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motive force and the current was first enunciated by Dr.

G. S. Ohm, of Berlin, in. 1827. It has since been known

as Ohm's La'.''.

If E be the E.M.F. between two points of a conductor

and I the current flowing through it, 'then if suitable

units be chosen,

where R is a quantity called the resistance of the con-

ductor; it is independent of the value and direction of the

current flowing, and depends only on the material of

the conductor, its length and sectional area, its tempera-

ture and state of strain.

The above equation is an expression of Ohm's law ; it is

usually written in the equivalent form.

'-5-

If the practical units now adopted internationally be

employed, this law may be expressed by saying that the

number of amperes flowing through a circuit is equal to

the number of volts of electromotive force divided by the

number of ohms of resistance.

When this formula is applied to the entire circuit, which

may contain several sources of E.M.F. of different signs,

274 ELECTRICITY AND MAGNETISM.

and both metallic and electrolytic resistances, it is not

quite so simple to apply. There are then several electro-

motive forces, some tending to produce a flow in one

direction and some in the other ; and a number of different

resistances each obstructing the flow, whether it takes place

in one direction or the other. Then

J_

~

Each E.M.F. must be taken with its proper sign. Resist-

ance is not a directed quantity. If, for example, there are

several voltaic cells in the circuit, some of them may be

connected in the wrong direction so that they oppose

the current ; or ' the circuit may include electrolytic or

storage cells or motors, which offer resistance in the form

of a counter E.M.F. All such electromotive forces must

be reckoned as negative.

. 220. Resistance. Resistance is that property of a

conductor in virtue of which the energy of a current is

converted into heat. It is independent of the direction

of the current, and the transformation into heat occasioned

by it is an irreversible one ; that is, there is no tendency for

the heat-energy to revert to the energy of an electric

current.

The practical unit of resistance is the ohm. It is repre-

srntcd by the resistance offered to an unvarying ele,c,tiic

current by a column of mercury at the temperature of

melting ice, 14.4521 grammes in mass, of a constant cross-

sectional area and of a length of 106.3 centimetres. This

statement is equivalent to a cross-sectional area of one

square millimetre.

OHM'S LAU' AND ITS APPLICATIONS. 275

221. Laws of Resistance. The resistances of diverse

conductors are found to conform to the following laws :

(1) The resistance of a uniform conductor is directly

proportional to its length.

(2) The resistance of a uniform conductor is inversely

proportional to its cross-sectional area. The resistances of

round wires are therefore inversely proportional to the

squares of their diameters.

(3) The resistance of a uniform conductor of given

length and cross-section depends upon the material of

which it consists. This property is called its specific re-

Distance.

222. Specific Resistance. A definite meaning may

be given to specific resistance by conceiving the material

to be in the form of a centi-

metre cube (Fig. Ill), a

cube whose edges are 1 era.

iu length. The specific re-

sistance is the resistance

which this cube opposes to Fig m

the passage of a current

from one face a to the opposite one b. If the conductor

is a cylinder 1 cm. long with parallel ends of one square

centimetre area, the resistance from a to b is the same

as that of the cube. The specific resistance may be rep-

resented by s. Then the following formula expresses all

the laws of resistance :

Is

r = -,

a

where I is the length of the conductor in centimetres, and

a its sectional area in square centimetres. A table of

specific resistances is given in the Appendix, Table IV.

/

276 ELECTRICITY AND MAGNETISM.

223. Conductivity. - - The inverse of a resistance is

called conductivity, or sometimes conductance. A conductor

whose resistance is r ohms has a conductivity equal to 1/r.

When a number of conductors are joined in parallel with

one another, the conductivity of the whole is the sum of

their several conductivities. Let two conductors of resist-

ances-ai|^tnd r.> be joined in parallel between the points A

anal* (Fig. 112). Let V l and V. 2 be the potentials of

A and B respectively. Then since P, V 2 equals the

E.M.F., we have by Ohm's law

_ht _ " _i_ -E

r iv .r/

The first member of this equation is the total current,

which is equal to the sum of the currents through the two

branches ; and r is the combined resistance of the two con-

ductors in parallel. Hence

Similar reasoning applies to any number of parallel con-

ductors.

From the last equation,

224. Effect of Heat on Resistance. - The resistance

of metallic conductors in general increases when the tern-

OHM'S LAW AND ITS APPLICATIONS. 277

perature rises. If R,) is the resistance of a conductor at

C. and R t at t C., then the equation

expresses the relation between the two through a consid-

erable range of temperature. The constant a is called the

temperature coefficient. For most pure metals it is about

(L_per cent for one degree C., or ^0 per cent for a range

of 100 degrees of temperature. The temperature coeffi-

cient for pure copper between 20 and 250 C. was found

by Kennelly and Fessenden to be 0.00406. Dewar and

Fleming have measured the resistances of pure metals in

liquid oxygen at temperatures of 182 and _-.197 C.,

and have shown that the resistance of all of them decreases

with fall of temperature as if it would become zero at

273 C., the zero of the absolute scale. They would

then offer no obstruction to the passage of a current, how-

ever great. Pure copper is the best known conductor, but

it is only slightly better than silver.

The temperature coefficient of alloys is smaller than

that of pure metals. German silver has a coefficient only

about one-tenth as great as that of copper; while that of

platinoid is only one-half as great as that of German silver.

