_Directing Couple acting on an Elongated_Encyclopaedia Britannica, 11th Edition, "McKinley, William" to "Magnetism, Terrestrial" Volume 17, Slice 3

field every volume-element of the body tends to move towards regions

of greater or less force according as the substance is paramagnetic or

diamagnetic, and the behaviour of the whole mass will be determined

chiefly by the tendency of its constituent elements. For this reason a

thin bar suspended at its centre of gravity between a pair of magnetic

poles will, if paramagnetic, set itself along the line joining the

poles, where the field is strongest, and if diamagnetic, transversely

to the line. These are the "axial" and "equatorial" positions of

Faraday. It can be shown[11] that in a uniform field an elongated

piece of any non-crystalline material is in stable equilibrium only

when its length is parallel to the lines of force; for diamagnetic

substances, however, the directing couple is exceedingly small, and it

would hardly be possible to obtain a uniform field of sufficient

strength to show the effect experimentally.

_Relative Magnetization._--A substance of which the real

susceptibility is [kappa] will, when surrounded by a medium having the

susceptibility [kappa]´, behave towards a magnet as if its

susceptibility were [kappa]_a = ([kappa] - [kappa]´)/(1 +

4[pi][kappa]´). Since 1 + 4[pi][kappa]´ can never be negative, the

apparent susceptibility [kappa]_a will be positive or negative

according as [kappa] is greater or less than [kappa]´. Thus, for

example, a tube containing a weak solution of an iron salt will appear

to be diamagnetic if it is immersed in a stronger solution of iron,

though in air it is paramagnetic.[12]

_Circular Magnetization._--An electric current i flowing uniformly

through a cylindrical wire whose radius is a produces inside the wire

a magnetic field of which the lines of force are concentric circles

around the axis of the wire. At a point whose distance from the axis

of the wire is r the tangential magnetic force is

H = 2ir/a² (39)

it therefore varies directly as the distance from the axis, where it

is zero.[13] If the wire consists of a ferromagnetic metal, it will

become "circularly" magnetized by the field, the lines of

magnetization being, like the lines of force, concentric circles. So

long as the wire (supposed isotropic) is free from torsional stress,

there will be no external evidence of magnetism.

_Magnetic Shielding._--The action of a hollow magnetized shell on a

point inside it is always opposed to that of the external magnetizing

force,[14] the resultant interior field being therefore weaker than

the field outside. Hence any apparatus, such as a galvanometer, may be

partially shielded from extraneous magnetic action by enclosing it in

an iron case. If a hollow sphere[15] of which the outer radius is R

and the inner radius r is placed in a uniform field H0, the field

inside will also be uniform and in the same direction as H0, and its

value will be approximately

H_i = ---------------------------- (40)

1 + (2/9)(µ - 2) (1 - r³/R³)

For a cylinder placed with its axis at right angles to the lines of

force,

H_i = ------------------------ (41)

1 + ¼(µ - 2) (1 - r²/R²)

These expressions show that the thicker the screen and the greater its

permeability µ, the more effectual will be the shielding action. Since

µ can never be infinite, complete shielding is not possible.

_Magneto-Crystallic Phenomenon._--In anisotropic bodies, such as

crystals, the direction of the magnetization does not in general

coincide with that of the magnetic force. There are, however, always

three _principal axes_ at right angles to one another along which the

magnetization and the force have the same direction. If each of these

axes successively is placed parallel to the lines of force in a

uniform field H, we shall have

I1 = [kappa]1H, I2 = [kappa]2H, I3 = [kappa]3H,

the three susceptibilities [kappa] being in general unequal, though in

some cases two of them may have the same value. For crystalline bodies

the value of [kappa] (+ or -) is nearly always small and constant, the

magnetization being therefore independent of the form of the body and

proportional to the force. Hence, whatever the position of the body,

if the field be resolved into three components parallel to the

principal axes of the crystal, the actual magnetization will be the

resultant of the three magnetizations along the axes. The body (or

each element of it) will tend to set itself with its axis of greatest

susceptibility parallel to the lines of force, while, if the field is

not uniform, each volume-element will also tend to move towards places

of greater or smaller force (according as the substance is

paramagnetic or diamagnetic), the tendency being a maximum when the

axis of greatest susceptibility is parallel to the field, and a

minimum when it is perpendicular to it. The phenomena may therefore be

exceedingly complicated.[16]

3. MAGNETIC MEASUREMENTS

_Magnetic Moment._--The moment M of a magnet may be determined in many

ways,[17] the most accurate being that of C. F. Gauss, which gives the

value not only of M, but also that of H, the horizontal component of the

earth's force. The product MH is first determined by suspending the

magnet horizontally, and causing it to vibrate in small arcs. If A is

the moment of inertia of the magnet, and t the time of a complete

vibration, MH = 4[pi]²A/t² (torsion being neglected). The ratio M/H is

then found by one of the magnetometric methods which in their simplest

forms are described below. Equation (44) shows that as a first

approximation.

M/H = (d² - l²) tan [theta]/2d,

where l is half the length of the magnet, which is placed in the

"broadside-on" position as regards a small suspended magnetic needle, d

the distance between the centre of the magnet and the needle, and

[theta] the angle through which the needle is deflected by the magnet.

We get therefore

M² = MH × M/H = 2[pi]²A(d² - l²)² tan [theta]/t²d (42)

H² = MH × H/M = 8[pi]²Ad/{t²(d² - l²)² tan [theta]} (43)

When a high degree of accuracy is required, the experiments and

calculations are less simple, and various corrections are applied. The

moment of a magnet may also be deduced from a measurement of the couple

exerted on the magnet by a uniform field H. Thus if the magnet is

suspended horizontally by a fine wire, which, when the magnetic axis

points north and south, is free from torsion, and if [theta] is the

angle through which the upper end of the wire must be twisted to make

the magnet point east and west, then MH = C[theta], or M = C[theta]/H,

where C is the torsional couple for 1°. A bifilar suspension is

sometimes used instead of a single wire. If P is the weight of the

magnet, l the length of each of the two threads, 2a the distance between

their upper points of attachment, and 2b that between the lower points,

then, approximately, MH = P(ab/l) sin [theta]. It is often sufficient to

find the ratio of the moment of one magnet to that of another. If two

magnets having moments M, M´ are arranged at right angles to each other

upon a horizontal support which is free to rotate, their resultant R

will set itself in the magnetic meridian. Let [theta] be the angle which

the standard magnet M makes with the meridian, then M´/R = sin [theta],

and M/R = cos [theta], whence M´ = M tan [theta].

[Illustration: FIG. 5.]

A convenient and rapid method of estimating a magnetic moment has been

devised by H. Armagnat.[18] The magnet is laid on a table with its north

pole pointing northwards, A compass having a very short needle is placed

on the line which bisects the axis of the magnet at right angles, and is

moved until a neutral point is found where the force due to the earth's

field H is balanced by that due to the magnet. If 2l is the distance

between the poles m and -m, d the distance from either pole to a point P

on the line AB (fig. 5), we have for the resultant force at P

R = -2 cos [theta] × m/d² = -2lm/d³ = -M/d³.

When P is the neutral point, H is equal and opposite to R; therefore M =

Hd³, or the moment is numerically equal to the cube of the distance from

the neutral point to a pole, multiplied by the horizontal intensity of

the earth's force. The distance between the poles may with sufficient

accuracy for a rough determination be assumed to be equal to five-sixths

of the length of the magnet.

_Measurement of Magnetization and Induction._--The magnetic condition

assumed by a piece of ferromagnetic metal in different circumstances is

determinable by various modes of experiment which may be classed as

magnetometric, ballistic, and traction methods. When either the

magnetization I or the induction B corresponding to a given magnetizing

force H is known, the other may be found by means of the formula B =

4[pi]I + H.

_Magnetometric Methods._--Intensity of magnetization is most directly

measured by observing the action which a magnetized body, generally a

long straight rod, exerts upon a small magnetic needle placed near it.

The magnetic needle may be cemented horizontally across the back of a

little plane or concave mirror, about ¼ or 3/8 in. in diameter, which is

suspended by a single fibre of unspun silk; this arrangement, when

enclosed in a case with a glazed front to protect it from currents of

air, constitutes a simple but efficient magnetometer. Deflections of the

suspended needle are indicated by the movement of a narrow beam of light

which the mirror reflects from a lamp and focusses upon a graduated

cardboard scale placed at a distance of a few feet; the angular

deflection of the beam of light is, of course, twice that of the needle.

The suspended needle is, in the absence of disturbing causes, directed

solely by the horizontal component of the earth's field of magnetic

force H_E, and therefore sets itself approximately north and south. The

magnetized body which is to be tested should be placed in such a

position that the force H_P due to its poles may, at the spot occupied

by the suspended needle, act in a direction at right angles to that due

to the earth--that is, east and west. The direction of the resultant

field of force will then make, with that of H_E, an angle [theta], such

that H_p/H_E = tan [theta], and the suspended needle will be deflected

through the same angle. We have therefore

H_P = H_E tan [theta].

The angle [theta] is indicated by the position of the spot of light upon

the scale, and the horizontal intensity of the earth's field H_E is

known; thus we can at once determine the value of H_P, from which the

magnetization I of the body under test may be calculated.

