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Font size Below the milled head there is a second pointer which, as the head
is turned, sweeps around the graduated circle and indicates the
angle through which the head has been turned.

When a current is flowing through the instrument, the movable
coil is urged in a counter-clockwise direction. The milled head is
turned in a clockwise direction and the torsion of the spiral spring,
which varies directly as the angle through which the milled head
is turned, tends to drag the coil back to its primary or zero position.
When the coil has finally been brought back to this position, the
pull exerted by the spring exactly balances the contrary moment
exerted by the current.

Consider a vertical portion of the wire in the movable coil and
an adjacent portion in the fixed coil. The force exerted between
these portions is directly proportional to their length, to the prod-
ucts of the currents flowing in them, and inversely proportional
to the simple distance between them (Par. 362). The length of the
portions is constant, so also is the distance between them, since
the coil is always brought back to its original position, therefore,
the force between the portions, and consequently the force be-
tween the coils themselves, varies as the square of the current and
also as the number of divisions of the scale over which the pointer

294 ELEMENTS OF ELECTRICITY,

attached to the milled head has been turned. From this it follows
that the current varies as the square root of the angle of torsion, or

I=KV5

d being the number of divi-
sions of the scale indicated by the pointer. The constant K is
different for different instruments but is easily determined by
passing through the instrument a known current / and noting the
corresponding torsion 5.

In addition to its use in measuring currents, this instrument, as
will be shown later (Chap. 36), may be used to measure electrical
power.

384. Ballistic Galvanometer. In the earlier attempts to meas-
ure the velocity of moving projectiles, use was made of a piece of
apparatus called a ballistic pendulum. This consisted of a large
pendulum with a very heavy and solid bob. The projectile was
fired against and embedded itself in the bob, the blow causing
the pendulum to swing through a certain angle which was recorded.
Knowing this angle, the vertical height through which the weight
had been lifted could be determined and, knowing the weight of
the projectile, the velocity with which it struck the pendulum
bob could be calculated. 4 -

When a charged body is discharged through a conductor, the
charge in its passage is a veritable current but its duration is only
momentary. If passed through a galvanometer, it gives to the
moving parts a sudden impulse or blow comparable to the blow
given to the pendulum by the bullet. If these moving parts be
somewhat heavy and therefore rather slow in vibration, the cur-
rent will have passed before any appreciable movement takes
place. It can be shown that the sine of half of the angle of the
first swing or "throw" of the needle is proportional to the charge
which has passed through the coil. (See Gray, Absolute Meas-
urements in Electricity, Vol. II, pp. 390-396.) Galvanometers
used in this manner are called ballistic galvanometers. They are
generally of the suspended coil type and must not be damped.

ELECTRO-MAGNETICS. 295

CHAPTER 31.

ELECTRIC MAGNETIZATION OF IRON AND STEEL.

385. Solenoid. A cylindrical coil of wire whose length is great
as compared to its diameter is called a solenoid, .the Greek word
solen meaning a tube. The successive turns of the coil are wrapped
as closely together as the thickness of the insulating covering will
permit but in diagrams it is usual, for the sake of clearness, to
represent these turns as somewhat widely separated. To give an
accurate shape to the coil it is generally wrapped upon a material
core, such as a wooden rod or a tube of glass or paper, which after-
wards may be withdrawn. In diagrams, to avoid confusion as
to the direction in which the coil is wrapped, it is preferable to
represent the core as in position.

The coil being a helix, the turns are inclined to the axis of the
cylinder but each is electrically equivalent to a turn at right angles

Fig. 172.

to the axis (Fig. 172) and a short portion parallel thereto and equal
in length to the pitch of the coil. The effect of these longitudinal
portions is neutralized if one end of the coil be brought back along
the axis of the coil, or if the wire, the circular direction of its wind-
ing being unchanged, be wound back to the starting point, thus
forming a second layer on top of the first.