Manganin, an alloy of manganese, copper, and nickel, has at

certain temperatures a small negative temperature coeffi-

cient ; that is, its resistance diminishes slightly as the 1

temperature rises. * {

The resistance of carbon and of electrolytes decreases

when the temperature rises. Thus, the resistance of an

incandescent lamp filament is only about half as great at

normal incandescence as when cold. Solutions of ZnS0 4

and of CuSO^ have a temperature coefficient somewhat

over 0.02, or 2 per cent for one degree C.

278 ELECTRICITY AND MAGNETISM.

225. Loss of Potential proportional to Resistance.

If V } arid V\ are the potentials of two points A and B

on a conductor, then by Ohm's law

It is obvious from this equation that the potential differ-

ence between any two points on a conductor through

which a constant current is flowing is proportional to the

resistance between them, provided the conductor is not

the seat of an E.M.F. Even when electromotive forces

are encountered, the loss of potential, when a given cur-

rent flows through a resistance,

is proportional to that resistance.

If another point be taken be-

tween A and B so situated that

the resistance between it and B

is one-half the resistance be-

tween A and B, then the poten-

tial difference between this point

' r and B is also reduced in the

Fig 113.

same ratio.

Let the distances measured along Or represent resist-

ances (Fig. 113), and those along Ov, potentials. Then

AP equals FJ and BQ, V z \ also PQ stands for the resist-

ance R between the points A and B on the conductor.

Join A and B and let BC be drawn parallel to Or; then

will AO be equal to Fi K, the potential difference

between the points A and B. The slope of the line AB

represents the rate at which the potential drops along the

resistance R. Moreover, since

it is evident that the tangent of the angle of slope equals

the strength of the current.

OHM'S LAM' AXD ITS APPLICATIONS.

279

The principle that the loss of potential is proportional

to the resistance passed over, when the current is constant,

is one of very frequent application in electrical measure-

ments.

226. Wheatstone's Bridge. The instrument known

as a Wheatstone's bridge illustrates the use made of the

principle of the last article. It is a combination of resist-

ances more commonly used than any other method for

the comparison of two of them. It consists of six conduc-

tors connecting four points ; in one of these conductors is

a source of E.M.F., and in another branch is a galvanom-

eter, or sensitive current detector.

Let A, B, (7, D (Fig. 114),

be the four points, B' the bat-

tery, and 6r the galvanometer.

Then since the fall of poten-

tial between A and D is the

same by the path ABD as by

ACD, there must be a point

B on the former which has

the same potential as the

point C on the latter. If the

circuit through the galvanometer is made to connect these

two equipotential points, no current will floAv through it.

Let /, be current through R { ; it will also be the current

through R-, because none flows through the galvanometer,

and the same quantity of electricity flows toward B as

away from it. Also, let I, be the current through the

branch ACD. Then, the potential difference between A

and B being the same as that between A and (7, we have

by Ohm's law (219)

(a)

Fig. 114.

280 ELECTRICITY AND MAGNETISM.

Similarly, R, I, = RJ, (6)

Dividing (a) by (6), ~ - .

This equation may be written

or R, : R 2 :: R, : R 4 .

When therefore the resistances are so adjusted that no

current flows through the galvanometer, the four form a

proportion. In practice three of the resistances are fixed,

and the adjustment for a balance is made by varying the

fourth. It is necessary to know only the ratio R^ / R A , for

example ; then the equation gives the relation between

R } and R 2 .

227. Cells joined in Series. Let there be n similar

voltaic cells, each having an electromotive force E and an

internal resistance between the terminals of the cell equal

to r. Then if R is the external resistance, by Ohm's law

_JP

: +

The n cells may be joined in series by connecting the

negative of the first with the positive of the second ; the

negative of the second with the positive of the third, and

so on. Then the total E.M.F. between the positive of the

first and the negative of the last will be nE, and the entire

internal resistance will be nr. Hence

/- nE

R + nr

If R is small in comparison with r, then 1= E/r nearly, or

OHM'S LA\V AM) ITS APPLICATIONS.

281

the current is no greater than could be obtained from one

cell. But if R is large in comparison with r, or even wr,

then the current is nearly n times as great as one cell alone

will yield.

228. Graphical Representation of Potentials for

Cells in Series. Let there be three cells in series ; and

let AB (Fig. 115) represent 3r, the internal resistance of

the three. Also let

BO equal the exter-

nal resistance R on

the same scale. Be-

ginning at A, erect a

perpendicular Al

equal to E, the E.^I.

F. of one of the cells.

Suppose the E.M.F.

to originate at the

surface of the zinc.

Then as the current flows across through the liquids over

the resistance r there will be a fall of potential represented

by the sloping line be. At , the zinc of the second cell, there

is a sudden rise of potential <?r7, equal to Ab, and then a fall

from d to e ; at e there is a third rise, represented by ef ;

then another drop from/ to g over the internal resistance

of the last cell. The potential difference between the ter-

minals of the battery is then Bg, and this is the loss of

potential over the external resistance R.

The line AD represents 3^7, and DF is the loss of poten-

tial in the three cells on account of their internal resist-

ance. Then

37? E'

Fig. 115.

282 ELECTRICITY AND MAGNETISM.

Since the tangent of the angle of slope is the numerical

value of the strength of current, it is evident that the

lines be, de, and/<? must slope at the same angle as DC, or

must be parallel to one another, because the current lias

the same value in every part of the circuit.