In order to fulfil the requirement that the field which a magnetized

rod produces at the magnetometer shall be at right angles to that of

the earth, the rod may be conveniently placed in any one of three

different positions with regard to the suspended needle.

[Illustration: FIG. 6.]

(1) The rod is set in a horizontal position level with the suspended

needle, its axis being in a line which is perpendicular to the

magnetic meridian, and which passes through the centre of suspension

of the needle. This is called the "end-on" position, and is indicated

in fig. 6. AB is the rod and C the middle point of its axis; NS is the

magnetometer needle; AM bisects the undeflected needle NS at right

angles. Let 2l = the length of the rod (or, more accurately, the

distance between its poles), v = its volume, m and -m the strength of

its poles, and let d = the distance CM. For most ordinary purposes the

length of the needle may be assumed to be negligible in comparison

with the distance between the needle and the rod. We then have

approximately for the field at M due to the rod

m m 4dl

H_P = -------- - -------- = m ----------.

(d - l)² (d + l)² (d² - l²)²

Therefore

(d² - l²)²H_P (d² - l²)² H_E tan [theta]

2ml = M = ------------- = --------------------------. (44)

2d 2d

And

M (d² - l²)²H_E

I = --- = ------------- tan [theta], (45)

v 2dv

whence we can find the values of I which correspond to different

angles of deflection.

(2) The rod may be placed horizontally east and west in such a

position that the direction of the undeflected suspended needle

bisects it at right angles. This is known as the "broadside-on"

position, and is represented in fig. 7. Let the distance of each pole

of the rod AB from the centre of the magnetometer needle = d. Then,

since H_P, the force at M due to m and -m, is the resultant of m/d²

and -m/d², we have

H_P 2l

--- = ---

m d

2ml

H_P = ---,

d³

the direction being parallel to AB.

And

d³ H_P d³ H_E

I = ------ = ------ tan [theta]. (46)

v v

[Illustration: FIG. 7.]

[Illustration: FIG. 8.]

(3) In the third position the test rod is placed vertically with one

of its poles at the level of the magnetometer needle, and in the line

drawn perpendicularly to the undeflected needle from its centre of

suspension. The arrangement is shown in fig. 8, where AB is the

vertical rod and M indicates the position of the magnetometer needle,

which is supposed to be perpendicular to the plane of the paper.

Denoting the distance AM by d1, BM by d2, and AB by l, we have for the

force at M due to the magnetism of the rod

m m

H_P = --- - horizontal component of ---

d1² d2²

/ 1 d1 \

= m ( --- - --- ).

\ d1² d2³ /

Therefore

H_P d1² H_E

m = --------- = ------------ tan [theta],

1 d1 / d1 \³

--- - --- 1 - ( ---- )

d1² d2³ \ d2 /

and

ld1²H_E

I = ------------------ tan [theta]. (47)

{ / d1 \³ }

v { 1 - ( ---- ) }

{ \ d2 / }

This last method of arrangement is called by Ewing the "one-pole"

method, because the magnetometer deflection is mainly caused by the

upper pole of the rod (_Magnetic Induction_, p. 40). For experiments

with long thin rods or wires it has an advantage over the other

arrangements in that the position of the poles need not be known with

great accuracy, a small upward or downward displacement having little

effect upon the magnetometer deflection. On the other hand, a

vertically placed rod is subject to the inconvenience that it is

influenced by the earth's magnetic field, which is not the case when

the rod is horizontal and at right angles to the magnetic meridian.

This extraneous influence may, however, be eliminated by surrounding

the rod with a coil of wire carrying a current such as will produce in

the interior a magnetic field equal and opposite to the vertical

component of the earth's field.

If the cardboard scale upon which the beam of light is reflected by

the magnetometer mirror is a flat one, the deflections as indicated by

the movement of the spot of light are related to the actual

deflections of the needle in the ratio of tan 2[theta] to [theta].

Since [theta] is always small, sufficiently accurate results may

generally be obtained if we assume that tan 2[theta] = 2 tan [theta].

If the distance of the mirror from the scale is equal to n scale

divisions, and if a deflection [theta] of the needle causes the

reflected spot of light to move over s scale divisions, we shall have

s/n = tan 2[theta] exactly,

s/2n = tan [theta] approximately.

We may therefore generally substitute s/2n for tan [theta] in the

various expressions which have been given for I.

Of the three methods which have been described, the first two are

generally the most suitable for determining the moment or the

magnetization of a permanent magnet, and the last for studying the

changes which occur in the magnetization of a long rod or wire when

subjected to various external magnetic forces, or, in other words, for

determining the relation of I to H. A plan of the apparatus as

arranged by Ewing for the latter purpose is shown diagrammatically in

fig. 9. The cardboard scale SS is placed above a wooden screen, having

in it a narrow vertical slit which permits a beam of light from the

lamp L to reach the mirror of the magnetometer M, whence it is

reflected upon the scale. A is the upper end of a glass tube, half a

metre or so in length, which is clamped in a vertical position behind

the magnetometer. The tube is wound over its whole length with two

separate coils of insulated wire, the one being outside the other. The

inner coil is supplied, through the intervening apparatus, with

current from the battery of secondary cells B1; this produces the

desired magnetic field inside the tube. The outer coil derives

current, through an adjustable resistance R, from a constant cell B2;

its object is to produce inside the tube a magnetic field equal and

opposite to that due to the earth's magnetism. C is a "compensating

coil" consisting of a few turns of wire through which the magnetizing

current passes; it serves to neutralize the effect produced upon the

magnetometer by the magnetizing coil, and its distance from the

magnetometer is so adjusted that when the circuit is closed, no

ferromagnetic metal being inside the magnetizing coil, the

magnetometer needle undergoes no deflection. K is a commutator for

reversing the direction of the magnetizing current, and G a

galvanometer for measuring it. The strength of the magnetizing current

is regulated by adjusting the position of the sliding contact E upon

the resistance DF. The current increases to a maximum as E approaches

F, and diminishes to almost nothing when E is brought up to D; it can

be completely interrupted by means of the switch H.

[Illustration: FIG. 9.]

The specimen upon which an experiment is to be made generally consists

of a wire having a "dimensional ratio" of at least 300 or 400; its

length should be rather less than that of the magnetizing coil, in

order that the field H0, to which it is subjected, may be

approximately uniform from end to end. The wire is supported inside

the glass tube A with its upper pole at the same height as the

magnetometer needle. Various currents are then passed through the

magnetizing coil, the galvanometer readings and the simultaneous

magnetometer deflections being noted. From the former we deduce H0,

and from the latter the corresponding value of I, using the formulae

H0 = 4[pi]in/l and

d1² H_E

I = ----------------------- × s, (48)

2n[pi]r² {1 - (d1/d2)³}

where s is the deflection in scale-divisions, n the distance in

scale-divisions between the scale and the mirror, and r the radius of

the wire.

[Illustration: FIG. 10.]

The curve, fig. 10, shows the result of a typical experiment made upon

a piece of soft iron (Ewing, _Phil. Trans._ vol. clxxvi. Plate 59),

the magnetizing field H0 being first gradually increased and then

diminished to zero. When the length of the wire exceeds 400 diameters,

or thereabouts, H0 may generally be considered as equivalent to H, the

actual strength of the field as modified by the magnetization of the

wire; but if greater accuracy is desired, the value of H_i (= NI) may

be found by the help of du Bois's table and subtracted from H0. For a

dimensional ratio of 400, N =0.00028, and therefore H = H0 - 0.00028I.

This correction may be indicated in the diagram by a straight line

drawn from 0 through the point at which the line of I = 1000

intersects that of H = 0.28 (Rayleigh, Phil. Mag. xxii. 175), the true

value of H for any point on the curve being that measured from the

sloping line instead of from the vertical axis. The effect of the ends

of the wire is, as Ewing remarks, to shear the diagram in the

horizontal direction through the angle which the sloping line makes

with the vertical.

Since the induction B is equal to H + 4[pi]I, it is easy from the

results of experiments such as that just described to deduce the

relation between B and H; a curve indicating such relation is called a

curve of induction. The general character of curves of magnetization

and of induction will be discussed later. A notable feature in both

classes of curves is that, owing to hysteresis, the ascending and

descending limbs do not coincide, but follow very different courses.

If it is desired to annihilate the hysteretic effects of previous

magnetization and restore the metal to its original condition, it may

be demagnetized by reversals. This is effected by slowly moving the

sliding contact E (fig. 9) from F to D, while at the same time the

commutator K is rapidly worked, a series of alternating currents of

gradually diminishing strength being thus caused to pass through the

magnetizing coil.

