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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



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