386. A Solenoid Equivalent to a Bar Magnet. If a current be
passed through a solenoid, application of the right hand rule will
reveal the fact, which indeed has already been shown (Par. 365),
that the successive turns combine in the production of a field in the
same direction. Thus (Fig. 173), all the lines of force inside of
the solenoid run in the direction shown by the long arrow. A
solenoid carrying a current is therefore magnetically equivalent

296 ELEMENTS OF ELECTRICITY.

to a bar magnet. It has poles, it will attract magnetic sub-
stances, it will attract or repel the pole of a bar magnet and, if

**. xv VS. y*v

A

nr"isr >

N

\

Fig. 173. \S

freely suspended, it will turn so as to place itself in the magnetic
meridian.

387. Intensity of Field on the Axis of a Solenoid. The inten-
sity of the field at a point on the axis of a circular coil is (Par. 354)

H =^ dynes (I)

in which / is the current in

absolute units, r is the radius of the coil, and x is the slant distance
from the coil to the point on the axis.

Let P, Fig. 174, be a point on the axis of a solenoid through
which a current / is flowing. If in each centimeter length of the

A B
oooooo -^ o op.9 o o o o

oooooo - - - - - - - - - - ooooooooo

Fig. 174.

solenoid there be N turns, we may consider that around such
unit of length there is flowing a sheet of current of strength NI.
The current over a small portion AB is therefore proportional to
the length of AB or N.I.dl.

Since - = sin 6, expression (I) can be written

rj 2wlr .

H = -^-sin dynes
The field at P due to the current on AB is, therefore,

, .. .

dH=- - sin B (II)

ELECTRO-MAGNETICS. 297

From the figure, = do, or AE = AP.de (III)

AB = AP.dQ. -

AB =dl=

From the similar triangles AEB and BPF
AE :AB = BF : BP

Hence . AB = AE *, BP (IV)

or, substituting from (III)

A,

and as A B decreases, AP approaches BP, hence

Substituting in (II)

dH~2arNI.sinO.dO (V)

Integrating

H = 2wNI(cos 0) + a constant

The field due to the entire solenoid is obtained by taking this
expression between the proper limits. If P be at the center of the
coil and if the coil be so long that = and 180, then

H = 4*rrNI dynes

If P be at the mouth of the solenoid so that is and 90,
H=2TrNI dynes

or the field at the mouth
of a long solenoid is one-half what it is at the center.

It is to be noted that since in these expressions for the field the
radius of the coil does not occur, the intensity of the field would
appear to be independent of the diameter of the solenoid. This,
however, is not correct unless the further condition be expressed,
a condition already introduced in the integration, that the various
solenoids are geometrically similar. Should the radius of the
solenoid be doubled or trebled, its length must be likewise doubled
or trebled.

The length of wire required in similar solenoids varies as the
square of their like dimensions and if the length be increased
the diameter of the wire must be increased to overcome the in-
crease in resistance, therefore, considerations of economy lead us
to make the coil fit its core as closely as possible.

298 ELEMENTS OF ELECTRICITY.

It has been shown that the field at the center of a long solenoid is
very uniform. If the solenoid be wrapped upon a circular core, so
as to return upon itself, the field at every cross-section is the same.

388. Ampere Turns. In the discussion in the preceding para-
graph the current / is in absolute units. If it be given in amperes
it must be reduced to absolute units by dividing by ten. The
expression for the field at the center of a long solenoid becomes in
this case A

H =^.N.I dynes

The field therefore varies with NI. This product remains a
constant if N and / vary reciprocally, hence three amperes mak-
ing five turns produce the same magnetic effect as five amperes
making three turns, or as one ampere making fifteen turns, or as
fifteen amperes making one turn. In any coil the product of the
total number of turns times the current flowing in the coil is called
the ampere turns, and this product appears as a factor in all ex-
pressions dealing with circular coils. We have already employed
it in the discussion of the tangent galvanometer (Par. 374).

The constant 4^/W is equal to 1.2566 + ; it is therefore suffi-
ciently accurate in ordinary calculations to say that H, the field,
or the number of lines of force per square centimeter at the center
of a long solenoid, is one and a quarter times the ampere turns per
unit of length.

