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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



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