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Font size line joining the two poles. Should these fields be intersected by
another, the resultant field would be obtained by compounding

106

ELEMENTS OF ELECTRICITY.

the two and would in general be unsymmetrical. The earth's
field most often produces distortion in others but its strength
being comparatively feeble, the distortion is not revealed in the
magnetic figures produced with filings. If, however, the field be
mapped as follows the effect of the earth's field becomes evident.
Place a bar magnet in the center of a sheet of paper and then in
contact with the magnet place one of the little compass needles one
centimeter in length and mounted in a glass-covered brass case.
With a pencil make a dot at the far end of the needle, then shift
the compass until the near end of the needle is over this dot and

Fig. 64.

r

again make a dot at the new position of the far end of the needle
and so on to the limits of the paper. Connect these dots by a
along the magnet and make a second chain of dots and so on until
the whole space about the magnet has been marked off. Figs. 64
and 65 represent fields traced in this way, Fig. 64 with the north
pole of the bar magnet pointing north, Fig. 65 with the north pole
pointing south. In each case immediately around the magnet
the strength of its field overpowers that of the earth but the
strength of the magnet's field falls off rapidly as the distance from
it increases while the earth's field is constant over a considerable

MAGNETISM.

107

area and at a distance from the magnet the earth's field has the
ascendancy. At the spots marked P these two forces neutralize
each other and the needle will vacillate and come to rest in any
position. These two figures, though different, are both symmet-
rical since the bar magnet was designedly placed in a north and
south position. Should it be placed in any oblique position the
symmetry will be destroyed.

'Fig. 65.

142. Properties of Magnetic Lines of Force. In some of
their properties magnetic lines of force are similar to electric lines
of force but in others they differ widely. They agree with electric
lines of force in having a tension along their length, or a tendency
to shorten, and also a pressure at right angles. They also never
intersect. They differ from electric lines of force in that they are
closed curves, that they penetrate all substances whether con-
ductors or not and that they do not necessarily, or even generally,
leave or enter a surface at right angles. Being a closed curve, a
complete magnetic line of force lies partly in the magnet and
partly in the surrounding medium. While the majority of these
lines emerge near the poles, many, as shown in Figs. 61 and 66,
emerge along the sides of the magnet. The lines within the magnet

108 ELEMENTS OF ELECTRICITY.

are designated collectively as the magnetic flux ana this flux is
evidently a maximum at the mid-section of the magnet. This
is sometimes otherwise expressed by saying that the intrinsic
magnetism is a maximum at this mid-section. The intrinsic
magnetism is of no effect on outside bodies. Magnetic effects
of attraction and repulsion are produced only by those lines of
force which emerge from the magnet. This is called the free
magnetism and is greatest in the neighborhood of the poles.

Fig. 66.

It must be noted here that that portion of these lines which lies
within the magnet does not conform to the definition of a line of
force as given in Par. 137, for which reason these internal lines
have been variously designated as lines of magnetization, lines
of magnetic induction, etc. An internal line of magnetization is,
however, always the continuation of an external line of force and
the above distinction, although academically correct, is without
practical importance. This fact will be brought out more clearly
in the subject of electro-magnetics.

That portion of a magnetic body from which lines of force
emerge is always a north pole and that portion of such body into
which they enter is always a south pole. To this rule there is but
one exception. The lines of force of the earth enter at the north
magnetic pole and come out at the south magnetic pole. The
reason for this exception has already been given (Par. 117).

143. Magnetic Lines Pass Preferably Through Magnetic
Substances. If in the space between the two dissimilar poles in
Fig. 62 there be inserted a soft iron block and a magnetic figure
be then taken, the lines of force which in Fig. 62 curved out widely
will now be seen to have drawn in, as shown in Fig. 67, and pass
through the iron block instead of through the air. A simple ex-
planation is that the iron block has become a magnet by induction
and the lines of force converge to the nearest poles, but it is some-
times conveniently explained by the statement that magnetic

MAGNETISM.

109

lines of force travel by preference through magnetic bodies and
will avail themselves of such a path whenever the opportunity
offers. This principle affords an explanation of certain phenomena
and is of considerable practical importance.

Fig. 67.

