Font size

the scale be graduated uniformly, the unit is immaterial, but a

millimeter scale running up to 2000 is very convenient. We will

tack the paper scale upon a board AB (Fig. 140), and stretch the

wire above it. To the end A of the wire we connect the negative

terminal of our cell C; the other terminal makes sliding contact

at P.

VOLTAIC ELECTRICITY. 249

If the E. M. F. of the cell be two volts, the difference of potential

between P and A before P is touched to the wire will be two volts

and after contact is made it will still be aboy^two volts. If it were

exactly two volts when P is at the 2000 division on the scale,

there would be a drop of two volts from P to A and each division

of the scale would correspond to a drop of one-thousandth of a

volt. If P be slid in towards A, these two volts will be spread

over a shorter length of the wire and each division on the scale

would correspond to a drop of more than one-thousandth of a volt.

On the other hand, if P be slid out from A, the scale divisions can

be made to correspond to less than one-thousandth of a volt,

therefore, by sliding P backwards and forwards we can vary the

drop over the scale and at one particular point this drop will be

exactly one-thousandth of a volt per millimeter. This point is

located as follows:

331. Calibration of Potentiometer. To the same end A of

our stretched wire we connect through a galvanometer G the

negative terminal of a standard cell S. If this be a Clark's cell

whose E. M. F. is 1.434 volts (Par. 212), we connect its positive

terminal to the wire at M, a point 1434 millimeters from A. If

M be at a higher potential than 1.434 volts a current will flow

from M to D, while if it be at a lower potential a current will flow

from D to M. In either case this flow will be indicated by a

deflection of the needle of the galvanometer G. If there be a flow,

we slide the contact P backwards or forwards until a point is

found where G indicates no current and we then know that the

potential of M is the same as that of D, that is, 1.434 volts, and

that consequently each division of the scale corresponds to a drop

of one-thousandth of a volt. The contact P is left at this point.

The instrument is now in adjustment so that the printed figures

on its scale read thousandths of a volt, in other words, it has been

calibrated.

332. Measurement with Potentiometer. To measure the

E. M. F. of a cell X, its negative terminal is connected through

the galvanometer H with A and its positive terminal is con-

nected to a contact T which is moved back and forth along the

wire until H indicates no current. Suppose this point to be

the 925th millimeter from A, then the E. M. F. of X is .925

volt.

250 ELEMENTS OF ELECTRICITY.

Instead of using a second galvanometer H, the negative ter-

minal of X could have been attached to G, that is, S and X can

use G in common.

333. Forms of Potentiometer. As in the case of the Wheat-

stone bridge, the actual instrument bears no resemblance at all

to the diagrammatic representation in Fig. 140. For example,

for the sake of compactness the long wire is wound in a helical

coil around an ebonite cylinder, etc., etc. There are numerous

forms of potentiometers but the principle of all is the same, that

is, they measure an unknown E. M. F. by balancing against it an

equal and opposite E. M. F. which latter is known.

VOLTAIC ELECTRICITY.

251

CHAPTER 28.

GROUPING OF CELLS IN BATTERIES.

334. Grouping of Cells. The cells composing a battery may

be connected up in several ways. If they are connected one after

the other they are said to be in series. If all of the positive poles

are connected to one common wire and all of the negative poles to

another, they are said to be in parallel. If they are divided into

groups, the cells in each group being connected in series and these

separate groups being then connected in parallel, the battery is

said to be grouped in multiple, or better, so many in parallel and

so many in series. For example, if we have ten cells we might

group them all in series, or all in parallel, or two abreast and five

deep, that is, two in parallel and five in series, or finally, five

abreast and two deep, that is, five in parallel and two in series.

Each of these arrangements is quite proper under certain condi-

tions but it will be shown in the following paragraphs that it is

not a matter of indifference which shall be employed.

335. Cells in Series. In Par. 192 we saw that in a voltaic cell

the copper or positive pole is at a higher potential than the zinc

or negative pole. Suppose that we have a number of simple cells,

each of an E. M. F. of one volt, and that we should arrange them

Fig. 141.

in series, the copper plate of each, as shown in Fig. 141, being

connected to the zinc plate of the adjoining one. The copper

plate of A and the zinc plate of B being connected are at a com-

252

ELEMENTS OF ELECTRICITY.

mon potential, therefore the zinc plate of B is one volt higher than

that of A. The copper plate of B being one volt higher than its

zinc plate is consequently two volts higher than the zinc plate of A.

Similarly, the copper plate of C is three volts higher than the zinc

plate of A, and in general the total E. M. F. of a number of similar

cells connected in series is equal to the E. M. F. of one cell multi-

plied by the number in series. This principle applies even though

the circuit includes cells of different kinds, electrical machines,

etc., and the most general statement is that in any electric circuit

containing several sources of E. M. F. in series the total E. M. F.

is the sum of the separate E. M. F.s.

336. Cells in Parallel. Fig. 142 represents three cells in

parallel. The three positive poles being brought together at a

common point A are all at the same potential, that is, one volt

higher than the three negative poles which are brought together

Cu

at B. This combination therefore has no greater E. M. F. than

has a single cell and it is in fact, as we have already seen (Par.