389. Variation of Field of Solenoid with Current. The fact
that the field on the axis of a solenoid varies directly with the

S
/N r> ,0

\r r r rr\

Fig. 175.

current may be shown experimentally as follows. In Fig. 175, S
represents a solenoid, B a battery, K a key, R a rheostat (Par.
302), A an ammeter (a current-measuring instrument), and G a
galvanometer with a short needle and long attached pointers
poised over a graduated circle and placed so that the axis of the

ELECTRO-MAGNETICS. 299

solenoid prolonged passes through the pivot of the needle and is
perpendicular to the magnetic meridian. By means of the rheo-
stat, the current through the solenoid may be varied at will. The
strength of the current is read direct from the ammeter. When
the key K is closed, permitting a current to flow, the needle of the
galvanometer is deflected. It will be seen that this is the case
discussed in Par. 146 and that the deflecting force (which is due
to the field of the solenoid) varies as the tangent of the angle of
deflection. If, therefore, we lay off on a horizontal axis distances
proportional to the current through the solenoid, the correspond-
ing ordinates laid off proportional to the tangent of the angle of
deflection will be proportional to the corresponding field. The
points so determined will lie on a straight line passing through
the origin (see OA, Fig. 176).

390. Effect of Material of Solenoid Core Upon the Field. With

the apparatus described in the preceding paragraph we may in-
vestigate the effect produced upon the field by varying the material
of which the core of the solenoid is composed. Using cores of
glass, rubber, wood, lead, copper, tubes of various gases or liquids
or even vacuous space, no perceptible variation of the field is
discovered, its strength remaining the same as when the solenoid
enclosed only air. If, however, we insert a core of steel, the deflec-
tion of the galvanometer needle will indicate that the field has
been increased several hundred times, that is, there are now several
hundred times more lines of force traversing the solenoid than
there were before the steel core was inserted. If the core be of
soft iron, the increase is still greater; if it be of nickel, it is less than
in the case of steel but much greater than in the case of air.

391. Permeability. The great increase in the density of the
magnetic flux (number of magnetic lines) when iron is inserted in
the coil has been explained by saying that iron is more permeable,
or has greater permeability than the other substances. When a
beam of light falls upon a sheet of clear glass many more rays go
through than when the beam falls upon a sheet of dark glass. We
may consider that in each case there is a force tending to drive the
rays through and that the dark glass offers greater resistance while
the clear glass offers less, or is more permeable. So also there is a
magnetizing force which tends to drive magnetic lines through
the field of the solenoid. Air, wood, etc., offer a magnetic resist-

300 ELEMENTS OF ELECTRICITY.

ance to this force and only a certain number of lines get through;
iron and steel offer much less resistance, or are much more per-
meable, and permit many more lines to pass. To this magnetic
resistance the name reluctance has been given. It follows that
reluctance is the reciprocal of permeability or that the two are
comparable to resistance and conductance, respectively.

392. Expression for Permeability. We have seen that the field
of a solenoid varies directly with the number of ampere turns per
unit of length. It follows that the magnetizing force varies in
the same manner, hence we may use H or 1.25 times these ampere
turns (Par. 388), as a measure of the magnetizing force. If the
magnetizing force which produced H lines per square centimeter
in air produces B lines per square centimeter in iron, then the per-
meability of the iron is Ej H. The accepted symbol for perme-
ability is the Greek letter mu, /*, hence

B

M = H

Hopkinson found that a magnetizing force which produced 10
lines of force per square centimeter in air produced 12,400 per
square centimeter in a specimen of wrought iron; the permeability
of the iron was therefore 1,240.

The permeability of air, glass, and other non-magnetic sub-
stances is unity; that of bismuth, the most diamagnetic substance,
differs from unity in the fourth place of decimals only.

393. Magnetic Saturation. The conception of permeability as
outlined in the preceding paragraphs loses some of its definiteness
when it is found that for magnetic substances it is not a constant
but is different for different magnetizing forces.

In Par. 389 it was stated that the field of a solenoid varies
directly with the current. This is shown by the line OA in Fig.
176, in which the abscissae are laid off proportional to the magnet-
izing current and the ordinates proportional to the corresponding
field. If we now insert in the solenoid a long soft-iron core, mag-
netically neutral, and gradually increase the current, we will notice
three stages in the field produced: (a) for small values of the cur-
rent it will increase slowly; (b) as the current is increased it will
rise suddenly until a certain point is reached, after which (c) it will
continue to increase but slowly. These stages are shown graphi-
cally in the curve OD. Since this curve represents the field pro-

ELECTRO-MAGNETICS.