It has already been noted (Par. 112) that filings cling to the
edges of a magnet rather than to the flat surfaces. This fact is
also clearly shown in Figs. 61, 62 and 63. In Fig. 68, a and b
represent end views of a bar magnet. If the lines of force radiated
equally from the internal pole they would emerge as shown in a
and there would be more filings just on top of the magnet than
elsewhere since this point is nearest to the pole and consequently

Fig. 68.

at this point the attraction would be strongest. But since the
lines prefer to travel as far as possible through the steel, their
actual path is as represented in b and the filings are thickest
where the lines of force are thickest, that is, along the edges.

An essential part of dynamos and motors consists in its simplest
form of two powerful magnetic poles embracing between them a
cylindrical opening. It is highly important that there should be a

110

ELEMENTS OF ELECTRICITY.

uniform field along the faces of these magnets but owing to the
principle above, the lines of force, as shown in Fig. 69 a, crowd
across at the top and bottom and leave the central portion of the
opening thin. If, however, there be inserted in this space a soft

Fig. 69.

iron cylinder, the lines will pass through this cylinder and, as
shown in b, will produce along the pole faces a dense and uniform
field.

If a magnet NS (Fig. 70) produces a deflection in a needle A,
the needle can be screened from this effect by interposing between

5 \

Fig. 70.

it and the magnet a comparatively thick iron plate B. The lines
of force from N, which formerly reached A, now travel through
B, as shown in the figure, and thence back to S. If A be placed
inside of an iron cylinder it may be entirely screened from outside
magnetic influences.

MAGNETISM.

Ill

If a pivoted iron bar AB (Fig. 71) be placed diagonally across
a magnetic field NS it will swing so as to place itself parallel to
the field. We may explain this motion as
follows. The lines of force of the field, from
what we have just seen, turn to one side and,
as shown in the figure, run lengthwise
through the bar. The tension along these
lines produces a couple on AB which pulls
it around to parallelism with NS. The law
under which this movement takes place is
that a magnetic body placed in a magnetic
field tends to move so that its longest axis
coincides in direction with the lines of force
of the field.

144. Law of Maximum Flux. In Fig. 72,
A is a piece of soft iron in a weak field to

one side of the strong field B. The lines of force of the strong
field move out, as shown in the figure, so as to pass through

A and if A be free to move it will be
drawn over into the denser field at B.
If A be a magnet placed obliquely to B
it will be both drawn over into B and
turned so that its own lines of force will
be parallel to and of the same direction
as those of B. It will therefore embrace
in lengthwise direction both its own
lines of force and those of the field B, or,
in general, a magnetic body placed in a
magnetic field tends to move so as to
embrace in one direction the maximum
number of lines of force. This is but a
particular case of Maxwell's Law, a prin-
ciple of great importance which will be discussed later (Par. 371).

145. Graphic Representation of Intensity of Magnetic Field.

For the same reasons as given in Par. 61, it has been agreed to
represent graphically the intensity of a magnetic field by the
number of lines of force per square centimeter taken perpendicular
to these lines. From this standpoint, a unit magnetic field is
defined as that field which contains one line of force per square

112

ELEMENTS OF ELECTRICITY.

centimeter of cross-section. A similar course of reasoning to that
given in Par. 63 will lead to the conclusion that there radiate
from a unit pole 4 TT lines of force.

146. Comparison of Magnetic Fields. Tangent Law. There
are a number of ways in which magnetic fields may be compared
by means of the deflection which they produce in a magnetic needle.
If a needle which is poised in the meridian be exposed to such a
field at right angles to the meridian, the needle will be deflected
through a certain angle. The field draws it to one side with a
decreasing moment; the horizontal component of the earth's mag-
netism pulls it back to the meridian with an increasing moment.
A position of equilibrium is finally reached and the angle of
deflection may be read from a scale placed beneath the needle.

In general, in instruments which operate thus by a needle
moving over a scale, the force which pulls the needle away from
its zero position is called the deflecting force; the force which tends
to restore it to the zero position is called the controlling force.