If the external resistance were made infinite by opening

the circuit, the line D would become horizontal, and the

current zero. Also, with any given external resistance,

the less the internal resistance the less the difference

between ZE and E'.

229. Cells joined in Parallel. A battery is said to

be connected in parallel, or in multiple, when all the pos-

itive terminals are joined together, and likewise all the

negatives. The chief object aimed at is the reduction

of the internal resistance. In the case of storage cells,

which have a very low resistance, they may be joined in

parallel when it is desired to use a larger current than the

normal discharge current for one cell. With several cells

in parallel, the current through the external circuit is

divided among them.

If n similar cells are connected in parallel, the E.M.F.

is the same as for a single cell, but there are n internal

paths of equal resistance through the cells, and the result-

ant internal resistance is r/n. Hence

n

In case R is small in comparison with r, the reduction of

the internal resistance secured by joining the n colls in

parallel results in a larger current, but no such result

follows for a large external resistance. For the latter

condition the cells should be in series.

OHM'S LAir AM) ITS APPLICATORS. 283

23O. Cells in Multiple Series. Let there be m series

of n cells each, the m series being joined in parallel. The X

whole number of cells is then mn. The current will be

/- nE E

m n m

To find the condition for a maximum current it may be

remarked that the product of the two terms in the denomi-

nator of the last expression is Rr/nm, a constant. R and

r are assumed to be constant, and nm is the whole number

of cells. But when the product of two terms is a con-

stant, their sum is least when they are equal to each other,

or when R/n r/m. For this condition

m

But R is the external resistance and nr/m is the internal

resistance. For the greatest steady current, therefore, the

cells should be so arranged that the resulting internal

resistance shall be equal to the external resistance. The

efficiency may then be said to be 50 per cent, since half

the energy is wasted internally and half may be utilized

externally. This relation does not hold if there is a

counter E.M.F. in the circuit.

231. Variation of Internal Resistance with Current.

- The internal resistance of a given cell is not a fixed

quantity. It changes with the operation of the cell, on

account of the chemical changes going on which alter the

composition of the liquids. It is also dependent on the

current drawn from the cell. The larger the current,

the smaller is the measured internal resistance. The

284

ELECTRICITY AND MAGNETISM.

curves of Fig. 116 represent graphically the relation be-

tween the internal resistance and the current for two

particular cells. The lower curve was made from obser-

Amperes

.02

.04 .06

.08

.10 .12 .14.

Fig. 116.

.16

.20

vations on an old " dry cell," and the upper one from

observations on a Daniell cell. The scale for the internal

resistance of the latter is twice as large as for the former.

The dry cell showed a most remarkable fall in the resist-

ance as the current increased.

PROBLEMS.

1. Three Daniell cells are connected in series; the E.M.F. of

each cell is 1.1 volts and the internal resistance 2 ohms; if the

external resistance is 5 ohms, find the current.

2. Two Leclanche cells are joined in parallel ; each has an E.M.F.

of 1.5 volts and an internal resistance of 4 ohms. If the external

resistance consists of two parallel conductors of 2 and 3 ohms

respectively, find the current through each branch

OHM'S LAW AM) ITS APPLICATION >. !>">

3. Deduce the formula for the resistance of three conductors in

parallel.

4. Three Bunsen cells are connected in series with one another

and with one copper oxide cell, the latter with its poles set the wrong

way round. If the internal resistance of the Bunsens is 0.5 ohm

each and that, of the other cell 0.2, find the current through an

external resistance of 3 ohms (201).

5. Two equal masses of copper are drawn into wire, one 10

metres long and the other 15 metres. If the resistance of the shorter

piece is 0.4 ohm, find that of the longer.

6. Three wires are joined in parallel ; their resistances are 30,

20, and 60 ohms. Find the resultant resistance.

7. The resistance between two points A and B of a circuit is

25 ohms ; on joining another wire in parallel between A and B the

resistance becomes 20 ohms. Find the resistance of the second wire.

8. The terminals of a battery of five Grove cells in series, the

total K.M.F. of which is 9.5 volts, are connected by three wires,

each of 12 ohms resistance. If the current through each wire is

one-third of an ampere, find the internal resistance of each cell.

9. Given 24 cells, each of 1 volt E.M.F. and 0.5 ohm internal

resistance. How should they be connected to give a maximum cur-

rent through an external resistance of 3 ohms? What will be the

current ?

10. What is the resistance of a column of mercury 212.6 cms.

long and 0.5 of a square millimetre in cross-section, at a tempera-

ture of 25 C. ? Temperature coefficient of mercury, 0.072 per cent

per degree C.

286 ELECTRICITY AND MAGNETISM.

CHAPTER XIX.

THERMAL RELATIONS.

232. Conversion of Electric Energy into Heat. -

Electric energy is readily convertible into other forms.

If an electric current encounters a back E.M.F. anywhere

in the circuit, work will be done by the passage of the

current against this opposing E.M.F. Such is the case in

electrolysis and in the storage battery. All the energy of

an electric current not so converted, or stored up in some

form of stress, is dissipated as heat. Heat appears

wherever the circuit offers resistance to the current. In a

simple circuit containing no devices for transforming and

storing energy, all of it is frittered away as heat. Part

of it disappears in heating the battery or other generator,

and the remainder in heating the external circuit.