The magnetometric method, except when employed in connexion with

ellipsoids, for which the demagnetizing factors are accurately known,

is generally less satisfactory for the exact determination of induction

or magnetization than the ballistic method. But for much important

experimental work it is better adapted than any other, and is indeed

sometimes the only method possible.[19]

_Ballistic Methods._--The so-called "ballistic" method of measuring

induction is based upon the fact that a change of the induction through

a closed linear conductor sets up in the conductor an electromotive

force which is proportional to the rate of change. If the conductor

consists of a coil of wire the ends of which are connected with a

suitable galvanometer, the integral electromotive force due to a sudden

increase or decrease of the induction through the coil displaces in the

circuit a quantity of electricity Q = [delta]BnsR, where [delta]B is the

increment or decrement of induction per square centimetre, s is the area

of the coil, n the number of turns of wire, and R the resistance of the

circuit. Under the influence of the transient current, the galvanometer

needle undergoes a momentary deflection, or "throw," which is

proportional to Q, and therefore to [delta]B, and thus, if we know the

deflection produced by the discharge through the galvanometer of a given

quantity of electricity, we have the means of determining the value of

[delta]B.

The galvanometer which is used for ballistic observations should have

a somewhat heavy needle with a period of vibration of not less than

five seconds, so that the transient current may have ceased before the

swing has well begun; an instrument of the d'Arsonval form is

recommended, not only because it is unaffected by outside magnetic

influence, but also because the moving part can be instantly brought

to rest by means of a short-circuit key, thus effecting a great saving

of time when a series of observations is being made. In practice it is

usual to standardize or "calibrate" the galvanometer by causing a

known change of induction to take place within a standard coil

connected with it, and noting the corresponding deflection on the

galvanometer scale. Let s be the area of a single turn of the standard

coil, n the number of its turns, and r the resistance of the circuit

of which the coil forms part; and let S, N and R be the corresponding

constants for a coil which is to be used in an experiment. Then if a

known change of induction [delta]B_a inside the standard coil is found

to cause a throw of d scale-divisions, any change of induction

[delta]B through the experimental coil will be numerically equal to

the corresponding throw D multiplied by snRB_a/SNrd. For a series of

experiments made with the same coil this fraction is constant, and we

may write [delta]B = kD. Rowland and others have used an earth coil

for calibrating the galvanometer, a known change of induction through

the coil being produced by turning it over in the earth's magnetic

field, but for several reasons it is preferable to employ an electric

current as the source of a known induction. A primary coil of length

l, having n turns, is wound upon a cylinder made of non-conducting and

non-magnetic material, and upon the middle of the primary a secondary

or induction coil is closely fitted. When a current of strength i is

suddenly interrupted in the primary, the increment of induction

through the secondary is sensibly equal to 4[pi]in/l units. All the

data required for standardizing the galvanometer can in this way be

determined with accuracy.

The ballistic method is largely employed for determining the relation of

induction to magnetizing force in samples of the iron and steel used in

the manufacture of electrical machinery, and especially for the

observation of hysteresis effects. The sample may have the form of a

closed ring, upon which are wound the induction coil and another coil

for taking the magnetizing current; or it may consist of a long straight

rod or wire which can be slipped into a magnetizing coil such as is used

in magnetometric experiments, the induction coil being wound upon the

middle of the wire. With these arrangements there is no demagnetizing

force to be considered, for the ring has not any ends to produce one,

and the force due to the ends of a rod 400 or 500 diameters in length is

quite insensible at the middle portion; H therefore is equal to H0.

E. Grassot has devised a galvanometer, or "fluxmeter," which greatly

alleviates the tedious operation of taking ballistic readings.[20] The

instrument is of the d'Arsonval type; its coil turns in a strong

uniform field, and is suspended in such a manner that torsion is

practically negligible, the swings of the coil being limited by

damping influences, chiefly electromagnetic. The index therefore

remains almost stationary at the limit of its deflection, and the

deflection is approximately the same whether the change of induction

occurs suddenly or gradually.

[Illustration: FIG. 11.]

[Illustration: FIG. 12.]

[Illustration: FIG. 13.]

_Induction and Hysteresis Curves._--Some typical induction curves,

copied from a paper by Ewing (_Proc. Inst. C.E._ vol. cxxvi.), are given

in figs. 11, 12 and 13. Fig. 11 shows the relation of B to H in a

specimen which has never before been magnetized. The experiment may be

made in two different ways: (1) the magnetizing current is increased by

a series of sudden steps, each of which produces a ballistic throw, the

value of B after any one throw being proportional to the sum of that and

all the previous throws; (2) the magnetizing current having been brought

to any desired value, is suddenly reversed, and the observed throw taken

as measuring twice the actual induction. Fig. 12 shows the nature of the

course taken by the curve when the magnetizing current, after having

been raised to the value corresponding to the point a, is diminished by

steps until it is nothing, and then gradually increased in the reverse

direction. The downward course of the curve is, owing to hysteresis,

strikingly different from its upward course, and when the magnetizing

force has been reduced to zero, there is still remaining an induction of

7500 units. If the operation is again reversed, the upward course will

be nearly, but not exactly, of the form shown by the line d e a, fig.

13. After a few repetitions of the reversal, the process becomes

strictly cyclic, the upward and downward curves always following with

precision the paths indicated in the figure. In order to establish the

cyclic condition, it is sufficient to apply alternately the greatest

positive and negative forces employed in the test (greatest H = about ±5

C.G.S. units in the case illustrated in the figure), an operation which

is performed by simply reversing the direction of the maximum

magnetizing current a few times.

The closed figure a c d e a is variously called a _hysteresis curve_ or

_diagram_ or _loop_. The area [int] H dB enclosed by it represents the

work done in carrying a cubic centimetre of the iron through the

corresponding magnetic cycle; expressed in ergs this work is (1/4)[pi]

[int] H dB.[21] To quote an example given by J. A. Fleming, it requires

about 18 foot-pounds of work to make a complete magnetic cycle in a

cubic foot of wrought iron, strongly magnetized first one way and then

the other, the work so expended taking the form of heat in the mass.

[Illustration: FIG. 14.]

Fig. 14 shows diagrammatically a convenient arrangement described by

Ewing (see _Proc. Inst. C.E._ vol. cxxvi., and _Phil. Trans._, 1893A,

p. 987) for carrying out ballistic tests by which either the simple

B-H curve (fig. 11) or the hysteresis curve (figs. 12 and 13) can be

determined. The sample under test is prepared in the form of a ring A,

upon which are wound the induction and the magnetizing coils; the

latter should be wound evenly over the whole ring, though for the sake

of clearness only part of the winding is indicated in the diagram. The

magnetizing current, which is derived from the storage battery B, is

regulated by the adjustable resistance R and measured by the

galvanometer G. The current passes through the rocking key K, which,

when thrown over to the right, places a in contact with c and b with

d, and when thrown over to the left, places a in contact with e and b

with f. When the switch S is closed, K acts simply as a commutator or

current-reverser, but if K is thrown over from right to left while S

is opened, not only is the current reversed, but its strength is at

the same time diminished by the interposition of the adjustable

resistance R2. The induction coil wound upon the ring is connected to

the ballistic galvanometer G2 in series with a large permanent

resistance R3. In the same circuit is also included the induction coil

E, which is used for standardizing the galvanometer; this secondary

coil is represented in the diagram by three turns of wire wound over a

much longer primary coil. The short-circuit key F is kept closed

except when an observation is about to be made; its object is to

arrest the swing of the d'Arsonval galvanometer G2. By means of the

three-way switch C the battery current may be sent either into the

primary of E, for the purpose of calibrating the galvanometer, or into

the magnetizing coil of the ring under test. When it is desired to

obtain a simple curve of induction, such as that in fig. 11, S is kept

permanently closed, and corresponding values of H and B are determined

by one of the two methods already described, the strength of the

battery-current being varied by means of the adjustable resistance R.

When a hysteresis curve is to be obtained, the procedure is as

follows: The current is first adjusted by means of R to such a

strength as will fit it to produce the greatest + and - values of the

magnetizing force which it is intended to apply in the course of the

cycle; then it is reversed several times, and when the range of the

galvanometer throws has become constant, half the extent of an

excursion indicates the induction corresponding to the extreme value

of H, and gives the point a in the curve fig. 12. The reversing key K

having been put over to the left side, the short-circuit key S is

suddenly opened; this inserts the resistance R, which has been

suitably adjusted beforehand, and thus reduces the current and

therefore the magnetizing force to a known value. The galvanometer

throw which results from the change of current measures the amount by

which the induction is reduced, and thus a second point on the curve

is found. In a similar manner, by giving different values to the

resistance R, any desired number of points between a and c in the

curve can be determined. To continue the process, the key K is turned

over to the right-hand side, and then, while S is open, is turned

back, thereby not only reversing the direction of the current, but

diminishing its strength by an amount depending upon the previous

adjustment of R2. In this way points can be found lying anywhere

between c and d of fig. 12, and the determination of the downward limb

of the curve is therefore completed. As the return curve, shown in

fig. 13, is merely an inverted copy of the other, no separate

determination of it is necessary.

[Illustration: FIG. 15.]

In fig. 15 (J. A. Fleming, _Magnets and Electric Currents_, p. 193) are

shown three very different types of hysteresis curves, characteristic of

the special qualities of the metals from which they were respectively

obtained. The distinguishing feature of the first is the steepness of

its outlines; this indicates that the induction increases rapidly in

relation to the magnetic force, and hence the metal is well suited for

the construction of dynamo magnets. The second has a very small area,

showing that the work done in reversing the magnetization is small; the

metal is therefore adapted for use in alternating current transformers.