294), equivalent to a single cell whose copper and zinc plates are

three times as large as those of the original cells.

337. Comparison of Series and Parallel Groupings. We may

by a concrete example best illustrate the different effect of the

two kinds of groupings. Suppose that we have a number of cells,

each of an E. M. F. of 2 volts and an internal resistance of .25 ohm.

From a single cell in a circuit of negligible external resistance the

current obtainable is

O25

= 8

VOLTAIC ELECTRICITY. 253

With two in series, the E. M. F. is twice as great (Par. 335),

but also the resistance is twice as great (Par. 286), therefore the

current is the same, and so on for any number, that is, with a cir-

cuit of negligible external resistance the effect of grouping cells

in series is to increase the voltage but not the current. Should,

however, the external resistance be great, a different state of

affairs results. For example, let R = 100 ohms (the resistance of

about 6 miles of iron telegraph wire), then for one cell

100 + .25

and for two cells

1 = 100 + .50

The difference in these denominators being negligible, we see that

in this second case we have doubled the current.

If, starting again with negligible external resistance, we arrange

two of these cells in parallel, the E. M. F. is no greater than for

one cell (Par. 336) but the resistances of the two cells being in

parallel, the total resistance is only one-half that of one cell (Par.

293), hence the current is doubled. For three cells it is trebled,

and so on, that is, with negligible external resistance, the effect

of grouping cells in parallel is to increase the current but not the

voltage.

With a large external resistance, the grouping of cells in parallel,

since it does not increase the E. M. F. nor change the total resist-

ance to any significant extent, does not alter the current.

We may sum up by saying that with a large external resistance

we increase the current by grouping the cells in series; with a

small external resistance, we increase it by grouping them in

parallel.

338. Analogy Between Voltaic Cells and Pumps. Difference

of potential has been compared to difference of water level (Par.

70). Since a difference of potential is produced in a cell, we may

continue the comparison by drawing an analogy between a cell

and a pump. In Fig. 143 the pumps A and B are analogous to

two cells in parallel; the two lift the water no higher than a single

pump but they lift twice the quantity. The pumps C, D and E

254

ELEMENTS OF ELECTRICITY.

are analogous to cells in series; they lift no more water than a

single pump but they raise it three times as high.

Fig. 143.

339. Parallel -Series Grouping. Suppose we have N cells,

each of an E. M. F. of e volts and an internal resistance of r ohms,

and suppose that they are arranged (Fig. 144) s in series and p in

Fig. 144.

parallel in a circuit of external resistance R. The resulting E. M,

F. is equal to the E. M. F. of one cell multiplied by the number in

series (Par. 335) or se. The resistance of one of the series is rs,

but since there are p rows in parallel, the total internal resistance

* "p"

The current produced by this arrangement is

/= -^-

340. Maximum Current. The question may arise, given N

cells, how should they be grouped to obtain the maximum current?

VOLTAIC ELECTRICITY. 255

The expression for the current is given in the preceding paragraph

and, since N = sp, can be written

T - Ne

NR

s

If this be differentiated and the first differential coefficient be

placed equal to zero, the resulting values of s will correspond to

maximum or minimum values of /. This differentiation is tedi-

ous. However, since Ne is a positive constant, / will be a maxi-

mum when h rs, the denominator of the expression, is a

s

minimum.

Place x = -f rs = NRs~ l + rs

s

Differentiating

Placing this equal to zero, we have

NR , A /NR

s 2 = , whence s = d= y

which is the value

sought. This will in general only approximate to the desired

arrangement since the mathematical supposition is that s and p

are continuous variables, while actually they are both discon-

tinuous or positive whole numbers. For example, if N be 10,

the only possible values of s are 1, 2, 5 and 10, yet the actual

solution will generally produce some mixed number. In such a

case we should make the calculation of the current from the two

groupings which come nearest to the one indicated by the solution

and select accordingly.

If in the above equation of condition for maximum current

S * = NR

r

we substitute for N its value

sp, we get _ S pR

~T

whence R= r-

256

ELEMENTS OF ELECTRICITY.

But (Par. 339) R is the external resistance and r- is the

internal resistance, whence we arrive at the important conclusion

that the current is a maximum when the battery is so grouped that

the internal and the external resistances are equal.

We saw (Par. 305) that the useful volts of a cell or battery are

given by IR, the lost volts by Ir. Since in the case of maximum

current R =r, the lost volts amount to one-half of the total E. M. F.

341. In Multiple Arrangements Equal E. M. F. is Required

of Groups in Series. In a parallel-series arrangement the series

groups must all have the same E. M. F. This requires that where

the cells are all of one kind there should be the same number in

each series group. The arrangement shown in Fig. 145 is not

Fig. 145.

permissible. The battery should be quiescent when the key K

is open but the three cells now constitute a closed circuit in which

the E. M. F. of the two upper cells acting in a clockwise direction

is not counterbalancedyby the opposing E. M. F. of the single

cell. If the E. M. F. of each cell be e and its resistance r, there

will flow through the single cell a reverse current whose strength

e

s =

3?