In the following method it is assumed that the action lines of
the magnetic force upon the poles of the needle are parallel and
that within the limits of the space over which the needle swings the
force is constant. In order to realize this condition as nearly as
possible the needle is made very small. If the scale of degrees
varied with the size of the needle it would become too small to be
read with much accuracy, therefore an auxiliary pointer of alum-
inum or of some other light substance
is fastened to the needle and a larger
scale may then be used.

In Fig. 73 let A B be a magnetic
needle of pole strength m, deflected
from the meridian NS through an
angle 8 by a magnetic field H ' act-
ing at right angles to the meridian.
At the pole A the controlling force
is mH, represented by AC. The
deflecting force is mH', represented
by AD. An exactly similar set of
forces act upon the pole B but to
consider them would simply be to
repeat what we shall prove for the set at A. The controlling
force may be divided into two components, one, AF, in the

Fig. 73.

>

MAGNETISM. 113

direction of the axis of the needle and of no effect so far as
rotation is concerned; the other, A E, perpendicular to the needle
and active in restoring it to the meridian. From Fig. 73,

AE = AC. sin 5 = mH.&iud

Similarly, the deflecting force may be divided into two com-
ponents, one in the direction AF, the other AG, which = AD. cos 6
=w# / .cos 5 and which is active in deflecting the needle from
the meridian. When the needle comes to rest these two active
components, A E and AG, are equal, hence

= mH .sinS

whence H f = H = H . tanS

cos 5

or, the magnetic

field which acting at right angles to the meridian produces in a
magnetic needle a deflection 5, is equal to the horizontal component
of the earth's magnetism at that point multiplied by the tangent
of the angle of deflection.

It follows direct from the foregoing that different magnetic
fields acting at right angles with the meridian will deflect a
needle through angles whose tangents are proportional to the
respective fields. This is known as the Tangent Law and is an
important principle in certain electrical measuring instruments to
be described later (Par. 373).

It will be noted that the deflection produced is independent of
the strength and of the length of the needle, or rather of the dis-
tance between the poles. Hence, as was stated in Par. 126, the
exact location of the poles is immaterial. If, however, the con-
trolling force be non-magnetic (as in Nobili's astatic galvanom-
eter) these factors are of importance.

147. The Sine Law. Should the deflecting field make a con-
stant angle with the needle instead of with the meridian, a different
state of affairs would result. Whatever the constant angle may be,
we can always divide the force into two components, one of which
is perpendicular to the needle and is the effective one in producing
deflection. We may therefore in Fig. 73 consider AG as repre-
senting the deflecting force mH'. When equilibrium is reached
AG = AE, or

mH' =mH .sin 5, or H' = H .sin 5

whence, magnetic fields acting at a constant angle with the needle

114 ELEMENTS OF ELECTRICITY.

are to each other as the sines of the respective angles of deflection.
This is known as the Sine Law and is the principle of one class of
galvanometers (Par. 376).

148. Determination of the Strength of a Magnetic Field. In
Pars. 129, 146 and 147, principles have been given which enable
us to compare magnetic fields among themselves, that is, to deter-
mine how many times stronger or weaker one field is than another,
but these do not enable us to make any absolute measurement.
The following method of determination of the absolute strength
of a magnetic field is due to Gauss.

In Par. 129 we saw that the time of oscillation of a simple
pendulum is given by the expression

T = 2

force

Multiplying the expression under the radical sign by I above
and below, it becomes

m.l 2

I X force

In mechanics, the sum of the product of the mass of each particle
of a rotating body into the square of the distance of the particle
from the axis of rotation is called the moment of inertia of the body.
In the above expression m is the concentrated mass of the pendu-
lum and I is its distance from the point of suspension, therefore,
m .I 2 is the moment of inertia of the pendulum. Representing this
by K, the above becomes

K

I X force

In the case of a needle in a magnetic field, the force is m . H, m
now representing the strength of the pole of the needle, and the

above may be written

K

l.m. H

But (Par. 130) l.m is the magnetic moment of the needle, hence
the expression becomes

K

M.H

The expression for the time of oscillation may therefore be
written

MAGNETISM. 115

In this, T can be determined by observation and K by calcula-
tion or by experiment.* We therefore have an equation involving
two unknown quantities M and H, one of which, H, we wish to
determine. If we can obtain another expression involving these
same two quantities, we may by combination determine H. The
obtaining of this second expression is explained in the two follow-
ing paragraphs.

