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substances are feebly so, so feebly however that the property can
be detected only by the most delicate apparatus and for practical
purposes may be neglected. Among these substances are some of
the salts of iron, and oxygen, especially when liquefied.

Gilbert carefully distinguished between magnets and magnetic
substances. A magnet exerts its attraction at certain portions
only, has polarity and its poles will repel similar poles of a second
magnet. A magnetic substance, such as soft iron, has no polarity,
attracts either pole of a magnet at any portion of its surface and
does not attract other magnetic substances.

122. Diamagnetism. It has long been known that some sub-
stances, notably bismuth and antimony, are repelled from the
poles of a magnet, the bodies placing themselves so that their
longer axis is at right angles to the magnet. Explanation of this
phenomenon can not be given until the subject of electro-magnetics
is reached (Par. 402). The repulsion is very feeble and delicate
instruments are required to detect it. Faraday investigated the
magnetic properties of many bodies and those which are attracted
he called paramagnetics; those which are repelled, diamagnetics.
The majority of liquids, except those containing in solution the
salts of iron, are feebly diamagnetic. The subject is of theoretical
interest only.



MAGNETISM.



93



CHAPTER 13.

MEASUREMENT OF MAGNETIC FORCES.

123. Coulomb's First Law. The first law of magnetic force
has already been given (Par. 115) and is that like poles repel and
unlike poles attract one another. The second law deals with the
variation of this force of attraction or repulsion. Before develop-
ing it, we must get some preliminary notion of what is meant by
the strength of magnets.

124. Lifting Power of Magnets. At first sight it might seem
that a simple way to determine and compare the strength of mag-
nets would be to ascertain the weight which they could support.
Various pieces of apparatus have been devised for this purpose.
For example (Fig. 56) the magnet is held vertically in a frame
and supports by its attraction an iron piece or armature A.
Attached to this armature is a hook from

which hangs a receptacle into which fine
sand is slowly poured. When the accumu-
lated weight reaches a certain point the
armature is torn away and with the recep-
tacle drops to a table placed just beneath
to receive it. The total weight supported N\ \
by the magnet is then determined by \\\
weighing. \\

When, however, the conditions of this \o[
experiment are varied, it will be seen that
these results are of but little value. The
following will make this clear.

(a) If one end of a bar magnet be
rounded and the other be squared, the
rounded end will lift a greater weight than

the squared end, and this although it can Fig. 56.

be shown that the two ends are of equal magnetism. The weight

lifted therefore varies with the shape of the pole.

(b) If the magnet be bent into a horseshoe shape so that both
poles concur in the lifting, instead of the weight being just twice




94 ELEMENTS OF ELECTRICITY.

what it was before for a single pole, it may be three or even four
times greater. The weight lifted therefore varies with the shape
of the magnet.

(c) If the weight be applied very gradually the magnet will
support more than it would if it were applied suddenly. If a
magnet be loaded to nearly the maximum point and the load be
left hi position for a day, the weight may then be gradually in-
creased until it considerably exceeds the original maximum. Once,
however, that the armature is torn away, the lifting power of the
magnet drops back to what it was formerly.

(d) Within certain limits, the larger the armature the greater
the weight lifted.

(e) The weight lifted varies with the character of the iron or
steel of which the armature is composed.

(f) Retaining the same weight and compositon, a greater
weight will be lifted if the armature be of a compact shape, such
as a cube, than if it be a flat disc. The weight lifted therefore
varies with the size and shape of the armature.

In the last four cases above we see that although the magnet
itself does not vary, the weight lifted fluctuates through a wide
range. We can not say which of these weights should be taken
to measure the strength of the magnet nor is it practicable to give
mathematical expression to the heterogeneous conditions enu-
merated and deduce formulae from which this strength might
be calculated. What we have done therefore is not to measure
the strength of the magnet but its lifting power under certain given
conditions.

A small bar magnet should lift from 15 to 25 times its own
weight. In the Paris Exhibition of 1882 there was shown a magnet
which supported 76 times its own weight. Thompson states that
the lifting power of a good steel magnet may amount to 40 pounds
per square inch of pole surface. Electro-magnets, to be described
later, are much more powerful, the lifting power reaching 200
pounds per square inch.