The elements of the two upper cells will therefore

consume away and the zinc plate of the single cell will have copper

deposited upon it which will cause local action. With K closed,

the loss is not so- great and it will diminish as the external resist-

ance decreases, but even in this case the elements of the single

cell will consume away much more rapidly than those of the two

in series.

342. Diagrams of Parallel-Series Grouping. A parallel-series

grouping represented as in Fig. 144 doubtless aids the beginner,

but in actual practice cells are seldom arranged in this geometrical

order. Especially is this the case with storage batteries in which

VOLTAIC ELECTRICITY.

257

the cells are very heavy and are placed in rows on shelves or

benches. Reflection will show that after all it is not necessary to

move the cells themselves but rather to shift the connecting

wires. Thus in Fig. 146 eight cells are represented in four different

el

1 1 1 1 1

K fl'l'l'lfl'l'l'l 1

Fig. 146.

groupings, the cells themselves not being disturbed. In A they

are all in series, in B all in parallel, in C four in series and two in

parallel, and in D two in series and four in parallel.

343. Cost of Power from Primary Cells. For the small and

irregular currents required in telegraphy and in operating tele-

phones, call bells, annunciators, alarms, etc., a battery of primary

cells is the most suitable and economical source of electrical

energy, but where the current is required to furnish appreciable

mechanical power through suitable machines, the cost is pro-

hibitive. The chemicals consumed in the cell correspond to the

fuel consumed in the boiler of a steam engine, and while one

pound of carbon burned in air evolves enough heat to raise 8080

pounds of water 1 C, in round numbers four pounds of zinc must

combine with six pounds of sulphuric acid to produce the same

amount of heat. With modern machines electrical energy may

be produced as cheaply as one cent per horse-power per hour but

the same energy supplied from primary cells costs from 30 to

50 times as much. Where many telephones or telegraph lines

are operated from a central station it is now the practice to use

storage batteries instead of the batteries of Daniell or gravity

cells formerly employed.

ELECTRO-MAGNETICS. 259

PART IV.

ELECTRO-MAGNETICS.

CHAPTER 29.

MAGNETIC FIELD ABOUT A WIRE CARRYING

A CURRENT.

344. Oerstedt's Discovery. In 1819, in the course of a lecture

on electricity, Oerstedt, Professor of Physics at Copenhagen,

observed that when a wire carrying a current was brought near a

magnetic needle a deflection of the needle was produced. He

recognized at once the importance of this discovery as demon-

strating what up to that time had been merely, a conjecture, that

is, that there existed some connection between electricity and

magnetism. He set to work immediately to investigate the matter

and soon discovered not only that an electric current produced a

deflection of a magnetic needle near it but that the direction of

this deflection depended both upon the direction in which the

current was flowing and upon the position of the conductor with

reference to the needle. His results were announced in 1820.

The news reached the French electrician Ampere on September 11

and was received by him with eagerness. Within one week there-

after he had repeated Oerstedt's experiments and had added to

the latter's discoveries; had confirmed by specially devised experi-

ments and had presented in a paper to the Academy a complete

theory of the new science of electro-dynamics (Par. 360) .

345. Right Hand Rule for Deflection of Needle. It is helpful

to the electrician, whether he be an advanced student or only a

beginner, to have some easy rule for determining, or some mechan-

ical way of remembering, in which direction certain phenomena

take place. Thus Ampere gave the rule of the "swimming man"

by which, the relative positions of the conductor and of the needle

and the direction of the current being given, the direction in

260

ELEMENTS OF ELECTRICITY.

which the north end of the needle would move could be predicted.

Other rules have been given by subsequent writers. Of these,

the following is thought to be the most useful, both because of

its simplicity, it being a true "rule of thumb," and because, as will

be shown later, of its applicability to a number of varied con-

ditions. It should be committed to memory.

Place the palm of the right hand upon the wire, the extended

fingers pointing in the direction of the flow of the current, the palm

turned towards the needle; the extended thumb will indicate the

direction in which the north pole of the needle will move.

Fig. 147 represents the application of this rule. The current

flowing in the wire in the direction indicated by the arrow will

Fig. 147.

cause the north pole of the needle to move out in the direction in

which the thumb is pointing.

If the wire be below, in order that the palm should be turned

towards the needle, the hand must be held back down, in which

case the thumb will point away from the observer and this is the

direction in which the north pole will actually move. In fact, the

rule is perfectly general and applies if the wire be vertical and

in front of either pole or if it be to either side of the needle, only

in this last case the needle must be capable of movement in a

vertical plane.

346. Magnetic Field About a Wire Carrying a Current. In

Pars. 143 and 144 it was shown that a needle in a magnetic field

tends to turn so as to place its longer axis and its own lines of

force parallel to the lines of force of the field. The needle in

Oerstedt's experiment turns for the same reason, that is, the cur-

rent flowing through the wire establishes about this wire a mag-

netic field with which the needle tends to coincide in direction.

ELECTRO-MAGNETICS.

261

This field may be studied in a similar manner to the other

magnetic fields already described. If a vertical wire, a portion

of an electric circuit, be passed through a hole in the center of a

horizontal sheet of cardboard

or of glass which has been

sprinkled with iron filings (Fig.