On the other hand, the form of the third curve, with its large

intercepts on the axes of H and B, denotes that the specimen to which it

relates possesses both retentiveness and coercive force in a high

degree; such a metal would be chosen for making good permanent magnets.

Several arrangements have been devised for determining hysteresis more

easily and expeditiously than is possible by the ballistic method. The

best known is J. A. Ewing's hysteresis-tester,[22] which is specially

intended for testing the sheet iron used in transformers. The sample,

arranged as a bundle of rectangular strips, is caused to rotate about

a central horizontal axis between the poles of an upright C-shaped

magnet, which is supported near its middle upon knife-edges in such a

manner that it can oscillate about an axis in a line with that about

which the specimen rotates; the lower side of the magnet is weighted,

to give it some stability. When the specimen rotates, the magnet is

deflected from its upright position by an amount which depends upon

the work done in a single complete rotation, and therefore upon the

hysteresis. The deflection is indicated by a pointer upon a graduated

scale, the readings being interpreted by comparison with two standard

specimens supplied with the instrument. G. F. Searle and T. G.

Bedford[23] have introduced the method of measuring hysteresis by

means of an electro-dynamometer used ballistically. The fixed and

suspended coils of the dynamometer are respectively connected in

series with the magnetizing solenoid and with a secondary wound upon

the specimen. When the magnetizing current is twice reversed, so as to

complete a cycle, the sum of the two deflections, multiplied by a

factor depending upon the sectional area of the specimen and upon the

constants of the apparatus, gives the hysteresis for a complete cycle

in ergs per cubic centimetre. For specimens of large sectional area it

is necessary to apply corrections in respect of the energy dissipated

by eddy currents and in heating the secondary circuit. The method has

been employed by the authors themselves in studying the effects of

tension, torsion and circular magnetization, while R. L. Wills[24] has

made successful use of it in a research on the effects of temperature,

a matter of great industrial importance.

C. P. Steinmetz (_Electrician_, 1891, 26, p. 261; 1892, 28, pp. 384,

408, 425) has called attention to a simple relation which appears to

exist between the amount of energy dissipated in carrying a piece of

iron or steel through a magnetic cycle and the limiting value of the

induction reached in the cycle. Denoting by W the work in ergs done

upon a cubic centimetre of the metal ( = 1/4[pi] [int] H dB or [int] H

dI), he finds W = [eta]B^(1.6) approximately, where [eta] is a number,

called the hysteretic constant, depending upon the metal, and B is the

maximum induction. The value of the constant [eta] ranges in different

metals from about 0.001 to 0.04; in soft iron and steel it is said to

be generally not far from 0.002. Steinmetz's formula may be tested by

taking a series of hysteresis curves between different limits of B,

measuring their areas by a planimeter, and plotting the logarithms of

these divided by 4[pi] as ordinates against logarithms of the

corresponding maximum values of B as abscissae. The curve thus

constructed should be a straight line inclined to the horizontal axis

at an angle [theta], the tangent of which is 1.6. Ewing and H. G.

Klaassen (_Phil. Trans._, 1893, 184, 1017) have in this manner

examined how nearly and within what range a formula of the type W =

[eta]B^[epsilon] may be taken to represent the facts. The results of

an example which they quote in detail may be briefly summarized as

follows:--

+-----------------+------------+-----------------+-----------+

| | [eta] | (= tan [theta]) | |

+-----------------+------------+-----------------+-----------+

| 200 to 500 | ... | 1.9 | 62.25 |

| 500 to 1,000 | ... | 1.68 | 59.25 |

| 1,000 to 2,000 | ... | 1.55 | 57.25 |

| 2,000 to 8,000 | 0.01 | 1.475 | 55.75 |

| 8,000 to 14,000 | 0.00134 | 1.70 | 59.50 |

+-----------------+------------+-----------------+-----------+

It is remarked by the experimenters that the value of the index

[epsilon] is by no means constant, but changes in correspondence with

the successive well-marked stages in the process of magnetization. But

though a formula of this type has no physical significance, and cannot

be accepted as an equation to the actual curve of W and B, it is,

nevertheless, the case that by making the index [epsilon] = 1.6, and

assigning a suitable value to [eta], a formula may be obtained giving

an approximation to the truth which is sufficiently close for the

ordinary purposes of electrical engineers, especially when the

limiting value of B is neither very great nor very small. Alexander

Siemens (_Journ. Inst. Eng._, 1894, 23, 229) states that in the

hundreds of comparisons of test pieces which have been made at the

works of his firm, Steinmetz's law has been found to be practically

correct.[25] An interesting collection of W-B curves embodying the

results of actual experiments by Ewing and Klaassen on different

specimens of metal is given in fig. 16. It has been shown by Kennelly

(_Electrician_, 1892, 28, 666) that Steinmetz's formula gives

approximately correct results in the case of nickel. Working with two

different specimens, he found that the hysteresis loss in ergs per

cubic centimetre (W) was fairly represented by 0.00125B^(1.6) and

0.00101B^(1.6) respectively, the maximum induction ranging from about

300 to 3000. The applicability of the law to cobalt has been

investigated by Fleming (_Phil. Mag._, 1899, 48, 271), who used a ring

of cast cobalt containing about 96% of the pure metal. The logarithmic

curves which accompany his paper demonstrate that within wide ranges

of maximum induction W = 0.01B^(1.6) = 0.527I^(1.62) very nearly.

Fleming rightly regards it as not a little curious that for materials

differing so much as this cast cobalt and soft annealed iron the

hysteretic exponent should in both cases be so near to 1.6. After

pointing out that, since the magnetization of the metal is the

quantity really concerned, W is more appropriately expressed in terms

of I, the magnetic moment per unit of volume, than of B, he suggests

an experiment to determine whether the mechanical work required to

effect the complete magnetic reversal of a crowd of small compass

needles (representative of magnetic molecules) is proportional to the

1.6th power of the aggregate maximum magnetic moment before or after

completion of the cycle.

[Illustration: FIG. 16.

a, Fine steel wire 0.257 mm. diam.

b, Fine iron wire 0.34 mm. diam.

c, Fine iron wire 0.2475 mm. diam.

d, Thin sheet iron 0.47 mm. thick.

e, Iron wire 0.602 mm. diam.

f, Iron wire 0.975 mm. diam.

g, Sheet iron 1.95 mm. thick.

h, Thin sheet iron 0.367 mm. thick.

i. Very soft iron wire.]

The experiments of K. Honda and S. Shimizu[26] indicate that

Steinmetz's formula holds for nickel and annealed cobalt up to B =

3000, for cast cobalt and tungsten steel up to B = 8000, and for

Swedish iron up to B = 18,000, the range being in all cases extended

at the temperature of liquid air.

[Illustration: FIG. 17.]

The diagram, fig. 17, contains examples of ascending induction curves

characteristic of wrought iron, cast iron, cobalt and nickel. These are

to be regarded merely as typical specimens, for the details of a curve

depend largely upon the physical condition and purity of the material;

but they show at a glance how far the several metals differ from and

resemble one another as regards their magnetic properties. Curves of

magnetization (which express the relation of I to H) have a close

resemblance to those of induction; and, indeed, since B = H + 4[pi]I,

and 4[pi]I (except in extreme fields) greatly exceeds H in numerical

value, we may generally, without serious error, put I = B/4[pi], and

transform curves of induction into curves of magnetization by merely

altering the scale to which the ordinates are referred. A scale for the

approximate transformation for the curves in fig. 12 is given at the

right-hand side of the diagram, the greatest error introduced by

neglecting H/4[pi] not exceeding 0.6%. A study of such curves as these

reveals the fact that there are three distinct stages in the process of

magnetization. During the first stage, when the magnetizing force is

small, the magnetization (or the induction) increases rather slowly with

increasing force; this is well shown by the nickel curve in the diagram,

but the effect would be no less conspicuous in the iron curve if the

abscissae were plotted to a larger scale. During the second stage small

increments of magnetizing force are attended by relatively large

increments of magnetization, as is indicated by the steep ascent of the

curve. Then the curve bends over, forming what is often called a "knee,"

and a third stage is entered upon, during which a considerable increase

of magnetizing force has little further effect upon the magnetization.

When in this condition the metal is popularly said to be "saturated."

Under increasing magnetizing forces, greatly exceeding those comprised

within the limits of the diagram, the magnetization does practically

reach a limit, the maximum value being attained with a magnetizing force

of less than 2000 for wrought iron and nickel, and less than 4000 for

cast iron and cobalt. The induction, however, continues to increase

indefinitely, though very slowly. These observations have an important

bearing upon the molecular theory of magnetism, which will be referred

to later.

The magnetic quality of a sample of iron depends very largely upon the

purity and physical condition of the metal. The presence of ordinary

impurities usually tends to diminish the permeability, though, as will

appear later, the addition of small quantities of certain other

substances is sometimes advantageous. A very pure form of iron, which

from the method of its manufacture is called "steel," is now extensively

used for the construction of dynamo magnets; this metal sometimes

contains not more than 0.3% of foreign substances, including carbon, and

is magnetically superior to the best commercial wrought iron. The

results of some comparative tests published by Ewing (_Proc. Inst.