The heat evolved by dissolving 33 gms. of zinc in sul-

phuric acid Favre found to be 18,682 calories. When the

same weight of zinc was consumed in a Smee cell, the heat

evolved in the entire circuit was 18,674 calories. These

operations were conducted by introducing the vessel con-

taining the zinc and acid in the first case, and the Smee cell

and its circuit in the second case, into a large calorimeter.

The two quantities are nearly identical, or the heat evolved

is the same whether the solution of the zinc produces a

current or not. When the electric current was employed

to do work in lifting a weight, the heat generated in the

circuit was diminished by the exact thermal equivalent of

THERM A L KEL A TIONS.

287

the work done. When, therefore, a definite amount of

chemical action takes place in a battery and no work is

done, the distribution of the heat is altered, but not its

amount.

233. Laws of the Development of Heat. The laws

of the development of heat in an electric circuit were dis-

covered experimentally by Joule and

Lenz. The latter experimented with a

simple calorimeter represented in Fig. 117.

A thin platinum wire, joined to two stout

conductors, was enclosed in a wide-

mouthed bottle containing alcohol. A

thermometer t was passed through a hole

in the insulating stopper of the bottle.

The resistance of the fine wire was known,

and the observations consisted in measur-

ing the current and noting the rise of

temperature. Joule found that the num-

U-r of units of heat generated in a con-

ductor is proportional

(1) To its resistance.

(2) To the square of the strength of the current.

(3) To the length of time the current flows.

234. The Heat Equivalent of a Current. Let the

potentials of two points A and B of a conductor be \\ and

i nd let Q units of electricity be transferred from A to

B in the time t. Then the work done, expressed in ergs,

will be

Fig. 117.

If all this work is converted into heat, W^JH, by (86) ;

if the current of strength T flows for time /, the quantity

288 ELECTRICITY AND MAGNETISM.

Q = It, since the strength of current is the quantity passing

any section of the conductor in one second; also Fj F* =

RL Substituting,

JH= PRt,

jr PRt PRt

and H=

J 4.19 xlO 7

The current strength and the resistance are expressed in

C.G.S. electromagnetic units (294). An ampere is 10~ l

C.G.S. unit ; an ohm, 10'*. If the measurements are made

in amperes and ohms, then for I- must be substituted

7 2 xlO~' 2 , and for 7, R x 10". Thy equation then becomes

ff= PU x :- 10 - t = PRt x 0.24.

4.19xlO :

The energy expended per second is the product of the cur-

rent strength and the electromotive force. If Zbe measured

in amperes and E in volts (a volt is 10 s C.G.S. units), then

W= IE x 10- 1 x 10 s = IEx 10 7 ergs per second,

or IE watts (I., 43). But

ff= I'Rx 0.24 = IE x 0.24 calories per second.

Therefore one watt is equivalent to 0.24 calorie per second.

235. Counter E.M.F. in a Circuit. The total activ-

ity, or rate at which a generator is supplying energy to

the circuit, is represented in part by the heat evolved in

accordance with Joule's law and in part by work done, such

as chemical decomposition by electrolysis, the mechanical

work of a motor, etc. In every case of doing work the

energy absorbed is a function of the current strength in-

stead of its square. We may therefore write for the whole

energy transformed in time t

THERMAL RELATION*. 289

The first term of the second member of this equation is the

waste in heat ; the second, the work done; A is a constant.

Dividing through by It and transposing,

T _E-A

~^r

R is the entire resistance of the circuit. It is evident from

the form of the equation that the quantity A is of the

nature of an E.M.F. Since it is affected by the negative

sign it is a counter E.M.F. The effective E.M.F. produc-

ing the current is the applied E.M.F. less the back E.M.F.

This counter E.M.F. is" a necessary phenomenon in every

case in which work is done by an electric current.

236. Division of the Energy in a Circuit. If the

counter E.M.F. be represented by E\ the equation for the

current by Ohm's law is

I= E-E'

II

But the heat waste in watts is

PR = I (E-E') = IE- IE'.

Now IE is the total activity in the portion of the circuit

considered. The heat generated in this same portion of

the circuit of resistance R is less than the entire activity

by IE' watts. Hence the energy spent per second in doing

work is the product of the current strength and the counter

E.M.F.

The ratio of the work done to the heat waste is

IE' E'

EE''

The efficiency with which electric energy is converted into

work increases therefore with the counter E.M.F.

290 ELECTRICITY AND MAGNETISM.

237. Applications of the Heating Effect of a Current.

Of the various applications of heating, the following are

some of the more important:

1. Electric Cautery. A thin platinum wire heated to

incandescence is sometimes employed in surgery instead

of a knife. Platinum is used because it is infusible, except

at a high temperature, and is not corrosive.

2. Safety Fuses. Advantage is taken of the low tem-

perature of fusion of some alloys, in which lead is a large

constituent, for the purpose of automatically interrupting

the circuit when for any reason the current becomes ex-

cessive. Some of these alloys, notably those containing

zinc, may oxidize on heating ; and if the current be in-

creased slowly the fused metal may become encased in

the oxide as an envelope, and be heated to redness with-

out breaking the circuit. Safety fuses should be mounted

on non-combustible bases; their length should be pro-

portioned to the voltage employed on the circuit in which

they are placed. Provision is sometimes made for an

automatic blast, produced by the explosive vaporization

of the metal, to blow out the arc which is formed between

the terminals when the fuse " blows."

3. Electric Heating. Electric street-cars are sometimes

heated by a current through suitable iron-wire resistance

embedded in cement, asbestos, or enamel. Similar de-

vices for cooking have now become articles of commerce.