148) and if the circuit be then

closed and the horizontal sheet

be tapped while the current is

flowing, the filings will be seen

to gather and form in more or

less distinct circles around the

wire as a center. The lines of

force of the field are circles,

and it was shown first by

Fig. 148.

Ampere that these circles lie

in planes perpendicular to the

wire. In Oerstedt's experiment as described in Par. 344, the needle

can never place itself at right angles to the wire, tor the controlling

force, the horizontal component of the earth's magnetism, is

always effective. However, if a perfectly balanced needle be

mounted so that its axis of rotation is parallel to the earth's field,

then this field has no influence upon its rotation and if Oerstedt's

experiment be now performed, the needle will always set itself

at right angles to the wire.

u

Fig. 149.

347. Direction of Field. The experiment with the iron filings

shows the lines of force of the field to be circles but does not

indicate their direction. This latter may be determined as follows.

262 ELEMENTS OF ELECTRICITY.

Using the same horizontal sheet and vertical wire as in the pre-

ceding experiment, distribute at equal distances apart on the

circumference of a circle whose center lies on the wire a number

of small compasses, A, B, C, D (Fig. 149). Before the circuit is

closed, these all point in the same direction. Let us assume that

this is the direction indicated by the needle A. Suppose now the

circuit to be closed and the current to flow down the wire. The

needle at A will not change its position, C will be entirely reversed,

B will point to the right and D to the left, that is, if we look at

the needles from above, they point around the circle in the direc-

tion of the motion of the hands of a clock. Had the current

flowed up, the needles would all have pointed in a counter-clock-

wise direction around the circle.

348. Clock Rule for Direction of Field. The foregoing experi-

ment suggests another simple rule for determining the direction

of the field about a wire carrying a current.

Suppose the eye placed so as to look along the wire in the direction

in which the current is flowing; the positive direction of the field

about the wire is the same as the direction of motion of the hands

of a clock.

Of course, if the current is flowing towards the eye, the field is

counter-clockwise. This rule should not supplant the right hand

rule given in Par. 345. Either one could be used to the exclusion

of the other but it is better to have both at command.

349. Wire Carrying a Current is not Itself a Magnet. Although

surrounded by a magnetic field, a wire carrying a current is not

itself a magnet. If a clean copper wire through which a current

is flowing be dipped into iron filings and then lifted, the filings

will cluster around the wire but will drop off when the current is

broken. At first sight this seems to indicate that the wire has

become magnetized but it can be shown that such is not the case.

When the wire is thrust into the filings they become magnetized,

since magnetic bodies placed in a magnetic field become magnets

(Par. 120), and if they surround the wire, or if any of them adhere

to it through stickiness, they cling together like the links of a

chain and really adhere to each other instead of to the wire. If

an elongated filing be placed at right angles to the wire and with

its ends lying upon one of the circular lines of force surrounding

the wire, one of these ends will be urged in one direction around

ELECTRO-MAGNETICS.

263

the circle, the other end in the opposite direction; the result is

that the filing will move broadside towards the wire. There is,

however, no radial component between a wire carrying a current

and a magnetic pole in its field. In this respect the field about a

conductor is unique. While all other forces exerted between

bodies act along the line joining the bodies, the force upon a pole

in a field about a wire acts at right angles to the line joining the

wire and the pole.

350. Rotation of a Magnetic Pole by a Current. In Par. 135

the positive direction of a magnetic field was defined as that direc-

tion in which a free north pole would move. Such a pole released

near the north end of a magnet would move off along a line of

force, curving around until it came to rest against the south face.

The statement was made (Par. 142) that a magnetic line of force

is a closed curve, but the moving

pole can not travel around a com-

plete orbit for its progress is arrested

by the material substance of the

magnet. In the field about a wire

carrying a current the case is dif-

ferent. Here the lines of force are

circles, return upon themselves and

do not necessarily pass through any

solid body. A pole released in such a

field should therefore rotate as long

as the field is maintained. Although

we can not obtain a free pole, we

can approximate to the theoretical

condition by arranging a circuit so

that only one pole of the magnet

lies in the field and we can thus

produce mechanical rotation.

Fig. 150 represents diagram-

niatically such an arrangement. NS

Fig. 150.

is a magnet bent in the center at an angle and placed upon the

pivot P about which it is free to rotate. B is a little cup of

mercury into which dips the conductor AB, thereby securing

movable electric contact with a minimum of friction. CD is an

annular cup of mercury surrounding but not touching the magnet.

From B a wire BD is carried over and bent down so as just to

264

ELEMENTS OF ELECTRICITY.

touch the surface of the mercury at D and to sweep along this

surface as the magnet rotates. DE is a conductor leading away

from the annular cup. If the current enters at A, it goes to B,

thence to D and out by E. It therefore passes by the pole N but

not by the pole S. According to the rule given in the preceding

paragraph, the field about AB, viewed from A, is clockwise. The

pole N will therefore spin 'around in the direction shown by the

dotted line. If the current be reversed the direction of rotation

is also reversed; so also if the magnet be inverted, the direction

of rotation is reversed.