C.E._, 1896) are given in the accompanying table. Those in the second

column are quoted from a paper by F. Lydall and A. W. Pocklington

(_Proc. Roy. Soc._, 1892, 52, 228) and relate to an exceptional specimen

containing nearly 99.9% of the pure metal.

+----------+-----------------------------------------+

| | Magnetic Induction. |

| Magnetic +--------+----------+----------+----------+

+----------+--------+----------+----------+----------+

| 5 | 12,700 | 10,900 | 12,300 | 9,600 |

| 10 | 14,980 | 13,120 | 14,920 | 13,050 |

| 15 | 15,800 | 14,010 | 15,800 | 14,600 |

| 20 | 16,300 | 14,580 | 16,280 | 15,310 |

| 30 | 16,950 | 15,280 | 16,810 | 16,000 |

| 40 | 17,350 | 15,760 | 17,190 | 16,510 |

| 50 | .. | 16,060 | 17,500 | 16,900 |

| 60 | .. | 16,340 | 17,750 | 17,180 |

| 70 | .. | 16,580 | 17,970 | 17,400 |

| 80 | .. | 16,800 | 18,180 | 17,620 |

| 90 | .. | 17,000 | 18,390 | 17,830 |

| 100 | .. | 17,200 | 18,600 | 18,030 |

+----------+--------+----------+----------+----------+

To secure the highest possible permeability it is essential that the

iron should be softened by careful annealing. When it is mechanically

hardened by hammering, rolling or wire-drawing its permeability may be

greatly diminished, especially under a moderate magnetizing force. An

experiment by Ewing showed that by the operation of stretching an

annealed iron wire beyond the limits of elasticity the permeability

under a magnetizing force of about 3 units was reduced by as much as

75%. Ewing has also studied the effect of vibration in conferring upon

iron an apparent or spurious permeability of high value; this effort

also is most conspicuous when the magnetizing force is weak. The

permeability of a soft iron wire, which was tapped while subjected to a

very small magnetizing force, rose to the enormous value of about 80,000

(_Magnetic Induction_, § 85). It follows that in testing iron for

magnetic quality the greatest care must be exercised to guard the

specimen against any accidental vibration.

Low hysteresis is the chief requisite for iron which is to be used for

transformer cores, and it does not necessarily accompany high

permeability. In response to the demand, manufacturers have succeeded in

producing transformer plate in which the loss of energy due to

hysteresis is exceedingly small. Tests of a sample supplied by Messrs.

Sankey were found by Ewing to give the following results, which,

however, are regarded as being unusually favourable. In a valuable

collection of magnetic data (_Proc. Inst. C.E._, cxxvi.) H. F. Parshall

quotes tests of six samples of iron, described as of good quality, which

showed an average hysteresis loss of 3070 ergs per c.cm. per cycle at an

induction of 8000, being 1.6 times the loss shown by Ewing's specimen at

the same induction.

+------------+----------------+-----------------+

| Limits of | Ergs per c.cm. | Watts per lb. |

| Induction. | per cycle. | Frequency, 100. |

+------------+----------------+-----------------+

| 2000 | 220 | 0.129 |

| 3000 | 410 | 0.242 |

| 4000 | 640 | 0.376 |

| 5000 | 910 | 0.535 |

| 6000 | 1200 | 0.710 |

| 7000 | 1520 | 0.890 |

| 8000 | 1900 | 1.120 |

| 9000 | 2310 | 1.360 |

+------------+----------------+-----------------+

The standard induction in reference to determinations of hysteresis is

generally taken as 2500, while the loss is expressed in watts per lb. at

a frequency of 100 double reversals, or cycles, per second. In many

experiments, however, different inductions and frequencies are employed,

and the hysteresis-loss is often expressed as ergs per cubic centimetre

per cycle and sometimes as horse-power per ton. In order to save

arithmetical labour it is convenient to be provided with conversion

factors for reducing variously expressed results to the standard form.

The rate at which energy is lost being proportional to the frequency, it

is obvious that the loss at frequency 100 may be deduced from that at

any other frequency n by simply multiplying by 100/n. Taking the density

of iron to be 7.7, the factor for reducing the loss in ergs per c.cm. to

watts per lb. with a frequency of 100 is 0.000589 (Ewing). Since 1

horse-power = 746 watts, and 1 ton = 2240 lb., the factor for reducing

horse-power per ton to watts per lb. is 746/2240, or just 1/3. The loss

for any induction B within the range for which Steinmetz's law holds may

be converted into that for the standard induction 2500 by dividing it by

B^(1.6)/2500^(1.6). The values of this ratio for different values of B,

as given by Fleming (_Phil. Mag._, 1897), are contained in the second

column of the annexed table. The third column shows the relative amount

of hysteresis deduced by Ewing as a general mean from actual tests of

many samples (_Journ. Inst. Elec. Eng._, 1895). Incidentally, these two

columns furnish an undesigned test of the accuracy of Steinmetz's law:

the greatest difference is little more than 1%.

+-----------+------------+-------------+

| Induction | B^(1.6) | Observed |

| B. | ---------- | relative |

| | 2500^(1.6) | Hysteresis. |

+-----------+------------+-------------+

| 2000 | 0.700 | 0.702 |

| 2500 | 1.000 | 1.000 |

| 3000 | 1.338 | 1.340 |

| 4000 | 2.118 | 2.128 |

| 5000 | 3.031 | 3.000 |

| 6000 | 4.058 | 4.022 |

| 7000 | 5.193 | 5.129 |

| 8000 | 6.430 | 6.384 |

+-----------+------------+-------------+

_Curves of Permeability and Susceptibility._--The relations of µ (= B/H)

to B, and of [kappa] (= I/H) to I may be instructively exhibited by

means of curves, a method first employed by H. A. Rowland.[27] The

dotted curve for µ and B in fig. 18 is copied from Rowland's paper. The

actual experiment to which it relates was carried only as the point

marked X, corresponding to a magnetizing force of 65, and an induction

of nearly 17,000. Rowland, believing that the curve would continue to

fall in a straight line meeting the horizontal axis, inferred that the

induction corresponding to the point B--about 17,500--was the highest

that could be produced by any magnetizing force, however great. It has,

however, been shown that, if the magnetizing force is carried far

enough, the curve always becomes convex to the axis instead of meeting

it. The full line shows the result of an experiment in which the

magnetizing force was carried up to 585,[28] but though the force was

thus increased ninefold, the induction only reached 19,800, and the

ultimate value of the permeability was still as much as 33.9.

[Illustration: FIG. 18.]

[Illustration: FIG. 19.]

_Ballistic Method with Yoke._--J. Hopkinson (_Phil. Trans._, 1885, 176,

455) introduced a modification of the usual ballistic arrangement which

presents the following advantages; (1) very considerable magnetizing

forces can be applied with ordinary means; (2) the samples to be tested,

having the form of cylindrical bars, are more easily prepared than rings

or wires; (3) the actual induction at any time can be measured, and not

only changes of induction. On the other hand, a very high degree of

accuracy is not claimed for the results. Fig. 19 shows the apparatus by

which the ends of the bar are prevented from exerting any material

demagnetizing force, while the permeance of the magnetic circuit is at

the same time increased. A A, called the "yoke," is a block of annealed

wrought iron about 18 in. long, 6½ in. wide and 2 in. thick, through

which is cut a rectangular opening to receive the two magnetizing coils

B B. The test bar C C, which slides through holes bored in the yoke, is

divided near the middle into two parts, the ends which come into contact

being faced true and square. Between the magnetizing coils is a small

induction coil D, which is connected with a ballistic galvanometer. The

induction coil is carried upon the end of one portion of the test bar,

and when this portion is suddenly drawn back the coil slips off and is

pulled out of the field by an india-rubber spring. This causes a

ballistic throw proportional to the induction through the bar at the

moment when the two portions were separated. With such an arrangement it

is possible to submit the sample to any series of magnetic forces, and

to measure its magnetic state at the end. The uncertainty with which the

results are affected depends chiefly upon the imperfect contact between

the bar and the yoke and also between the ends of the divided bar. It is

probable that Hopkinson did not attach sufficient importance to the

demagnetizing action of the cut (cf. Ewing, _Phil. Mag._, Sept. 1888, p.

274), and that the values which he assigned to H are consequently

somewhat too high. He applied his method with good effect, however, in

testing a large number of commercial specimens of iron and steel, the

magnetic constants of which are given in a table accompanying his paper.

When it is not required to determine the residual magnetization there is

no necessity to divide the sample bar, and ballistic tests may be made

in the ordinary way--by steps or by reversals--the source of error due

to the transverse cut thus being avoided. Ewing (_Magnetic Induction_, §

194) has devised an arrangement in which two similar test bars are

placed side by side; each bar is surrounded by a magnetizing coil, the

two coils being connected to give opposite directions of magnetization,

and each pair of ends is connected by a short massive block of soft iron

having holes bored through it to fit the bars, which are clamped in

position by set-screws. Induction coils are wound on the middle parts of

both bars, and are connected in series. With this arrangement it is

possible to find the actual value of the magnetizing force, corrected

for the effects of joints and other sources of error. Two sets of

observations are taken, one when the blocks are fixed at the ends of the

bars, and another when they are nearer together, the clear length of the

bars between them and of the magnetizing coils being reduced to

one-half. If H1 and H2 be the values of 4[pi]in/l and 4[pi]i´[(n/2) /

(l/2)] for the same induction B, it can be shown that the true

magnetizing force is H = H1 - (H2 - H1). The method, though tedious in

operation, is very accurate, and is largely employed for determining the

magnetic quality of bars intended to serve as standards.