G. S. Ohm, of Berlin, in. 1827. It has since been known

as Ohm's La'.''.

If E be the E.M.F. between two points of a conductor

and I the current flowing through it, 'then if suitable

units be chosen,

where R is a quantity called the resistance of the con-

ductor; it is independent of the value and direction of the

current flowing, and depends only on the material of

the conductor, its length and sectional area, its tempera-

ture and state of strain.

The above equation is an expression of Ohm's law ; it is

usually written in the equivalent form.

'-5-

If the practical units now adopted internationally be

employed, this law may be expressed by saying that the

number of amperes flowing through a circuit is equal to

the number of volts of electromotive force divided by the

number of ohms of resistance.

When this formula is applied to the entire circuit, which

may contain several sources of E.M.F. of different signs,

274 ELECTRICITY AND MAGNETISM.

and both metallic and electrolytic resistances, it is not

quite so simple to apply. There are then several electro-

motive forces, some tending to produce a flow in one

direction and some in the other ; and a number of different

resistances each obstructing the flow, whether it takes place

in one direction or the other. Then

J_

~

Each E.M.F. must be taken with its proper sign. Resist-

ance is not a directed quantity. If, for example, there are

several voltaic cells in the circuit, some of them may be

connected in the wrong direction so that they oppose

the current ; or ' the circuit may include electrolytic or

storage cells or motors, which offer resistance in the form

of a counter E.M.F. All such electromotive forces must

be reckoned as negative.

. 220. Resistance. Resistance is that property of a

conductor in virtue of which the energy of a current is

converted into heat. It is independent of the direction

of the current, and the transformation into heat occasioned

by it is an irreversible one ; that is, there is no tendency for

the heat-energy to revert to the energy of an electric

current.

The practical unit of resistance is the ohm. It is repre-

srntcd by the resistance offered to an unvarying ele,c,tiic

current by a column of mercury at the temperature of

melting ice, 14.4521 grammes in mass, of a constant cross-

sectional area and of a length of 106.3 centimetres. This

statement is equivalent to a cross-sectional area of one

square millimetre.

OHM'S LAU' AND ITS APPLICATIONS. 275

221. Laws of Resistance. The resistances of diverse

conductors are found to conform to the following laws :

(1) The resistance of a uniform conductor is directly

proportional to its length.

(2) The resistance of a uniform conductor is inversely

proportional to its cross-sectional area. The resistances of

round wires are therefore inversely proportional to the

squares of their diameters.

(3) The resistance of a uniform conductor of given

length and cross-section depends upon the material of

which it consists. This property is called its specific re-

Distance.

222. Specific Resistance. A definite meaning may

be given to specific resistance by conceiving the material

to be in the form of a centi-

metre cube (Fig. Ill), a

cube whose edges are 1 era.

iu length. The specific re-

sistance is the resistance

which this cube opposes to Fig m

the passage of a current

from one face a to the opposite one b. If the conductor

is a cylinder 1 cm. long with parallel ends of one square

centimetre area, the resistance from a to b is the same

as that of the cube. The specific resistance may be rep-

resented by s. Then the following formula expresses all

the laws of resistance :

Is

r = -,

a

where I is the length of the conductor in centimetres, and

a its sectional area in square centimetres. A table of

specific resistances is given in the Appendix, Table IV.

/

276 ELECTRICITY AND MAGNETISM.

223. Conductivity. - - The inverse of a resistance is

called conductivity, or sometimes conductance. A conductor

whose resistance is r ohms has a conductivity equal to 1/r.

When a number of conductors are joined in parallel with

one another, the conductivity of the whole is the sum of

their several conductivities. Let two conductors of resist-

ances-ai|^tnd r.> be joined in parallel between the points A

anal* (Fig. 112). Let V l and V. 2 be the potentials of

A and B respectively. Then since P, V 2 equals the

E.M.F., we have by Ohm's law

_ht _ " _i_ -E

r iv .r/

The first member of this equation is the total current,

which is equal to the sum of the currents through the two

branches ; and r is the combined resistance of the two con-

ductors in parallel. Hence

Similar reasoning applies to any number of parallel con-

ductors.

From the last equation,

224. Effect of Heat on Resistance. - The resistance

of metallic conductors in general increases when the tern-

OHM'S LAW AND ITS APPLICATIONS. 277

perature rises. If R,) is the resistance of a conductor at

C. and R t at t C., then the equation

expresses the relation between the two through a consid-

erable range of temperature. The constant a is called the

temperature coefficient. For most pure metals it is about

(L_per cent for one degree C., or ^0 per cent for a range

of 100 degrees of temperature. The temperature coeffi-

cient for pure copper between 20 and 250 C. was found

by Kennelly and Fessenden to be 0.00406. Dewar and

Fleming have measured the resistances of pure metals in

liquid oxygen at temperatures of 182 and _-.197 C.,

and have shown that the resistance of all of them decreases

with fall of temperature as if it would become zero at

273 C., the zero of the absolute scale. They would

then offer no obstruction to the passage of a current, how-

ever great. Pure copper is the best known conductor, but

it is only slightly better than silver.

The temperature coefficient of alloys is smaller than

that of pure metals. German silver has a coefficient only

about one-tenth as great as that of copper; while that of

platinoid is only one-half as great as that of German silver.