351. Rotation of a Current by a Magnetic Pole. The reaction

between the pole and the field being mutual, it follows that if

the pole be fixed and the conductor be free

to move, the latter may be made to rotate

about the former. This may be shown by

the apparatus represented in Fig. 151. NS

millimeter scale running up to 2000 is very convenient. We will

tack the paper scale upon a board AB (Fig. 140), and stretch the

wire above it. To the end A of the wire we connect the negative

terminal of our cell C; the other terminal makes sliding contact

at P.

VOLTAIC ELECTRICITY. 249

If the E. M. F. of the cell be two volts, the difference of potential

between P and A before P is touched to the wire will be two volts

and after contact is made it will still be aboy^two volts. If it were

exactly two volts when P is at the 2000 division on the scale,

there would be a drop of two volts from P to A and each division

of the scale would correspond to a drop of one-thousandth of a

volt. If P be slid in towards A, these two volts will be spread

over a shorter length of the wire and each division on the scale

would correspond to a drop of more than one-thousandth of a volt.

On the other hand, if P be slid out from A, the scale divisions can

be made to correspond to less than one-thousandth of a volt,

therefore, by sliding P backwards and forwards we can vary the

drop over the scale and at one particular point this drop will be

exactly one-thousandth of a volt per millimeter. This point is

located as follows:

331. Calibration of Potentiometer. To the same end A of

our stretched wire we connect through a galvanometer G the

negative terminal of a standard cell S. If this be a Clark's cell

whose E. M. F. is 1.434 volts (Par. 212), we connect its positive

terminal to the wire at M, a point 1434 millimeters from A. If

M be at a higher potential than 1.434 volts a current will flow

from M to D, while if it be at a lower potential a current will flow

from D to M. In either case this flow will be indicated by a

deflection of the needle of the galvanometer G. If there be a flow,

we slide the contact P backwards or forwards until a point is

found where G indicates no current and we then know that the

potential of M is the same as that of D, that is, 1.434 volts, and

that consequently each division of the scale corresponds to a drop

of one-thousandth of a volt. The contact P is left at this point.

The instrument is now in adjustment so that the printed figures

on its scale read thousandths of a volt, in other words, it has been

calibrated.

332. Measurement with Potentiometer. To measure the

E. M. F. of a cell X, its negative terminal is connected through

the galvanometer H with A and its positive terminal is con-

nected to a contact T which is moved back and forth along the

wire until H indicates no current. Suppose this point to be

the 925th millimeter from A, then the E. M. F. of X is .925

volt.

250 ELEMENTS OF ELECTRICITY.

Instead of using a second galvanometer H, the negative ter-

minal of X could have been attached to G, that is, S and X can

use G in common.

333. Forms of Potentiometer. As in the case of the Wheat-

stone bridge, the actual instrument bears no resemblance at all

to the diagrammatic representation in Fig. 140. For example,

for the sake of compactness the long wire is wound in a helical

coil around an ebonite cylinder, etc., etc. There are numerous

forms of potentiometers but the principle of all is the same, that

is, they measure an unknown E. M. F. by balancing against it an

equal and opposite E. M. F. which latter is known.

VOLTAIC ELECTRICITY.

251

CHAPTER 28.

GROUPING OF CELLS IN BATTERIES.

334. Grouping of Cells. The cells composing a battery may

be connected up in several ways. If they are connected one after

the other they are said to be in series. If all of the positive poles

are connected to one common wire and all of the negative poles to

another, they are said to be in parallel. If they are divided into

groups, the cells in each group being connected in series and these

separate groups being then connected in parallel, the battery is

said to be grouped in multiple, or better, so many in parallel and

so many in series. For example, if we have ten cells we might

group them all in series, or all in parallel, or two abreast and five

deep, that is, two in parallel and five in series, or finally, five

abreast and two deep, that is, five in parallel and two in series.

Each of these arrangements is quite proper under certain condi-

tions but it will be shown in the following paragraphs that it is

not a matter of indifference which shall be employed.

335. Cells in Series. In Par. 192 we saw that in a voltaic cell

the copper or positive pole is at a higher potential than the zinc

or negative pole. Suppose that we have a number of simple cells,

each of an E. M. F. of one volt, and that we should arrange them

Fig. 141.

in series, the copper plate of each, as shown in Fig. 141, being

connected to the zinc plate of the adjoining one. The copper

plate of A and the zinc plate of B being connected are at a com-

252

ELEMENTS OF ELECTRICITY.

mon potential, therefore the zinc plate of B is one volt higher than

that of A. The copper plate of B being one volt higher than its

zinc plate is consequently two volts higher than the zinc plate of A.

Similarly, the copper plate of C is three volts higher than the zinc

plate of A, and in general the total E. M. F. of a number of similar

cells connected in series is equal to the E. M. F. of one cell multi-

plied by the number in series. This principle applies even though

the circuit includes cells of different kinds, electrical machines,

etc., and the most general statement is that in any electric circuit

containing several sources of E. M. F. in series the total E. M. F.

is the sum of the separate E. M. F.s.

336. Cells in Parallel. Fig. 142 represents three cells in

parallel. The three positive poles being brought together at a

common point A are all at the same potential, that is, one volt

higher than the three negative poles which are brought together

Cu

at B. This combination therefore has no greater E. M. F. than

has a single cell and it is in fact, as we have already seen (Par.