125. Strength of Magnets. If we examine the force with
which one magnet attracts or repels another, the two being at
some distance apart, we find that it is not affected by shape of
poles or length of exposure to each other's influence, etc., but is
to a great extent independent of the varying conditions mentioned
in the preceding paragraph. The force with which magnetic poles,



MAGNETISM.



95



interact is therefore selected as the measure of their strength. We
are thus naturally led to enquire what is precisely a magnetic pole
and how is the force between two poles measured.

126. Magnetic Pole Defined. In the preceding pages we have
used the word pole to designate rather vaguely the terminal por-
tions of a magnet, the regions in which the magnetic force is most
marked. In mathematical discussions it is desirable to treat a
pole as if it were a focus or the point of application of the resultant
of the magnetic forces at that particular end of the magnet. This
point may be approximately located as follows. In a bar magnet
the magnetic forces are symmetrically distributed around its axis
and the pole must consequently lie upon this axis. In Fig. 57 let
MN represent one-half of the bar magnet which is supported



N




Fig. 57.



horizontally. With a pencil mark off this half in equal divisions.
Cut a small soft iron wire into a number of short pieces of equal
length (and hence of equal weight). Apply the end of one of these
pieces to one of the divisions of the bar and then other pieces to
the first piece until the accumulated cluster drops off of its own
weight. Note the particular division and the corresponding
number of pieces. Repeat this for each of the divisions, then
construct a curve MEN in which the divisions are the abscissae
and the ordinates are laid off to a scale to represent the number of
pieces of wire supported. The pole is on the axis of the magnet
and approximately opposite the center of gravity of the tri-
angular figure MEN.



96 ELEMENTS OF ELECTRICITY.

In short thick magnets the poles are distributed over a con-
siderable area but for long slender bars they approach the ends
and approximate the hypothetical point or focus. According to
Fleming, the poles of a bar magnet are about one-twelfth of its
length from the ends. For shorter and thicker bars this distance
may amount to one-sixth or even one-fifth.

It will be shown later (Par. 146) that for certain magnetic
measurements the exact location of the pole is immaterial.

Although it is impossible to get a magnetic pole unaccompanied
by an equal and opposite pole, yet in a long slender magnet the
poles are so far apart that in many experiments the effect of the
more distant one may be neglected and the results are as if we
were dealing with a single or "free" pole.

127. Measurement of Magnetic Forces. The measurement of
magnetic forces is not entirely a simple matter. Two magnets,
A and B, exposed so each other's influence are each acted upon by
four forces. The north pole of A is repelled by the north pole of
B and attracted by its south pole; the south pole of A is repelled
by the south pole of B and attracted by its north pole. In addi-
tion, each magnet is acted upon by the earth's magnetic poles so
that each is subject to eight forces.

In most cases the forces are comparatively feeble. They must
therefore be measured by comparing them with, or by balancing
them against, other forces, likewise feeble, whose variation follows
some readily determined law. The forces used for comparison
are

(a) A known magnetic force, usually that of the earth. There
are two methods of comparison, both of which will shortly be
described (Pars. 129, 146).

(b) The torsion of a suspending thread, as in Coulomb's torsion
balance, already described (Par. 52). The law in this case is that
the force varies directly as the angle through which the thread is
twisted.

(c) The force of gravity applied through a bifilar suspension.
A magnet is suspended in a horizontal position by two parallel
threads. If it be deflected the threads must be twisted from a
vertical to an oblique position and the magnet must therefore be
raised. The law in this case is that the force varies directly as the
sine of the angle through which the magnet is twisted.



MAGNETISM. 97

128. Coulomb's Second Law. The second law of magnetic
force comprises two statements. First, the force exerted between
two magnetic poles varies directly with the product of their
strengths, and second, this force varies inversely as the square of
the distance separating them. The distance between the poles is
supposed to be so great that they may be regarded as points.

The first of these statements hardly requires proof since it
follows at once from the fact that the action between poles is
mutual and that if we double or treble the strength of either one
we double or treble the force exerted between them. Its truth
may easily be shown experimentally. The second statement
is proved experimentally by one of the methods now to be
described.