301

duced by the solenoid and the core in conjunction, if we subtract
from its ordinates the corresponding ordinates of OA, we will get
the curve of magnetization of the core alone. The result is the
curve OE. The upper portion of this being very nearly parallel to

B

A t^'Q *

f KO^ C_-f- """"

*^^ ^- -T ' ^" "

,U'^

. '

^T-

c?/f
/ i .-4

-

1

__!__

]

/

xt 2. ATI

)N

OF

CO, RE

z

>/ /

/^"

p

w

^ i

N

/r

i

l__

//

u

//

z

/

|

Z

/

|

^ !S

^v

. -

b sj_

-

-

/ ' ^

of[i
1

1

O

MAG\NETIZINC\ FORCE
Fig. 176.

the horizontal axis indicates that the magnetization of the core
would be but slightly increased by a further increase in the
magnetizing current ; in other words, the core is now magnetically
saturated.

394. Curves of Magnetization. As will shortly be shown, the
designer of electrical machines and apparatus is frequently called
upon to solve problems such as the following: Given an iron core
of a certain size, shape and quality; required the number of ampere
turns to produce in this core a flux of a certain strength. Among
the data needed for the solution is not simply the permeability of
the particular kind of iron of which the core is constructed but its
permeability when the magnetic flux is of the strength called for
in the problem. Such information is contained in tables but is
more striking when presented graphically in the form of curves of
magnetization. Fig. 177 represents these curves for five different
qualities of iron and steel, whence it is seen that soft annealed iron
may be both most easily and most highly magnetized and that
hard steel is most difficult of magnetization. From the figure it
is seen that for a magnetizing force of 5 the magnetization of soft

302

ELEMENTS OF ELECTRICITY.

iron is 10,000, or the permeability is 2000, while for a force of 50
the magnetization is 16,000, or the permeability is only 320.

It will be noted that these curves all exhibit the three stages as
described in Par. 393.

\ *NNEM-ED_ IRON
15,000

10.000

5.000

10

30 35 40 45

Fig. 177.

395. Ewing's Theory of Molecular Magnetism. The accepted
explanation of these phenomena is that advanced by Ewing and
has already been given in part in Par. 153. The molecules of mag-
netic substances are inherently magnets but ordinarily exhibit no
magnetic effects since they are grouped so as to mutually satisfy
their individual polarities. Application of a magnetizing force
disturbs this grouping, and exercises a directive effect upon the mo-

b c

Fig. 178.

lecular magnets, causing them to take approximately a common
direction so that they combine in the production of a magnetic
flux. His theory, as the following will show, satisfactorily ac-
counts for the three stages in the curves of magnetization. Let us
take the simplest possible hypothetical case, that of two molecular
magnets, and let the two small needles in a, Fig. 178, represent
these molecules. If they be remote from other magnetic bodies

ELECTRO-MAGNETICS.

303

they will take up a position of equilibrium with their axes lying
upon a common line. Let them now, as shown in 6, be subjected
to a magnetizing force H. If H be feeble the needles will move
slightly but will not swing entirely to the right because they are
pulled back by their mutual attraction. However, as H increases,
this attraction will finally be overcome and the needles will then
whirl suddenly to the right as shown in c. This corresponds to the
stage of saturation. The needles, because of their action upon
each other, are not strictly parallel nor can they ever become so.
Further increase of H can only pull them a little more nearly
parallel. If the magnetizing force be discontinued, the needles
will not fly back at once to their original position but will linger
and may require a slight jar to cause them to turn back.

Swing's theory has been corroborated experimentally. A great
many small magnetic needles were distributed side by side upon
a long board which was then inserted in a coil and the needles
allowed to come to a position of equilibrium. The arrangement
was then subjected to a gradually increasing magnetizing force
and the resulting fields were determined and plotted as described
above. The result was a curve showing the three stages of the
usual magnetization curves. Furthermore, when subjected to a
demagnetizing force the curve went through the cyclic changes
described in the following paragraphs.