_Traction Methods._--The induction of the magnetization may be measured

by observing the force required to draw apart the two portions of a

divided rod or ring when held together by their mutual attraction. If a

transverse cut is made through a bar whose magnetization is I and the

two ends are placed in contact, it can be shown that this force is

2[pi]I² dynes per unit of area (Mascart and Joubert, _Electricity and

Magnetism_, § 322); and if the magnetization of the bar is due to an

external field H produced by a magnetizing coil or otherwise, there is

an additional force equal to HI. Thus the whole force, when the two

portions of the bar are surrounded by a loosely-fitting magnetizing

coil, is

F = 2[pi]I² + HI

expressed as dynes per square centimetre. If each portion of the bar has

an independent magnetizing coil wound tightly upon it, we have further

to take into account the force due to the mutual action of the two

magnetizing coils, which assists the forces already considered. This is

equal to H²8[pi] per unit of sectional area. In the case supposed

therefore the total force per square centimetre is

H²

F = 2[pi]I² + HI + -----

8[pi]

(4[pi]I + H)³

= -------------

8[pi]

B²

= -----.

8[pi]

The equation F = B²/8[pi] is often said to express "Maxwell's law of

magnetic traction" (Maxwell, _Electricity and Magnetism_, §§ 642-646).

It is, of course, true for permanent magnets, where H = 0, since then F

= 2[pi]I²; but if the magnetization is due to electric currents, the

formula is only applicable in the special case when the mutual action of

the two magnets upon one another is supplemented by the electromagnetic

attraction between separate magnetizing coils rigidly attached to

them.[29]

The traction method was first employed by S. Bidwell (_Proc. Roy. Soc._,

1886, 40, 486), who in 1886 published an account of some experiments in

which the relation of magnetization to magnetic field was deduced from

observations of the force in grammes weight which just sufficed to tear

asunder the two halves of a divided ring electromagnet when known

currents were passing through the coils. He made use of the expression

F = Wg = 2[pi]I² + HI,

where W is the weight in grammes per square centimetre of sectional

area, and g is the intensity of gravity which was taken as 981. The term

for the attraction between the coils was omitted as negligibly small

(see _Phil. Mag._, 1890, 29, 440). The values assigned to H were

calculated from H = 2ni/r, and ranged from 3.9 to 585, but inasmuch as

no account was taken of any demagnetizing action which might be due to

the two transverse cuts, it is probable that they are somewhat too high.

The results, nevertheless, agree very well with those for annealed

wrought iron obtained by other methods. Below is given a selection from

Bidwell's tables, showing corresponding values of magnetizing force,

weight supported, magnetization, induction, susceptibility and

permeability:--

+--------+---------+-------+--------+----------+--------+

| H. | W. | I. | B. | [kappa]. | µ. |

+--------+---------+-------+--------+----------+--------+

| 3.9 | 2,210 | 587 | 7,390 | 151.0 | 1889.1 |

| 5.7 | 3,460 | 735 | 9,240 | 128.9 | 1621.3 |

| 10.3 | 5,400 | 918 | 11,550 | 89.1 | 1121.4 |

| 22.2 | 8,440 | 1147 | 14,450 | 51.7 | 650.9 |

| 40 | 9,680 | 1226 | 15,460 | 30.7 | 386.4 |

| 115 | 12,170 | 1370 | 17,330 | 11.9 | 150.7 |

| 208 | 13,810 | 1452 | 18,470 | 7.0 | 88.8 |

| 362 | 14,740 | 1489 | 19,080 | 4.1 | 52.7 |

| 465 | 15,275 | 1508 | 19,420 | 3.2 | 41.8 |

| 585 | 15,905 | 1530 | 19,820 | 2.6 | 33.9 |

+--------+---------+-------+--------+----------+--------+

[Illustration: FIG. 20.]

A few months later R. H. M. Bosanquet (_Phil. Mag._, 1886, 22, 535)

experimented on the relation of tractive force to magnetic induction.

Instead of a divided ring he employed a divided straight bar, each half

of which was provided with a magnetizing coil. The joint was surrounded

by an induction coil connected with a ballistic galvanometer, an

arrangement which enabled him to make an independent measurement of the

induction at the moment when the two portions of the bar were separated.

He showed that there was, on the whole, a fair agreement between the

values determined ballistically and those given by the formula B =

[root](8[pi]F). The greatest weight supported in the experiments was

14,600 grammes per square cm., and the corresponding induction 18,500

units. Taylor Jones subsequently found a good agreement between the

theoretical and the observed values of the tractive force in fields

ranging up to very high intensities (_Phil. Mag._, 1895, 39, 254, and

1896, 41, 153).

_Permeameters._--Several instruments in which the traction method is

applied have been devised for the rapid measurement of induction or of

magnetization in commercial samples of iron and steel. The earliest of

these is S. P. Thompson's _permeameter_ (_Journ. Sci. Arts_, 1890, 38,

885), which consists of a rectangular block of iron shaped like

Hopkinson's yoke, and slotted out in the same way to receive a

magnetizing coil (fig. 20); the block is bored through at the upper

end only, and its inner face opposite the hole is made quite flat and

smooth. The sample has the form of a thin rod, one end of which is

faced true; it is slipped into the magnetizing coil from above, and

when the current is turned on its smooth end adheres tightly to the

surface of the yoke. The force required to detach it is measured by a

registering spring balance, which is clamped to the upper end of the

rod, and thence the induction or the magnetization is deduced by

applying the formula

(B - H)²/8[pi] = 2[pi]I² = Pg/S,

where P is the pull in grammes weight, S the sectional area of the rod

in square cm., and g = 981. If the pull is measured in pounds and the

area in square inches, the formula may be written B = 1317 ×

[root](P/S) + H. The instrument exhibited by Thompson would, without

undue heating, take a current of 30 amperes, which was sufficient to

produce a magnetizing force of 1000 units. A testing apparatus of a

similar type devised by Gisbert Kapp (_Journ. Inst. Elec. Eng._ xxiii.

199) differs only in a few details from Thompson's permeameter. Ewing

has described an arrangement in which the test bar has a soft-iron

pole piece clamped to each of its ends; the pole pieces are joined by

a long well-fitting block of iron, which is placed upon them (like the

"keeper" of a magnet), and the induction is measured by the force

required to detach the block. In all such measurements a correction

should be made in respect of the demagnetizing force due to the joint,

and unless the fit is very accurate the demagnetizing action will be

variable. In the _magnetic balance_ of du Bois (_Magnetic Circuit_, p.

346) the uncertainty arising from the presence of a joint is avoided,

the force measured being that exerted between two pieces of iron

separated from each other by a narrow air-gap of known width. The

instrument is represented diagrammatically in fig. 21. The test-piece

A, surrounded by a magnetizing coil, is clamped between two soft-iron

blocks B, B´. Y Y´ is a soft iron yoke, which rocks upon knife-edges K

and constitutes the beam of the balance. The yoke has two projecting

pieces C, C´ at unequal distances from the knife-edges, and separated

from the blocks B, B´ by narrow air-gaps. The play of the beam is

limited by a stop S and a screw R, the latter being so adjusted that

when the end Y of the beam is held down the two air-gaps are of equal

width. W is a weight capable of sliding from end to end of the yoke

along a graduated scale. When there is no magnetization, the yoke is

in equilibrium; but as soon as the current is turned on the block C is

drawn downwards as far as the screw R will allow, for, though the

attractive forces F between B and C and between B´ and C´ are equal,

the former has a greater moment. The weight W is moved along the scale

until the yoke just tilts over upon the stop S; the distance of W from

its zero position is then, as can easily be shown, proportional to F,

and therefore to B², and approximately to I². The scale is graduated

in such a manner that by multiplying the reading by a simple factor

(generally 10 or 2) the absolute value of the magnetization is

obtained. The actual magnetizing force H is of course less than that

due to the coil; the corrections required are effected automatically

by the use of a set of demagnetization lines drawn on a sheet of

celluloid which is supplied with the instrument. The celluloid sheet

is laid upon the squared paper, and in plotting a curve horizontal

distances are reckoned from the proper demagnetization line instead of

from the vertical axis. An improved but somewhat more complex form of

the instrument is described in _Ann. d. Phys._, 1900, 2, 317.

[Illustration: FIG. 21.]

