuit, a difference of potential E, there will be a flow of electricity

from the point of higher potential to that of lower. The value of

this current as given by Ohm's law is I = E/R, whence E=IR,

which last expression, as we have already seen (Par. 298), may be

interpreted as expressing the fact that E is the electro-motive

force required to drive a current of strength / through the con-

ductor of resistance R.

234 ELEMENTS OF ELECTRICITY.

To drive the same current through a resistance only one-half as

great requires only one-half as much E. M. F., or, if the resistance

of AM be one-half of the total resistance between A and B, then

one-half of the total E. M. F. will be expended in driving the cur-

rent from A to M, and the difference of potential between M and

B is only one-half of that between A and B. In more general

terms, for a constant current, the expenditure of E. M. F., or the

drop of potential, is directly proportional to the resistance passed

over.

310. Measurement of Resistance by Drop of Potential.

Should we have at our disposal a known resistance and an instru-

ment for measuring difference of potential (Chapter 34), the fore-

going affords us a means of measuring the resistance between any

two points in a circuit. For example, suppose that the resistance

R between A and M, Fig. 128, be known and that we desire to

determine the resistance x between M and B. We have simply

to measure with our instrument the drop E' between A and M,

and the drop E" between M and B. From the preceding paragraph

E' :E" = R:x

E"

whence x = -^7 R

This method supposes the current to be constant during the two

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46