294), equivalent to a single cell whose copper and zinc plates are

three times as large as those of the original cells.

337. Comparison of Series and Parallel Groupings. We may

by a concrete example best illustrate the different effect of the

two kinds of groupings. Suppose that we have a number of cells,

each of an E. M. F. of 2 volts and an internal resistance of .25 ohm.

From a single cell in a circuit of negligible external resistance the

current obtainable is

O25

= 8

VOLTAIC ELECTRICITY. 253

With two in series, the E. M. F. is twice as great (Par. 335),

but also the resistance is twice as great (Par. 286), therefore the

current is the same, and so on for any number, that is, with a cir-

cuit of negligible external resistance the effect of grouping cells

in series is to increase the voltage but not the current. Should,

however, the external resistance be great, a different state of

affairs results. For example, let R = 100 ohms (the resistance of

about 6 miles of iron telegraph wire), then for one cell

100 + .25

and for two cells

1 = 100 + .50

The difference in these denominators being negligible, we see that

in this second case we have doubled the current.

If, starting again with negligible external resistance, we arrange

two of these cells in parallel, the E. M. F. is no greater than for

one cell (Par. 336) but the resistances of the two cells being in

parallel, the total resistance is only one-half that of one cell (Par.

293), hence the current is doubled. For three cells it is trebled,

and so on, that is, with negligible external resistance, the effect

of grouping cells in parallel is to increase the current but not the

voltage.

With a large external resistance, the grouping of cells in parallel,

since it does not increase the E. M. F. nor change the total resist-

ance to any significant extent, does not alter the current.

We may sum up by saying that with a large external resistance

we increase the current by grouping the cells in series; with a

small external resistance, we increase it by grouping them in

parallel.

338. Analogy Between Voltaic Cells and Pumps. Difference

of potential has been compared to difference of water level (Par.

70). Since a difference of potential is produced in a cell, we may

continue the comparison by drawing an analogy between a cell

and a pump. In Fig. 143 the pumps A and B are analogous to

two cells in parallel; the two lift the water no higher than a single

pump but they lift twice the quantity. The pumps C, D and E

254

ELEMENTS OF ELECTRICITY.

are analogous to cells in series; they lift no more water than a

single pump but they raise it three times as high.

Fig. 143.

339. Parallel -Series Grouping. Suppose we have N cells,

each of an E. M. F. of e volts and an internal resistance of r ohms,

and suppose that they are arranged (Fig. 144) s in series and p in

Fig. 144.

parallel in a circuit of external resistance R. The resulting E. M,

F. is equal to the E. M. F. of one cell multiplied by the number in

series (Par. 335) or se. The resistance of one of the series is rs,

but since there are p rows in parallel, the total internal resistance

* "p"

The current produced by this arrangement is

/= -^-

340. Maximum Current. The question may arise, given N

cells, how should they be grouped to obtain the maximum current?

VOLTAIC ELECTRICITY. 255

The expression for the current is given in the preceding paragraph

and, since N = sp, can be written

T - Ne

NR

s

If this be differentiated and the first differential coefficient be

placed equal to zero, the resulting values of s will correspond to

maximum or minimum values of /. This differentiation is tedi-

ous. However, since Ne is a positive constant, / will be a maxi-

mum when h rs, the denominator of the expression, is a

s

minimum.

Place x = -f rs = NRs~ l + rs

s

Differentiating

Placing this equal to zero, we have

NR , A /NR

s 2 = , whence s = d= y

which is the value

sought. This will in general only approximate to the desired

arrangement since the mathematical supposition is that s and p

are continuous variables, while actually they are both discon-

tinuous or positive whole numbers. For example, if N be 10,

the only possible values of s are 1, 2, 5 and 10, yet the actual

solution will generally produce some mixed number. In such a

case we should make the calculation of the current from the two

groupings which come nearest to the one indicated by the solution

and select accordingly.

If in the above equation of condition for maximum current

S * = NR

r

we substitute for N its value

sp, we get _ S pR

~T

whence R= r-

256

ELEMENTS OF ELECTRICITY.

But (Par. 339) R is the external resistance and r- is the

internal resistance, whence we arrive at the important conclusion

that the current is a maximum when the battery is so grouped that

the internal and the external resistances are equal.

We saw (Par. 305) that the useful volts of a cell or battery are

given by IR, the lost volts by Ir. Since in the case of maximum

current R =r, the lost volts amount to one-half of the total E. M. F.

341. In Multiple Arrangements Equal E. M. F. is Required

of Groups in Series. In a parallel-series arrangement the series

groups must all have the same E. M. F. This requires that where

the cells are all of one kind there should be the same number in

each series group. The arrangement shown in Fig. 145 is not

Fig. 145.

permissible. The battery should be quiescent when the key K

is open but the three cells now constitute a closed circuit in which

the E. M. F. of the two upper cells acting in a clockwise direction

is not counterbalancedyby the opposing E. M. F. of the single

cell. If the E. M. F. of each cell be e and its resistance r, there

will flow through the single cell a reverse current whose strength

e

s =

3?

The elements of the two upper cells will therefore

consume away and the zinc plate of the single cell will have copper

deposited upon it which will cause local action. With K closed,

the loss is not so- great and it will diminish as the external resist-

ance decreases, but even in this case the elements of the single

cell will consume away much more rapidly than those of the two

in series.