129. Method by Oscillations. From mechanics, the time of
oscillation of a simple pendulum, its angular displacement being
small, is given by the equation

I
9



-*v



in which I is the length of

the pendulum and g is the acceleration due to gravity. The force
acting upon the pendulum is mg, m being its mass. The above
expression may be written



V mg V force
whence

Force = ^~ = constant X ^

But \ is the number of oscillations n in a unit of time, hence
Force = constant Xft 2 ,

or the force producing

pendular vibrations is proportional to the square of the number
of vibrations executed in a unit of time. Any convenient inter-
val of time may be taken as the unit.

If a magnetic needle AB (Fig. 58), whose position of rest is
along the magnetic meridian NS, be pushed aside through an
angle 5 and then released, it will be acted upon by forces tending




98 ELEMENTS OF ELECTRICITY.

to return it to its primary position but in swinging back it will
acquire a momentum which will carry it very nearly an equal
angular distance beyond NS and will then swing
in the opposite direction and so on, that is, the
needle will act as a double pendulum in a hori-
zontal plane and if 8 be small will execute oscil-
lations whose period is practically constant. The
forces which tend to restore the needle to its posi-
tion in the meridian are due to the interaction of
the poles of the magnet and those of the earth.
The earth being a sphere and its poles being
located beneath its surface, their action lines are
oblique to the plane of oscillation of the needle
and only the horizontal component of the forces
along these lines affects the oscillations. This
horizontal component of the earth's magnetism
is usually designated by the letter H. Within the
limits of space covered by the average experiment H is constant.
If the strength of the poles of the needle be represented by m, the
force acting upon each pole will be mH, represented in Fig. 58 by
AD and BC and, from what we have seen above, this force is pro-
portional to the square of the number of oscillations executed by
the needle in a given time. How this principle may be utilized in
measurements will be shown in Par. 131.

130. Magnetic Moment. The active components of the forces
AD and BC (Fig. 58) are AE and BF, each of which is equal
to m. H.sm8. These constitute a couple whose moment is
m.H.smd.l, I being the distance between the two poles. The
product ml is called the magnetic moment of the needle and in
formulae is represented by M. Although the exact position of
the poles, and consequently the distance I, is most often unknown,
M itself may be determined by experiment and is used in certain
magnetic measurements to be described later (Pars. 148, 149, 150).

131. Experimental Proof of Law of Inverse Squares. In

Fig. 59, A is a very small magnet, less than half an inch in length,
suspended in a paper stirrup by a single fibre of unspun silk and
at rest in the magnetic meridian. The resistance of the silk fibre
being very slight, if the magnet be started in oscillation it will
continue so for from five to ten minutes. It is given a slight im-






MAGNETISM. 99

pulse and the number of oscillations executed in a given interval,
say one minute, is counted. Suppose this number to be 10. A
slender bar magnet B is now placed in the same meridian, its axis
in the prolongation of the axis of A. (This experiment may also
be performed with the magnet B in a vertical position, its pole
being in the meridian and horizontal plane of A.) B must be
placed at such distance from A that for small angular deviations
of A the action lines of B are sensibly parallel. This is also one of
the reasons for keeping A very small, the other being that if A be
small its poles are more nearly the same distance from the pole of
B. A is again set in motion and if the poles of the bar magnet




Fig. 59.

coincide in direction with those of A, the oscillations will be more
rapid. Suppose that now 12 are executed in one minute. The
force due to the horizontal component of the earth's magnetism
is to the combined force of this component and that of the pole of
B as 100 is to 144. Let B now be pushed up towards A until the
distance between its pole and that of A has been halved. A set in
motion will now be found to execute about 16.5 oscillations per
minute. The total force upon A in the first place is to that in the
second as (12) 2 is to (16.5) 2 or as 144 is to 272. The force due to B
alone is as (144-100) is to (272-100) or as 44 is to 172, which is
very nearly as 1 is to 4. In other words, as the distance is halved
the force is quadrupled, which is in accordance with the law of
inverse squares.