In Ewing's _magnetic balance_ (_Journ. Inst. Elec. Eng._ 1898, 27,

526), the value of the magnetic induction corresponding to a single

stated magnetizing force is directly read off on a divided scale. The

specimen, which has the form of a turned rod, 4 in. long and ¼ in. in

diameter, is laid across the poles of a horseshoe electromagnet,

excited by a current of such strength as to produce in the rod a

magnetizing force H = 20. One pole has a V-shaped notch for the rod to

rest in; the surface of the other is slightly rounded, forming a

portion of a cylinder, the axis of which is perpendicular to the

direction of the length of the rod. The rod touches this pole at a

single point, and is pulled away from it by the action of a lever, the

long arm of which is graduated and carries a sliding weight. The

position of the weight at the moment when contact is broken indicates

the induction in the rod. The standard force H = 20 was selected as

being sufficiently low to distinguish between good and bad specimens,

and at the same time sufficiently high to make the order of merit the

same as it would be under stronger forces.

_Permeability Bridges._--Several pieces of apparatus have been

invented for comparing the magnetic quality of a sample with that of a

standard iron rod by a zero method, such as is employed in the

comparison of electrical resistances by the Wheatstone bridge. An

excellent instrument of the class is Ewing's _permeability bridge_.

The standard rod and the test specimen, which must be of the same

dimensions, are placed side by side within two magnetizing coils, and

each pair of adjacent ends is joined by a short rectangular block or

"yoke" of soft iron. An iron bar shaped like an inverted L projects

upwards from each of the yokes, the horizontal portions of the bars

being parallel to the rods, and nearly meeting at a height of about 8

in. above them (thus [symbol]). A compass needle placed in the gap

serves to detect any flow of induction that may exist between the bent

bars. For simplicity of calculation, the clear length of each rod

between the yokes is made 12.56 (= 4[pi]) centimetres, while the coil

surrounding the standard bar contains 100 turns; hence the magnetizing

force due to a current of n amperes will be 10n C.G.S. units. The

effective number of turns in the coil surrounding the test rod can be

varied by means of three dial switches (for hundreds, tens and units),

which also introduce compensating resistances as the number of

effective turns in the coil is reduced, thus keeping the total

resistance of the circuit constant. The two coils are connected in

series, the same current passing through both. Suppose the switches to

be adjusted so that the effective number of turns in the variable coil

is 100; the magnetizing forces in the two coils will then be equal,

and if the test rod is of the same quality as the standard, the flow

of induction will be confined entirely to the iron circuit, the two

yokes will be at the same magnetic potential, and the compass needle

will not be affected. If, however, the permeability of the test rod

differs from that of the standard, the number of lines of induction

flowing in opposite directions through the two rods will differ, and

the excess will flow from one yoke to the other, partly through the

air, and partly along the path provided by the bent bars, deflecting

the compass needle. But a balance may still be obtained by altering

the effective number of turns in the test coil, and thus increasing or

decreasing the magnetizing force acting on the test rod, till the

induction in the two rods is the same, a condition which is fulfilled

when reversal of the current has no effect on the compass needle. Let

m be the number of turns in use, and H1 and H2 the magnetizing forces

which produce the same induction B in the test and the standard rods

respectively; then H1 = H2 × m/100. The value of B which corresponds

to H2m/100 can be found from the (B, H) curve for the standard, which

is assumed to have been determined; and this same value corresponds to

the force H in the case of the test bar. Thus any desired number of

corresponding values of H and B can be easily and quickly found.

_Measurement of Field Strength. Exploring Coil._--Since in air B = H,

the ballistic method of measuring induction described above is also

available for determining the strength of a magnetic field, and is more

often employed than any other. A small coil of fine wire, connected in

series with a ballistic galvanometer, is placed in the field, with its

windings perpendicular to the lines of force, and then suddenly reversed

or withdrawn from the field, the integral electromotive force being

twice as great in the first case as in the second. The strength of the

field is proportional to the swing of the galvanometer-needle, and, when

the galvanometer is calibrated, can be expressed in C.G.S. units.

Convenient arrangements have been introduced whereby the coil is

reversed or withdrawn from the field by the action of a spring.

_Bismuth Resistance._--The fact, which will be referred to later, that

the electrical resistance of bismuth is very greatly affected by a

magnetic field has been applied in the construction of apparatus for

measuring field intensity. A little instrument, supplied by Hartmann and

Braun, contains a short length of fine bismuth wire wound into a flat

double spiral, half an inch or thereabouts in diameter, and attached to

a long ebonite handle. Unfortunately the effects of magnetization upon

the specific resistance of bismuth vary enormously with changes of

temperature; it is therefore necessary to take two readings of the

resistance, one when the spiral is in the magnetic field, the other when

it is outside.

_Electric Circuit._--If a coil of insulated wire is suspended so that it

is in stable equilibrium when its plane is parallel to the direction of

a magnetic field, the transmission of a known electric current through

the coil will cause it to be deflected through an angle which is a

function of the field intensity.

One of the neatest applications of this principle is that described by

Edser and Stansfield (_Phil. Mag._, 1893, 34, 186), and used by them

to test the stray fields of dynamos. An oblong coil about an inch in

length is suspended from each end by thin strips of rolled German

silver wire, one of which is connected with a spiral spring for

regulating the tension, the other being attached to a torsion-head.

Inside the torsion-head is a commutator for automatically reversing

the current, so that readings may be taken on each side of zero, and

the arrangement is such that when the torsion-head is exactly at zero

the current is interrupted. To take a reading the torsion-head is

turned until an aluminium pointer attached to the coil is brought to

the zero position on a small scale; the strength of the field is then

proportional to the angular torsion. The small current required is

supplied to the coil from a single dry cell. The advantages of

portability, very considerable range (from H = 1 upwards), and fair

accuracy are claimed for the instrument.

_Polarized Light._--The intensity of a field may be measured by the

rotation of the plane of polarization of light passing in the direction

of the magnetic force through a transparent substance. If the field is

uniform, H = [theta]/[omega]d, where [theta] is the rotation, d the

thickness of the substance arranged as a plate at right angles to the

direction of the field, and [omega] Verdet's constant for the substance.

For the practical measurement of field intensity du Bois has used

plates of the densest Jena flint glass. These are preferably made

slightly wedge-shape, to avoid the inconvenience resulting from

multiple internal reflections, and they must necessarily be rather

thin, so that double refractions due to internal strain may not exert

a disturbing influence. Since Verdet's constant is somewhat uncertain

for different batches of glass even of the same quality, each plate

should be standardized in a field of known intensity. As the source of

monochromatic light a bright sodium burner is used, and the rotation,

which is exactly proportional to H, is measured by an accurate

polarimeter. Such a plate about 1 mm. in thickness is said to be

adapted for measuring fields of the order of 1000 units. A part of one

surface of the plate may be silvered, so that the polarized ray, after

having once traversed the glass, is reflected back again; the rotation

is thus doubled, and moreover, the arrangement is, for certain

experiments, more convenient than the other.

4. MAGNETIZATION IN STRONG FIELDS

_Fields due to Coils._--The most generally convenient arrangement for

producing such magnetic fields as are required for experimental

purposes is undoubtedly a coil of wire through which an electric current

can be caused to flow. The field due to a coil can be made as nearly

uniform as we please throughout a considerable space; its intensity,

when the constants of the coil are known, can be calculated with ease

and certainty and may be varied at will through wide ranges, while the

apparatus required is of the simplest character and can be readily

constructed to suit special purposes. But when exceptionally strong

fields are desired, the use of a coil is limited by the heating effect

of the magnetizing current, the quantity of heat generated per unit of

time in a coil of given dimensions increasing as the square of the

magnetic field produced in its interior. In experiments on magnetic

strains carried out by H. Nagaoka and K. Honda (_Phil. Mag._, 1900, 49,

329) the intensity of the highest field reached in the interior of a

coil was 2200 units; this is probably the strongest field produced by a

coil which has hitherto been employed in experimental work. In 1890 some

experiments in which a coil was used were made by du Bois (_Phil. Mag._,

1890, 29, 253, 293) on the magnetization of iron, nickel, and cobalt

under forces ranging from about 100 to 1250 units. Since the

demagnetizing factor was 0.052, the strongest field due to the coil was

about 1340; but though arrangements were provided for cooling the

apparatus by means of ice, great difficulty was experienced owing to

heating. Du Bois's results, which, as given in his papers, show the

relation of H to the magnetic moment per unit of mass, have been reduced

by Ewing to the usual form, and are indicated in fig. 22, the earlier

portions of the curves being sketched in from other data.

[Illustration: FIG. 22.]

_Fields due to Electromagnets._--The problem of determining the

magnetization of iron and other metals in the strong fields formed

between the poles of an electromagnet was first attacked by J. A. Ewing

and W. Low. An account of their preliminary experiments by what they

call the _isthmus method_ was published in 1887 (_Proc. Roy. Soc._ 42,

200), and in the following year they described a more complete and

perfect series (_Phil. Trans._, 1889, 180, 221).

[Illustration: FIG. 23.]