149. Turning Moment of One Magnet upon Another. Let

ns (Fig. 74) be a small magnetic needle, its center lying on the
prolongation of the axis of the magnet NS and its length perpen-
dicular to this prolongation. Let m' be the strength of the poles

A

n

o

3 - - - -

[Fig. 74.

S

of ns and let m be that of the poles of NS. Let the distance
between the poles of ns be 21 and that between the poles of NS
be2L. LeiOC = Da,udNn = d. From the figure d = Vl*+(D -L) 2 .
The repulsion between N and n is m . m'/d 2 and the moment of this
force upon ns is

The triangles NOn and OnA are similar since they are both
right angled and have a common angle NnO, hence

OA : On = NO : Nn
On X NO

Hence OA =

^-~

Nn

l(D - L)
d
l(D-L)

(D- L)

* The moment of inertia of a bar magnet is

R= / (length)' +(breadth)\ x magg

that of a cylindrical magnet is

116 ELEMENTS OF ELECTRICITY.

Hence the above moment is

m . m' .1 (D - L)
[I* + (D- L) 2 ]?

The moment of N on s is the same and the total moment due
to AT is

2m.m'.l(D-L)
[P + (D - L) 2 ]*

The moment due to S is found in the same manner and may be
obtained direct from the preceding expression by substituting
D+L for D L. Since it acts in the opposite direction to that
due to N, the resultant component is

2.m.m / .l(D - L) _ 2.m.m' .l(D + L)

[P + (D - L) 2 ]* [Z 2 + (D + L) 2 ]*

If ns be very small, Z 2 can be neglected in comparison to (D L) 2 ,
and consequently also in comparison to (Z)+L) 2 , and the above
expression can be written

I I

2.m.m'.l

or

(D - L) 2 (D + L) 2
DL

Finally, if the distance between the two magnets be so great
that we may neglect L 2 as compared to D 2 , the foregoing becomes

Sm.m'.l.L

But 2mL is the magnetic moment of NS and 2m'l is that of ns,
hence the turning moment reduces to

2M.M'

If, in accordance with what we have assumed above, ns be very
small in comparison to Z), the field about ns is uniform, and if ns
be deflected through an angle 5, the turning moment becomes

2M.M'

150. Measurement of Strength of Magnetic Field. From
Par. 130 we have seen that if a needle of magnetic moment M' be
placed in a field of strength H and be deflected through an angle
5, the moment which tends to restore it to the meridian is

M'.H.sind

MAGNETISM. 117

and from the preceding paragraph we have seen that when such a
needle, ns, and a magnet, NS, are placed in the relative positions

N

Fig. 75.

as shown in Fig. 75, the turning moment due to the magnet is

2M.M'

When equilibrium is reached these two moments are equal,
hence yp . cos 5 = M' . H . sin 5

hence -jj = -FT tan 5

D and 5 may be measured directly and we thus obtain a second
expression involving M and H, which, when combined with the
expression deduced in Par. 148, enables us to determine H.

The needle ns being very small, a graduated circle over which
its ends might travel could not be read with much accuracy, there-
fore, the angle 6 is usually determined by observing the movement
over a scale of a beam of light reflected from a tiny mirror attached
to the needle. This method of reading the deflection of a needle
will be more fully explained in the description of the mirror gal-
vanometer (Par. 377).

118 ELEMENTS OF ELECTRICITY.

CHAPTER 15.
THEORY OF MAGNETISM.

151. Magnetism. As in the case of electricity, we must at the
outset admit that we do not know what magnetism is. It is not
matter yet in its manifestations it must always be associated with
matter. Gilbert called attention to the fact that with one magnet
we could make hundreds of others and yet the strength and weight
of the original magnet would be unaltered. We can conceive of
no form of matter which could thus be dipped out or drawn from
indefinitely and yet the original source of supply be undiminished.
It is not electricity. A charged body placed in a magnetic field
is not on account of its charge acted upon in any different way, nor
is a magnetized body placed in an electric field attracted or re-
pelled in any different manner on account of its magnetism. An
electric current does however produce certain magnetic effects,
and mechanical energy expended in moving or varying magnetic
fields may be transmuted into electric energy. We may say then
that with electric currents we can produce magnetism and from
magnetism we can produce electric currents.