Manganin, an alloy of manganese, copper, and nickel, has at

certain temperatures a small negative temperature coeffi-

cient ; that is, its resistance diminishes slightly as the 1

temperature rises. * {

The resistance of carbon and of electrolytes decreases

when the temperature rises. Thus, the resistance of an

incandescent lamp filament is only about half as great at

normal incandescence as when cold. Solutions of ZnS0 4

and of CuSO^ have a temperature coefficient somewhat

over 0.02, or 2 per cent for one degree C.

278 ELECTRICITY AND MAGNETISM.

225. Loss of Potential proportional to Resistance.

If V } arid V\ are the potentials of two points A and B

on a conductor, then by Ohm's law

It is obvious from this equation that the potential differ-

ence between any two points on a conductor through

which a constant current is flowing is proportional to the

resistance between them, provided the conductor is not

the seat of an E.M.F. Even when electromotive forces

are encountered, the loss of potential, when a given cur-

rent flows through a resistance,

is proportional to that resistance.

If another point be taken be-

tween A and B so situated that

the resistance between it and B

is one-half the resistance be-

tween A and B, then the poten-

tial difference between this point

' r and B is also reduced in the

Fig 113.

same ratio.

Let the distances measured along Or represent resist-

ances (Fig. 113), and those along Ov, potentials. Then

AP equals FJ and BQ, V z \ also PQ stands for the resist-

ance R between the points A and B on the conductor.

Join A and B and let BC be drawn parallel to Or; then

will AO be equal to Fi K, the potential difference

between the points A and B. The slope of the line AB

represents the rate at which the potential drops along the

resistance R. Moreover, since

it is evident that the tangent of the angle of slope equals

the strength of the current.

OHM'S LAM' AXD ITS APPLICATIONS.

279

The principle that the loss of potential is proportional

to the resistance passed over, when the current is constant,

is one of very frequent application in electrical measure-

ments.

226. Wheatstone's Bridge. The instrument known

as a Wheatstone's bridge illustrates the use made of the

principle of the last article. It is a combination of resist-

ances more commonly used than any other method for

the comparison of two of them. It consists of six conduc-

tors connecting four points ; in one of these conductors is

a source of E.M.F., and in another branch is a galvanom-

eter, or sensitive current detector.

Let A, B, (7, D (Fig. 114),

be the four points, B' the bat-

tery, and 6r the galvanometer.

Then since the fall of poten-

tial between A and D is the

same by the path ABD as by

ACD, there must be a point

B on the former which has

the same potential as the

point C on the latter. If the

circuit through the galvanometer is made to connect these

two equipotential points, no current will floAv through it.

Let /, be current through R { ; it will also be the current

through R-, because none flows through the galvanometer,

and the same quantity of electricity flows toward B as

away from it. Also, let I, be the current through the

branch ACD. Then, the potential difference between A

and B being the same as that between A and (7, we have

by Ohm's law (219)

(a)

Fig. 114.

280 ELECTRICITY AND MAGNETISM.

Similarly, R, I, = RJ, (6)

Dividing (a) by (6), ~ - .

This equation may be written

or R, : R 2 :: R, : R 4 .

When therefore the resistances are so adjusted that no

current flows through the galvanometer, the four form a

proportion. In practice three of the resistances are fixed,

and the adjustment for a balance is made by varying the

fourth. It is necessary to know only the ratio R^ / R A , for

example ; then the equation gives the relation between

R } and R 2 .

227. Cells joined in Series. Let there be n similar

voltaic cells, each having an electromotive force E and an

internal resistance between the terminals of the cell equal

to r. Then if R is the external resistance, by Ohm's law

_JP

: +

The n cells may be joined in series by connecting the

negative of the first with the positive of the second ; the

negative of the second with the positive of the third, and

so on. Then the total E.M.F. between the positive of the

first and the negative of the last will be nE, and the entire

internal resistance will be nr. Hence

/- nE

R + nr

If R is small in comparison with r, then 1= E/r nearly, or

OHM'S LA\V AM) ITS APPLICATIONS.

281

the current is no greater than could be obtained from one

cell. But if R is large in comparison with r, or even wr,

then the current is nearly n times as great as one cell alone

will yield.

228. Graphical Representation of Potentials for

Cells in Series. Let there be three cells in series ; and

let AB (Fig. 115) represent 3r, the internal resistance of

the three. Also let

BO equal the exter-

nal resistance R on

the same scale. Be-

ginning at A, erect a

perpendicular Al

equal to E, the E.^I.

F. of one of the cells.

Suppose the E.M.F.

to originate at the

surface of the zinc.

Then as the current flows across through the liquids over

the resistance r there will be a fall of potential represented

by the sloping line be. At , the zinc of the second cell, there

is a sudden rise of potential <?r7, equal to Ab, and then a fall

from d to e ; at e there is a third rise, represented by ef ;

then another drop from/ to g over the internal resistance

of the last cell. The potential difference between the ter-

minals of the battery is then Bg, and this is the loss of

potential over the external resistance R.

The line AD represents 3^7, and DF is the loss of poten-

tial in the three cells on account of their internal resist-

ance. Then

37? E'

Fig. 115.

282 ELECTRICITY AND MAGNETISM.

Since the tangent of the angle of slope is the numerical

value of the strength of current, it is evident that the

lines be, de, and/<? must slope at the same angle as DC, or

must be parallel to one another, because the current lias

the same value in every part of the circuit.

If the external resistance were made infinite by opening

the circuit, the line D would become horizontal, and the

current zero. Also, with any given external resistance,

the less the internal resistance the less the difference

between ZE and E'.