342. Diagrams of Parallel-Series Grouping. A parallel-series

grouping represented as in Fig. 144 doubtless aids the beginner,

but in actual practice cells are seldom arranged in this geometrical

order. Especially is this the case with storage batteries in which

VOLTAIC ELECTRICITY.

257

the cells are very heavy and are placed in rows on shelves or

benches. Reflection will show that after all it is not necessary to

move the cells themselves but rather to shift the connecting

wires. Thus in Fig. 146 eight cells are represented in four different

el

1 1 1 1 1

K fl'l'l'lfl'l'l'l 1

Fig. 146.

groupings, the cells themselves not being disturbed. In A they

are all in series, in B all in parallel, in C four in series and two in

parallel, and in D two in series and four in parallel.

343. Cost of Power from Primary Cells. For the small and

irregular currents required in telegraphy and in operating tele-

phones, call bells, annunciators, alarms, etc., a battery of primary

cells is the most suitable and economical source of electrical

energy, but where the current is required to furnish appreciable

mechanical power through suitable machines, the cost is pro-

hibitive. The chemicals consumed in the cell correspond to the

fuel consumed in the boiler of a steam engine, and while one

pound of carbon burned in air evolves enough heat to raise 8080

pounds of water 1 C, in round numbers four pounds of zinc must

combine with six pounds of sulphuric acid to produce the same

amount of heat. With modern machines electrical energy may

be produced as cheaply as one cent per horse-power per hour but

the same energy supplied from primary cells costs from 30 to

50 times as much. Where many telephones or telegraph lines

are operated from a central station it is now the practice to use

storage batteries instead of the batteries of Daniell or gravity

cells formerly employed.

ELECTRO-MAGNETICS. 259

PART IV.

ELECTRO-MAGNETICS.

CHAPTER 29.

MAGNETIC FIELD ABOUT A WIRE CARRYING

A CURRENT.

344. Oerstedt's Discovery. In 1819, in the course of a lecture

on electricity, Oerstedt, Professor of Physics at Copenhagen,

observed that when a wire carrying a current was brought near a

magnetic needle a deflection of the needle was produced. He

recognized at once the importance of this discovery as demon-

strating what up to that time had been merely, a conjecture, that

is, that there existed some connection between electricity and

magnetism. He set to work immediately to investigate the matter

and soon discovered not only that an electric current produced a

deflection of a magnetic needle near it but that the direction of

this deflection depended both upon the direction in which the

current was flowing and upon the position of the conductor with

reference to the needle. His results were announced in 1820.

The news reached the French electrician Ampere on September 11

and was received by him with eagerness. Within one week there-

after he had repeated Oerstedt's experiments and had added to

the latter's discoveries; had confirmed by specially devised experi-

ments and had presented in a paper to the Academy a complete

theory of the new science of electro-dynamics (Par. 360) .

345. Right Hand Rule for Deflection of Needle. It is helpful

to the electrician, whether he be an advanced student or only a

beginner, to have some easy rule for determining, or some mechan-

ical way of remembering, in which direction certain phenomena

take place. Thus Ampere gave the rule of the "swimming man"

by which, the relative positions of the conductor and of the needle

and the direction of the current being given, the direction in

260

ELEMENTS OF ELECTRICITY.

which the north end of the needle would move could be predicted.

Other rules have been given by subsequent writers. Of these,

the following is thought to be the most useful, both because of

its simplicity, it being a true "rule of thumb," and because, as will

be shown later, of its applicability to a number of varied con-

ditions. It should be committed to memory.

Place the palm of the right hand upon the wire, the extended

fingers pointing in the direction of the flow of the current, the palm

turned towards the needle; the extended thumb will indicate the

direction in which the north pole of the needle will move.

Fig. 147 represents the application of this rule. The current

flowing in the wire in the direction indicated by the arrow will

Fig. 147.

cause the north pole of the needle to move out in the direction in

which the thumb is pointing.

If the wire be below, in order that the palm should be turned

towards the needle, the hand must be held back down, in which

case the thumb will point away from the observer and this is the

direction in which the north pole will actually move. In fact, the

rule is perfectly general and applies if the wire be vertical and

in front of either pole or if it be to either side of the needle, only

in this last case the needle must be capable of movement in a

vertical plane.

346. Magnetic Field About a Wire Carrying a Current. In

Pars. 143 and 144 it was shown that a needle in a magnetic field

tends to turn so as to place its longer axis and its own lines of

force parallel to the lines of force of the field. The needle in

Oerstedt's experiment turns for the same reason, that is, the cur-

rent flowing through the wire establishes about this wire a mag-

netic field with which the needle tends to coincide in direction.

ELECTRO-MAGNETICS.

261

This field may be studied in a similar manner to the other

magnetic fields already described. If a vertical wire, a portion

of an electric circuit, be passed through a hole in the center of a

horizontal sheet of cardboard

or of glass which has been

sprinkled with iron filings (Fig.

148) and if the circuit be then

closed and the horizontal sheet

be tapped while the current is

flowing, the filings will be seen

to gather and form in more or

less distinct circles around the

wire as a center. The lines of

force of the field are circles,

and it was shown first by

Fig. 148.