Magnetic forces pass with equal ease through the hardest sub-
stances, the thinnest gases and a vacuum. The medium concerned
in this propagation is therefore considered to be the ether.

152. Molecular Magnetism. We have already seen (Par. Ill)
that if a bar magnet be broken across, one surface of this fracture
will be of north polarity, the other of south, and this no matter at
what point the bar be broken nor how the line of fracture runs
across. If these portions be again broken, the resulting fragments
will still possess polarity and even if the final result be dust the
ultimate particles will still be little magnets. The inevitable con-
clusion is that the individual molecules are themselves magnets
and that they are arranged with their like poles all pointing in one
direction. This affords a satisfactory explanation of the fact that
the free magnetism resides mainly at the poles, for at any inter-
mediate cross-section the magnetism on one side of the section is

MAGNETISM. 119

exactly balanced and neutralized by that on the other, consequent-
ly, the end layers are the only ones free to cause external effect.

153. Ewing's Theory. Two hypotheses may be advanced to
account for the arrangement of the molecular magnets. First,
before a steel bar is magnetized its molecules are unmagnetized
and the act of magnetizing imparts to them their magnetism and
arrangement. This throws but little light on the matter. Second,
the molecules possess magnetism as an inherent property and are
always magnetized but are indiscriminately arranged or rather
are arranged in little groups satisfying each other's polarity and
thus neutralizing each other's magnetic effects and producing
little or no external magnetism. The act of magnetizing simply
turns these molecules until their like poles point in one direction.
The maximum effect would be produced when all of the molecules
had been turned and the magnet is then said to be saturated.

This theory was first advanced by Weber and later elaborated
by Ewing, whose name it bears. The latter showed that it satis-
factorily accounts for the known facts of magnetization, especially,
as will be seen later (Par. 395), for the varying rate of change of
magnetism accompanying a constant rate of change of the mag-
netizing force. Certain corroborating phenomena are described
in the following paragraphs.

154. Magnetization is Accompanied by Molecular Movement.

(a) If a small glass tube be filled with steel filings and then
subjected to magnetization, the filings will be seen to arrange
themselves .-end to end and thereafter the tube will act as a magnet.
This is thought to be analogous to what takes place among the
molecules of a magnetic body during magnetization. If the filings
be shaken up and disarranged the magnetism disappears.

(b) When an iron bar is suddenly magnetized by an electric
current a metallic clink is heard. This could be produced only
by a vibration among the molecules of the bar.

(c) When an iron bar is rapidly magnetized and demagnetized
it grows hot. This heat could be produced only by internal move-
ment among the molecules.

(d) Magnetization is accompanied by a change in the dimen-
sions of the magnetic substance. An iron bar strongly magnetized
increases 1/720,000 of its length but if still more strongly magnet-
ized it contracts again. A bar of cobalt at first diminishes and

120 ELEMENTS OF ELECTRICITY.

then increases. Nickel diminishes from the first. Later inves-
tigations show that iron and steel contract along the lines of
magnetization and expand across these lines. The change in
dimensions must result from movement among the molecules.

155. Freedom of Molecular Movement Facilitates Magneti-
zation. When a magnetic body is placed in a magnetic field, we
may consider that a force is tugging at each of the little molecular
magnets endeavoring to turn them so that their like poles will
point in one direction. This turning is impeded by the crowding
of the molecules or by what may be designated molecular friction.
If this crowding or friction be relieved in any way, as by vibration,
by heating or by liquefaction, magnetization is rendered much
easier.

It has long been known that hard steel is very much more
difficult to magnetize than soft iron but that once magnetized it
retains its magnetism much better, or, as this last is usually
expressed, its retentivity is much greater than that of iron. We
may consider that the molecules of the rigid steel offer more
resistance to turning than do those of the soft iron, and on account
of this same rigidity they remain more persistently in the position
into which they have been turned. A piece of pure iron loses its
magnetism as soon as the magnetizing force is discontinued. The
iron generally used in electrical machinery is not absolutely pure
and some traces of magnetism persist after the cessation of the
magnetizing force. This residual magnetism plays an important
part in certain electric machines.

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