396. Hysteresis. Suppose that beginning with a magnetically
neutral specimen of soft iron and apply-
force we should determine and plot the
corresponding curve of magnetization.
Suppose that having reached a point
where a magnetizing force OD (Fig. 179)
produces a magnetization DA, we should
reduce the magnetizing force to zero. It
will be found that the magnetization is
by no means reduced to zero but persists
or lingers after the withdrawal of the
force and has some such value as OC.
That portion of the curve representing
the change from A to C is concave to the

Fig. 179.

horizontal axis. If now the magnetizing force be reapplied, the
curve of magnetization will not retrace the path A NC but there

304 ELEMENTS OF ELECTRICITY.

will be a tendency for the magnetization to linger at the value
OC and it will increase at first at a slower rate than it decreased,
the corresponding portion CM A of the curve of magnetization
being convex to the horizontal axis. If at some other point E
the magnetizing force be again reduced to zero and then reapplied,
a similar loop EQ KPE will be traced, and so on, the magnetiza-
tion always hold ng back or conforming reluctantly to the changes
in the magnetizing force. To this phenomenon the term hysteresis,
a lag or lagging, is applied.

397. Further Data on Permeability. The magnetizing force
OF, Fig. 179, produces the various degrees of magnetization cor-
responding to FG, FM, FN, FP, FQ, FR, etc. Which of these is
to be taken in determining the permeability of the specimen? It
is seen that the notion of permeability is even more indefinite than
was pointed out in Par. 393, and that in order that it may be of
any practical use we must know the previous magnetic history
of the specimen with which we are dealing. It can easily be shown
that even though a specimen be magnetically neutral, its perme-
ability, if it has recently been demagnetized by a single reversal
of the current, is very different from what it is if it has never been
magnetized at all. The usual understanding, therefore, is that
when the permeability of iron or steel of a certain quality is given,
it refers to a specimen which has not previously been magnetized
and, furthermore, the permeability has been determined by the
application of a continually increasing magnetizing force without
reversals.

398. Cycle of Magnetization. If a specimen of soft iron be
magnetized, then demagnetized, then magnetized to an equal
degree in the opposite direction, then demagnetized, and finally
again subjected to the original magnetizing force, it will pass
through a cycle of magnetization represented by the curve in Fig.
180. When the magnetizing force has first been reduced to zero
the magnetization of the specimen is still proportional to OC. In
order to remove this residual magnetism an opposite or negative
magnetizing force OF must be applied. Since after the with-
drawal of the magnetizing force the iron still retains a portion
of the magnetism, we may say that the iron clings to this
magnetism with a force equal to the force OF which must be
employed to cause its relinquishment. The force which must

ELECTRO-MAGNETICS.

305

be applied to remove the residual magnetism is called the coercive
force.

The broken curve in Fig. 180 represents a cycle of magnetization
of a specimen of hard steel, whence it is seen that the coercive

/ F

/

/

.-'"' /' H

/

/

Fig. 180.

force is very much greater than in the case of iron. This has
already been shown in Par. 155.

399. Energy Loss Due to Hysteresis. In raising a weight a
certain amount of work must be performed. If after the weight
is raised it be released, it will in its fall restore the same amount
of energy. In magnetizing a bar of iron or steel work is likewise
performed but when the magnetizing force is withdrawn the entire
amount of energy is not given back, in other words, there is a
loss.

In Par. 358 we saw that the work expended in changing the field
within a coil carrying a current is IN ergs, in which / is the
current in absolute units and N the increase or decrease in the num-
ber of lines embraced. If there be n turns in the coil, the expres-
sion becomes nIN ergs, but n being a constant the work is always
proportional to the product of the current by the change in the
number of lines embraced.

In Fig. 181, AL is the average magnetizing force as the number
of lines embraced by the coil increased from OE to OF. But we
have seen that the magnetizing force is proportional to the
current, therefore AL is proportional to the current and the

306

ELEMENTS OF ELECTRICITY.

H

area of the rectangle ALxEF is proportional to IN or to the
energy expended while the magnetization increased from OE
to OF. In a like manner the area of the rectangle FM is pro-
portional to the energy expended while the magnetism increased
from OF to K. The sum of these elementary rectangles, or the
B area SDGO, represents the total energy ex-

^ pended in magnetizing the iron to the stage
OG. It follows that the area FDG repre-
sents the energy restored as the magnetiza-
tion falls to OF, and the difference between
these two areas represents lost energy.

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