229. Cells joined in Parallel. A battery is said to

be connected in parallel, or in multiple, when all the pos-

itive terminals are joined together, and likewise all the

negatives. The chief object aimed at is the reduction

of the internal resistance. In the case of storage cells,

which have a very low resistance, they may be joined in

parallel when it is desired to use a larger current than the

normal discharge current for one cell. With several cells

in parallel, the current through the external circuit is

divided among them.

If n similar cells are connected in parallel, the E.M.F.

is the same as for a single cell, but there are n internal

paths of equal resistance through the cells, and the result-

ant internal resistance is r/n. Hence

n

In case R is small in comparison with r, the reduction of

the internal resistance secured by joining the n colls in

parallel results in a larger current, but no such result

follows for a large external resistance. For the latter

condition the cells should be in series.

OHM'S LAir AM) ITS APPLICATORS. 283

23O. Cells in Multiple Series. Let there be m series

of n cells each, the m series being joined in parallel. The X

whole number of cells is then mn. The current will be

/- nE E

m n m

To find the condition for a maximum current it may be

remarked that the product of the two terms in the denomi-

nator of the last expression is Rr/nm, a constant. R and

r are assumed to be constant, and nm is the whole number

of cells. But when the product of two terms is a con-

stant, their sum is least when they are equal to each other,

or when R/n r/m. For this condition

m

But R is the external resistance and nr/m is the internal

resistance. For the greatest steady current, therefore, the

cells should be so arranged that the resulting internal

resistance shall be equal to the external resistance. The

efficiency may then be said to be 50 per cent, since half

the energy is wasted internally and half may be utilized

externally. This relation does not hold if there is a

counter E.M.F. in the circuit.

231. Variation of Internal Resistance with Current.

- The internal resistance of a given cell is not a fixed

quantity. It changes with the operation of the cell, on

account of the chemical changes going on which alter the

composition of the liquids. It is also dependent on the

current drawn from the cell. The larger the current,

the smaller is the measured internal resistance. The

284

ELECTRICITY AND MAGNETISM.

curves of Fig. 116 represent graphically the relation be-

tween the internal resistance and the current for two

particular cells. The lower curve was made from obser-

Amperes

.02

.04 .06

.08

.10 .12 .14.

Fig. 116.

.16

.20

vations on an old " dry cell," and the upper one from

observations on a Daniell cell. The scale for the internal

resistance of the latter is twice as large as for the former.

The dry cell showed a most remarkable fall in the resist-

ance as the current increased.

PROBLEMS.

1. Three Daniell cells are connected in series; the E.M.F. of

each cell is 1.1 volts and the internal resistance 2 ohms; if the

external resistance is 5 ohms, find the current.

2. Two Leclanche cells are joined in parallel ; each has an E.M.F.

of 1.5 volts and an internal resistance of 4 ohms. If the external

resistance consists of two parallel conductors of 2 and 3 ohms

respectively, find the current through each branch

OHM'S LAW AM) ITS APPLICATION >. !>">

3. Deduce the formula for the resistance of three conductors in

parallel.

4. Three Bunsen cells are connected in series with one another

and with one copper oxide cell, the latter with its poles set the wrong

way round. If the internal resistance of the Bunsens is 0.5 ohm

each and that, of the other cell 0.2, find the current through an

external resistance of 3 ohms (201).

5. Two equal masses of copper are drawn into wire, one 10

metres long and the other 15 metres. If the resistance of the shorter

piece is 0.4 ohm, find that of the longer.

6. Three wires are joined in parallel ; their resistances are 30,

20, and 60 ohms. Find the resultant resistance.

7. The resistance between two points A and B of a circuit is

25 ohms ; on joining another wire in parallel between A and B the

resistance becomes 20 ohms. Find the resistance of the second wire.

8. The terminals of a battery of five Grove cells in series, the

total K.M.F. of which is 9.5 volts, are connected by three wires,

each of 12 ohms resistance. If the current through each wire is

one-third of an ampere, find the internal resistance of each cell.

9. Given 24 cells, each of 1 volt E.M.F. and 0.5 ohm internal

resistance. How should they be connected to give a maximum cur-

rent through an external resistance of 3 ohms? What will be the

current ?

10. What is the resistance of a column of mercury 212.6 cms.

long and 0.5 of a square millimetre in cross-section, at a tempera-

ture of 25 C. ? Temperature coefficient of mercury, 0.072 per cent

per degree C.

286 ELECTRICITY AND MAGNETISM.

CHAPTER XIX.

THERMAL RELATIONS.

232. Conversion of Electric Energy into Heat. -

Electric energy is readily convertible into other forms.

If an electric current encounters a back E.M.F. anywhere

in the circuit, work will be done by the passage of the

current against this opposing E.M.F. Such is the case in

electrolysis and in the storage battery. All the energy of

an electric current not so converted, or stored up in some

form of stress, is dissipated as heat. Heat appears

wherever the circuit offers resistance to the current. In a

simple circuit containing no devices for transforming and

storing energy, all of it is frittered away as heat. Part

of it disappears in heating the battery or other generator,

and the remainder in heating the external circuit.

The heat evolved by dissolving 33 gms. of zinc in sul-

phuric acid Favre found to be 18,682 calories. When the

same weight of zinc was consumed in a Smee cell, the heat

evolved in the entire circuit was 18,674 calories. These

operations were conducted by introducing the vessel con-

taining the zinc and acid in the first case, and the Smee cell

and its circuit in the second case, into a large calorimeter.