In the above proof the distance between the poles of the two
magnets must be known and as the exact position of the poles
themselves is not precise, their distance apart is apparently un-
certain; however, this distance may be assumed as nearly as pos-
sible and one or two trial experiments thereafter will show what
the correct distance should be.

132. Proof of Law of Inverse Squares by Coulomb's Balance.

Coulomb also proved this law by means of the balance which
bears his name. A description of the actual experiment would be



100 ELEMENTS OF ELECTRICITY.

somewhat long and it will suffice to say that in the apparatus as
represented in Fig. 22, a slender bar magnet took the place of the
shellac needle G and a second one took that of KH. The instru-
ment was set up so that with no torsion on the suspending fibre,
the horizontal needle and the opening K in the glass cover lay in
the same magnetic meridian. The experiment was then conducted
as explained in Par. 52, due allowance being made for the effect of
the earth's magnetism.

133. Unit Magnetic Pole. Magnetic poles differ in strength.
We may consider that there is more magnetism concentrated at
the stronger pole or may assume that there are magnetic poles of
unit strength and that a greater number of these are gathered at
the stronger pole. A definite conception of a unit pole may be
obtained from the following. Coulomb's second law may be
expressed thus,

f _ m X m'
J '' ~~~

in which, since we are

using the C. G. S. system, /is the force in dynes between the poles,
m and m f the strength of the respective poles and d their distance
apart in centimeters. If the poles be of equal strength this
becomes



Finally, if / becomes one dyne and d one centimeter, we have
w = l, or a unit magnetic pole is that pole which when placed at a
distance of one centimeter from a similar and equal pole repels it
with a force of one dyne.



MAGNETISM. 101



CHAPTER 14.
THE MAGNETIC FIELD.

134. Magnetic Field. In the space around a magnet all poles
experience forces of attraction and of repulsion and this space is
called the field of the magnet. As we recede from the magnet
these forces diminish in accordance with the law of inverse squares
and, to fix its limits more definitely, we define a magnetic field as
that space surrounding a magnet in which magnetic force due to
this magnet is perceptible.

135. Direction of Magnetic Field. As an aid to the conception
of a magnetic field we may resort to the same analogy as in the case
of the electric field (Par. 58) and compare it to a current of water.
In a magnetic field there is no matter in actual movement but
there is in a certain sense a flow of force and magnetic poles, placed
in the field, are swept along just as light objects are carried by a
stream. Since free north poles would be carried along in one
direction and free south poles in the opposite direction we by
convention define the positive direction of a magnetic field as that
direction in which a free north pole would move.

136. Intensity of Magnetic Field. Just as we might measure
the strength of a current by the force with which it pushes a board
of unit area placed in it, so we agree to measure the intensity of a
magnetic field by the force with which it acts upon a unit pole
placed in it and we define a unit magnetic field as that field which
acts with a force of one dyne upon a unit pole placed in it. If we
say that a magnetic field has a strength of three, we mean that it
will act with a force of three dynes upon a unit pole placed in it.
If the strength of the field be H and that of the pole be m, the
force with which the field acts upon the pole is Hm dynes. From
the foregoing and from Par. 128 it follows that the field at a dis-
tance d from a pole of strength m is m/d 2 .

137. Magnetic Lines of Force. In Fig. 60 let P be a point in
the field of the bar magnet NS and for simplicity of construction
suppose that at this point the distance SP is twice the distance



102 ELEMENTS OF ELECTRICITY.

NP. Suppose a free north pole to be placed at the point P. It
will be repelled from N along NP and attracted towards S along
PS. In the case assumed the distance NP being only one-half of
PS, by the law of inverse squares the repulsion along NP is four




J Fig. 60. V

times as great as the attraction along PS. Lay off PB any con-
venient distance and PA four times as great and complete the
parallelogram. PR is the resultant at the point P of the magnetic
forces of the two poles N and S or, in other words, the free north
pole at P will be urged along the resultant PR with a force pro-
portional to PR.