Ampere that these circles lie

in planes perpendicular to the

wire. In Oerstedt's experiment as described in Par. 344, the needle

can never place itself at right angles to the wire, tor the controlling

force, the horizontal component of the earth's magnetism, is

always effective. However, if a perfectly balanced needle be

mounted so that its axis of rotation is parallel to the earth's field,

then this field has no influence upon its rotation and if Oerstedt's

experiment be now performed, the needle will always set itself

at right angles to the wire.

u

Fig. 149.

347. Direction of Field. The experiment with the iron filings

shows the lines of force of the field to be circles but does not

indicate their direction. This latter may be determined as follows.

262 ELEMENTS OF ELECTRICITY.

Using the same horizontal sheet and vertical wire as in the pre-

ceding experiment, distribute at equal distances apart on the

circumference of a circle whose center lies on the wire a number

of small compasses, A, B, C, D (Fig. 149). Before the circuit is

closed, these all point in the same direction. Let us assume that

this is the direction indicated by the needle A. Suppose now the

circuit to be closed and the current to flow down the wire. The

needle at A will not change its position, C will be entirely reversed,

B will point to the right and D to the left, that is, if we look at

the needles from above, they point around the circle in the direc-

tion of the motion of the hands of a clock. Had the current

flowed up, the needles would all have pointed in a counter-clock-

wise direction around the circle.

348. Clock Rule for Direction of Field. The foregoing experi-

ment suggests another simple rule for determining the direction

of the field about a wire carrying a current.

Suppose the eye placed so as to look along the wire in the direction

in which the current is flowing; the positive direction of the field

about the wire is the same as the direction of motion of the hands

of a clock.

Of course, if the current is flowing towards the eye, the field is

counter-clockwise. This rule should not supplant the right hand

rule given in Par. 345. Either one could be used to the exclusion

of the other but it is better to have both at command.

349. Wire Carrying a Current is not Itself a Magnet. Although

surrounded by a magnetic field, a wire carrying a current is not

itself a magnet. If a clean copper wire through which a current

is flowing be dipped into iron filings and then lifted, the filings

will cluster around the wire but will drop off when the current is

broken. At first sight this seems to indicate that the wire has

become magnetized but it can be shown that such is not the case.

When the wire is thrust into the filings they become magnetized,

since magnetic bodies placed in a magnetic field become magnets

(Par. 120), and if they surround the wire, or if any of them adhere

to it through stickiness, they cling together like the links of a

chain and really adhere to each other instead of to the wire. If

an elongated filing be placed at right angles to the wire and with

its ends lying upon one of the circular lines of force surrounding

the wire, one of these ends will be urged in one direction around

ELECTRO-MAGNETICS.

263

the circle, the other end in the opposite direction; the result is

that the filing will move broadside towards the wire. There is,

however, no radial component between a wire carrying a current

and a magnetic pole in its field. In this respect the field about a

conductor is unique. While all other forces exerted between

bodies act along the line joining the bodies, the force upon a pole

in a field about a wire acts at right angles to the line joining the

wire and the pole.

350. Rotation of a Magnetic Pole by a Current. In Par. 135

the positive direction of a magnetic field was defined as that direc-

tion in which a free north pole would move. Such a pole released

near the north end of a magnet would move off along a line of

force, curving around until it came to rest against the south face.

The statement was made (Par. 142) that a magnetic line of force

is a closed curve, but the moving

pole can not travel around a com-

plete orbit for its progress is arrested

by the material substance of the

magnet. In the field about a wire

carrying a current the case is dif-

ferent. Here the lines of force are

circles, return upon themselves and

do not necessarily pass through any

solid body. A pole released in such a

field should therefore rotate as long

as the field is maintained. Although

we can not obtain a free pole, we

can approximate to the theoretical

condition by arranging a circuit so

that only one pole of the magnet

lies in the field and we can thus

produce mechanical rotation.

Fig. 150 represents diagram-

niatically such an arrangement. NS

Fig. 150.

is a magnet bent in the center at an angle and placed upon the

pivot P about which it is free to rotate. B is a little cup of

mercury into which dips the conductor AB, thereby securing

movable electric contact with a minimum of friction. CD is an

annular cup of mercury surrounding but not touching the magnet.

From B a wire BD is carried over and bent down so as just to

264

ELEMENTS OF ELECTRICITY.

touch the surface of the mercury at D and to sweep along this

surface as the magnet rotates. DE is a conductor leading away

from the annular cup. If the current enters at A, it goes to B,

thence to D and out by E. It therefore passes by the pole N but

not by the pole S. According to the rule given in the preceding

paragraph, the field about AB, viewed from A, is clockwise. The

pole N will therefore spin 'around in the direction shown by the

dotted line. If the current be reversed the direction of rotation

is also reversed; so also if the magnet be inverted, the direction

of rotation is reversed.

351. Rotation of a Current by a Magnetic Pole. The reaction

between the pole and the field being mutual, it follows that if

the pole be fixed and the conductor be free

to move, the latter may be made to rotate

about the former. This may be shown by

the apparatus represented in Fig. 151. NS

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