The two quantities are nearly identical, or the heat evolved

is the same whether the solution of the zinc produces a

current or not. When the electric current was employed

to do work in lifting a weight, the heat generated in the

circuit was diminished by the exact thermal equivalent of

THERM A L KEL A TIONS.

287

the work done. When, therefore, a definite amount of

chemical action takes place in a battery and no work is

done, the distribution of the heat is altered, but not its

amount.

233. Laws of the Development of Heat. The laws

of the development of heat in an electric circuit were dis-

covered experimentally by Joule and

Lenz. The latter experimented with a

simple calorimeter represented in Fig. 117.

A thin platinum wire, joined to two stout

conductors, was enclosed in a wide-

mouthed bottle containing alcohol. A

thermometer t was passed through a hole

in the insulating stopper of the bottle.

The resistance of the fine wire was known,

and the observations consisted in measur-

ing the current and noting the rise of

temperature. Joule found that the num-

U-r of units of heat generated in a con-

ductor is proportional

(1) To its resistance.

(2) To the square of the strength of the current.

(3) To the length of time the current flows.

234. The Heat Equivalent of a Current. Let the

potentials of two points A and B of a conductor be \\ and

i nd let Q units of electricity be transferred from A to

B in the time t. Then the work done, expressed in ergs,

will be

Fig. 117.

If all this work is converted into heat, W^JH, by (86) ;

if the current of strength T flows for time /, the quantity

288 ELECTRICITY AND MAGNETISM.

Q = It, since the strength of current is the quantity passing

any section of the conductor in one second; also Fj F* =

RL Substituting,

JH= PRt,

jr PRt PRt

and H=

J 4.19 xlO 7

The current strength and the resistance are expressed in

C.G.S. electromagnetic units (294). An ampere is 10~ l

C.G.S. unit ; an ohm, 10'*. If the measurements are made

in amperes and ohms, then for I- must be substituted

7 2 xlO~' 2 , and for 7, R x 10". Thy equation then becomes

ff= PU x :- 10 - t = PRt x 0.24.

4.19xlO :

The energy expended per second is the product of the cur-

rent strength and the electromotive force. If Zbe measured

in amperes and E in volts (a volt is 10 s C.G.S. units), then

W= IE x 10- 1 x 10 s = IEx 10 7 ergs per second,

or IE watts (I., 43). But

ff= I'Rx 0.24 = IE x 0.24 calories per second.

Therefore one watt is equivalent to 0.24 calorie per second.

235. Counter E.M.F. in a Circuit. The total activ-

ity, or rate at which a generator is supplying energy to

the circuit, is represented in part by the heat evolved in

accordance with Joule's law and in part by work done, such

as chemical decomposition by electrolysis, the mechanical

work of a motor, etc. In every case of doing work the

energy absorbed is a function of the current strength in-

stead of its square. We may therefore write for the whole

energy transformed in time t

THERMAL RELATION*. 289

The first term of the second member of this equation is the

waste in heat ; the second, the work done; A is a constant.

Dividing through by It and transposing,

T _E-A

~^r

R is the entire resistance of the circuit. It is evident from

the form of the equation that the quantity A is of the

nature of an E.M.F. Since it is affected by the negative

sign it is a counter E.M.F. The effective E.M.F. produc-

ing the current is the applied E.M.F. less the back E.M.F.

This counter E.M.F. is" a necessary phenomenon in every

case in which work is done by an electric current.

236. Division of the Energy in a Circuit. If the

counter E.M.F. be represented by E\ the equation for the

current by Ohm's law is

I= E-E'

II

But the heat waste in watts is

PR = I (E-E') = IE- IE'.

Now IE is the total activity in the portion of the circuit

considered. The heat generated in this same portion of

the circuit of resistance R is less than the entire activity

by IE' watts. Hence the energy spent per second in doing

work is the product of the current strength and the counter

E.M.F.

The ratio of the work done to the heat waste is

IE' E'

EE''

The efficiency with which electric energy is converted into

work increases therefore with the counter E.M.F.

290 ELECTRICITY AND MAGNETISM.

237. Applications of the Heating Effect of a Current.

Of the various applications of heating, the following are

some of the more important:

1. Electric Cautery. A thin platinum wire heated to

incandescence is sometimes employed in surgery instead

of a knife. Platinum is used because it is infusible, except

at a high temperature, and is not corrosive.

2. Safety Fuses. Advantage is taken of the low tem-

perature of fusion of some alloys, in which lead is a large

constituent, for the purpose of automatically interrupting

the circuit when for any reason the current becomes ex-

cessive. Some of these alloys, notably those containing

zinc, may oxidize on heating ; and if the current be in-

creased slowly the fused metal may become encased in

the oxide as an envelope, and be heated to redness with-

out breaking the circuit. Safety fuses should be mounted

on non-combustible bases; their length should be pro-

portioned to the voltage employed on the circuit in which

they are placed. Provision is sometimes made for an

automatic blast, produced by the explosive vaporization

of the metal, to blow out the arc which is formed between

the terminals when the fuse " blows."

3. Electric Heating. Electric street-cars are sometimes

heated by a current through suitable iron-wire resistance

embedded in cement, asbestos, or enamel. Similar de-

vices for cooking have now become articles of commerce.

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