Suppose this free north pole to move along PR a very small
distance. In doing so it will move away from N more rapidly
than it does from S. This will cause the repulsion from N to grow
weaker and the attraction to S to grow relatively stronger and the
path of the pole will bend around towards S. In its successive
positions therefore, the pole will follow a curve which at every
point indicates by its direction the direction of the resultant of
the magnetic forces at that point. This curve is called a magnetic
line of force.

If instead of a free north pole a free south pole had been placed
at P, it would have been urged with an exactly equal force in an
exactly opposite direction PR', and in its path would have traced
out the same curved line but in a reverse direction. By convention
(Par. 135) we define the positive direction of a magnetic line of
force as that direction in which a free north pole would move. In



MAGNETISM.



103



our diagrams the positive direction of these lines is always in-
dicated by an arrowhead placed upon the lines.

138. Mapping Lines of Force. If at any point P (Fig. 60)
there be placed a very small magnetic needle, its north pole would
be urged in the direction PR, its south pole in the direction PR',
and the needle will take up a position approximately tangent to
the line of force at the point P. If a sufficient number of these
little needles be placed one after the other, as shown in Fig. 60,






) -. . ' ; : /^ :> .^VV:\s ~-r\vT: - * '//'-> ',#> ,>;':' A; ' . ' "^

. *:/:;. .;\-J : Aii> ^ z-* -'^;*- : ^ *'*-?*.'&*..&; 4 \^* '

Fig. 61.




the successive tangents which they indicate will serve as an enve-
lope and will mark out the line of force, approximating more and
more closely to it as their length is decreased and number increased.
Finally, if the entire space about the magnet were strewn closely
with the little needles a number of lines of force would be
traced.

This condition may be realized practically as follows. A sheet
of glass, of stiff paper or of any non-magnetic body is placed upon
a magnet and is then sprinkled with fine iron filings. From what
we have already seen (Par. 120) each individual filing becomes for
the time being a magnet, but these little magnets are not free to



104 ELEMENTS OF ELECTRICITY.

move since their weight holds them with friction against the sur-
face over which they are sprinkled. If the sheet be given a gentle
tap the filings are jarred and for a minute interval of time are
bounced up into the air. Being now freed from the friction which
held them in place, they move under the influence of the magnetic
forces and after a few repetitions of the jarring they gather along
well marked lines as shown in Fig. 61.

139. Permanent Record of Magnetic Figures. Several ways
have been described by which these magnetic figures, or curves
traced by the filings, may be recorded permanently. The following
is simple and convenient. Upon a soft pine board about a foot
square the magnet is placed and its outline is traced with a pencil.
With a chisel this outline is then hollowed out until when in posi-
tion the upper surface of the magnet is on a level with that of the
board. The board is then taken into a subdued light and there is
pinned upon it, prepared surface up, a sheet of blue-print paper
about 8 XlO inches. Iron filings are then sprinkled over this paper
and the board is tapped on the under side until the magnetic
figures come out as desired. Better results are obtained if before
using the filings they are passed through two sieves, one to separate
the dust-like particles and the other those of too large size. The
board with the filings in position is then exposed in a strong
sunlight for from three to five minutes, the rays falling as
nearly perpendicular to the paper as possible. It is then carried
back to the subdued light, the filings poured off and the paper
thoroughly washed in clear water. The resulting blue-print is
then dried.

140. Use of Magnetic Figures. These magnetic figures are of
assistance in the study of magnetic fields and often enable us to
grasp at a glance conditions which might otherwise require con-
siderable mathematical analysis to develop. For example, they
show in a striking manner how the field between two mutually
attracting poles differs from that between two that mutually
repel. Fig. 62 represents the field between two dissimilar poles.
In this the lines of force are seen to pass from one to the other as
if pulling them together. At the same time these lines are bowed
out revealing the existence of the crosswise pressure causing them
to separate. Fig. 63 shows the field between two similar poles and
it does not require a great stretch of the imagination to conceive



MAGNETISM.



105



of the lines of force as hands placed palm against palm and pushing
each other back. Further examples of the use of these figures will
be met in subsequent pages.




Fig. 62.




Fig. 63.

141. Compounding Magnetic Fields. The magnetic fields
hitherto considered are those surrounding a single pole or pair of
poles and are symmetrical with respect to the single pole or to the



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