The sample to be inserted between the magnet poles was prepared in the

form of a bobbin resembling an ordinary cotton reel, with a short

narrow neck (constituting the "isthmus") and conical ends. Upon the

central neck was wound a coil consisting of one or two layers of very

fine wire, which was connected with a ballistic galvanometer for

measuring the induction in the iron; outside this coil, and separated

from it by a small and accurately determined distance, a second coil

was wound, serving to measure the induction in the iron, together with

that in a small space surrounding it. The difference of the ballastic

throws taken with the two coils measured the intensity of the field in

the space around the iron, and it also enabled a correction to be made

for the non-ferrous space between the iron neck and the centre of the

thickness of the inner coil. The pole pieces of the electromagnet (see

fig. 23) were furnished with a pair of truncated cones _b b_, of soft

iron forming an extension of the conical ends of the bobbin c. The

most suitable form for the pole faces is investigated in the paper,

and the conclusion arrived at is that to produce the greatest

concentration of force upon the central neck, the cones should have a

common vertex in the middle of the neck with a semi-vertical angle of

54° 44´, while the condition for a uniform field is satisfied when the

cones have a semi-vertical angle of 39° 14´; in the latter case the

magnetic force in the air just outside is sensibly equal to that

within the neck. A pair of cones having a semi-vertical angle of 45°

were considered to combine high concentrative power with a sufficient

approximation to uniformity of field. In most of the experiments the

measurements were made by suddenly withdrawing the bobbin from its

place between the pole pieces. Two groups of observations were

recorded, one giving the induction in the inner coil and the other

that in the outer coil. The value of the residual induction which

persisted when the bobbin was drawn out was added to that of the

induction measured, and thus the total induction in the iron was

determined. The highest induction reached in these experiments was

45,350 units, more than twice the value of any previously recorded.

The corresponding intensity of the outside field was 24,500, but,

owing to the wide angle of the cones used (about 2 × 63°), this was

probably greater than the value of the magnetic force within the

metal. The following table shows some results of other experiments in

which H was believed to have sensibly the same value inside as outside

the metal. Values of I are derived from (B - H)/4[pi] and of µ from

B/H.

+--------------+--------+--------+------+-------+

| Metal. | H | B | I | µ |

+--------------+--------+--------+------+-------+

| | 1,490 | 22,650 | 1680 | 15.20 |

| | 6,070 | 27,130 | 1680 | 4.47 |

| Swedish Iron | 8,600 | 30,270 | 1720 | 3.52 |

| | 19,450 | 40,820 | 1700 | 2.10 |

| | 19,880 | 41,140 | 1700 | 2.07 |

+--------------+--------+--------+------+-------+

| | 4,560 | 20,070 | 1230 | 4.40 |

| Cast Iron | 13,460 | 28,710 | 1210 | 2.13 |

| | 16,200 | 30,920 | 1170 | 1.91 |

| | 16,900 | 31,760 | 1180 | 1.88 |

+--------------+--------+--------+------+-------+

| | 6,210 | 25,480 | 1530 | 4.10 |

| | 9,970 | 29,650 | 1570 | 2.97 |

| Tool Steel | 12,170 | 31,620 | 1550 | 2.60 |

| | 14,660 | 34,550 | 1580 | 2.36 |

| | 15,530 | 35,820 | 1610 | 2.31 |

+--------------+--------+--------+------+-------+

| | 2,220 | 7,100 | 390 | 3.20 |

| | 4,440 | 9,210 | 380 | 2.09 |

| Hard Nickel | 7,940 | 12,970 | 400 | 1.63 |

| | 14,660 | 19,640 | 400 | 1.34 |

| | 16,000 | 21,070 | 400 | 1.32 |

+--------------+--------+--------+------+-------+

| | 1,350 | 16,000 | 1260 | 12.73 |

| Cobalt | 4,040 | 18,870 | 1280 | 4.98 |

| | 8,930 | 23,890 | 1290 | 2.82 |

| | 14,990 | 30,210 | 1310 | 2.10 |

+--------------+--------+--------+------+-------+

These results are of extreme interest, for they show that under

sufficiently strong magnetizing forces the intensity of magnetization I

reaches a maximum value, as required by W. E. Weber's theory of

molecular magnetism. There appears to be no definite limit to the value

to which the induction B may be raised, but the magnetization I attains

a true saturation value under magnetizing forces which are in most cases

comparatively moderate. Thus the magnetization which the sample of

Swedish iron received in a field of 1490 was not increased (beyond the

limits of experimental error) when the intensity of the field was

multiplied more than thirteen-fold, though the induction was nearly

doubled. When the saturation value of I has been reached, the relation

of magnetic induction to magnetic force may be expressed by

B = H + constant.

The annexed table gives the saturation values of I for the particular

metals examined by Ewing and Low:--

Saturation

Value of I

Wrought iron 1,700

Cast iron 1,240

Nickel (0.75% iron) 515

" (0.56% " ) 400

Cobalt (1.66% " ) 1,300

It is shown in the paper that the greatest possible force which the

isthmus method can apply at a point in the axis of the bobbin is

F = 11.137 I_s log 10 b/a,

I_s being the saturation value of the magnet poles, a the radius of the

neck on which the cones converge, and b the radius of the bases of the

cones.

Some experiments made by H. du Bois (_Phil. Mag._, 1890, 29, 293) with

an electromagnet specially designed for the production of strong

fields, confirm Ewing's results for iron, nickel and cobalt. The

method employed did not admit of the production of such high

magnetizing forces, but was of special interest in that both B and I

were measured optically--B by means of the rotation of a polarized ray

inside a glass plate, as before described, and I by the rotation of a

polarized ray reflected from the polished surface of the magnetized

metal (see "Kerr's constant," MAGNETO-OPTICS). H(= B - 4[pi]I) was

calculated from corresponding values of I and B. Taylor Jones (_Wied.

Ann._, 1896, 57, 258, and _Phil. Mag._, 1896, 41, 153), working with

du Bois's electromagnet and using a modification of the isthmus

method, succeeded in pushing the induction B up to 74,200 with H =

51,600, the corresponding value of I being 1798, and of µ only 1.44.

The diameter of the isthmus was 0.241 mm., and the electromagnet was

excited by a current of 40 amperes.

_Tractive Force of a Magnet._--Closely connected with the results just

discussed is the question what is the greatest tractive force that can

be exerted by a magnet. In the year 1852 J. P. Joule (_Phil. Mag._,

1852, 3, 32) expressed the opinion that no "force of current could give

an attraction equal to 200 lb. per sq. in.," or 14,000 grms. per square

centimetre, and a similar view prevailed among high authorities more

than twenty years later. For the greatest possible "lifting power" of

permanent magnets this estimate is probably not very far from the truth,

but it is now clearly understood that the force which can be exerted by

an electromagnet, or by a pair of electromagnets with opposite poles in

contact, is only limited by the greatest value to which it is

practically possible to raise the magnetizing force H. This is at once

evident when the tractive force due to magnetization is expressed as

2[pi]I²+ HI. For fields of moderate intensity the first term of the

expression is the more important, but when the value of H exceeds 12,000

or thereabouts, the second preponderates, and with the highest values

that have been actually obtained, HI is several times greater than

2[pi]I². If H could be increased without limit, so also could the

tractive force. The following table shows the greatest "lifting powers"

experimentally reached at the dates mentioned:--

+-----------+-----------+---------+------+

+-----------+-----------+---------+------+

| Joule | 12.3 | 175 | 1852 |

| Bidwell | 15.9 | 226 | 1886 |

| Wilde | 26.8 | 381 | 1891 |

| T. Jones | 114.9 | 1634 | 1896 |

+-----------+-----------+---------+------+

5. MAGNETIZATION IN VERY WEAK FIELDS

Some interesting observations have been made of the effects produced by

very small magnetic forces. It was first pointed out by C. Baur (_Wied.

Ann._, 1880, 11, 399) that in weak fields the relation of the

magnetization I to the magnetizing force H is approximately expressed by

an equation of the form

I = aH + bH²,

[kappa] = I/H = a + bH,

whence it appears that within the limits of Baur's experiments the

magnetization curve is a parabola, and the susceptibility curve an

inclined straight line, [kappa] being therefore a known function of H.

If these equations could be assumed to hold when H is indefinitely

small, it would follow that [kappa] has a finite initial value, from

which there would be no appreciable deviation in fields so weak that bH

was negligibly small in comparison with a. Such an assumption could not,

however, without dangerous extrapolation, be founded upon the results of

Baur's experiments, which did not go far enough to justify it. In some

experiments carried out in 1887, Lord Rayleigh (_Phil. Mag._, 1887, 23,

225) approached very much more nearly than Baur to the zero of magnetic

force. Using an unannealed Swedish iron wire, he found that when H was

gradually diminished from 0.04 to 0.00004 C.G.S. unit, the ratio of

magnetization to magnetizing force remained sensibly constant at 6.4,

which may therefore with great probability be assumed to represent the

initial value of [kappa] for the specimen in question. Experiments with

annealed iron gave less satisfactory results, on account of the slowness

with which the metal settled down into a new magnetic state, thus

causing a "drift" of the magnetometer needle, which sometimes persisted

for several seconds. Apart from this complication, it appeared that I

was proportional to H when the value of H was less than 0.02.

The observations of Baur and Rayleigh have been confirmed and

discussed by (amongst others) W. Schmidt (_Wied. Ann._, 1895, 54,

655), who found the limiting values of [kappa] to be 7.5 to 9.5 for

iron, and 11.2 to 13.5 for steel, remaining constant up to H = .06; by

P. Culmann (_Elekt. Zeit._, 1893, 14, 345; _Wied. Ann._, 1895, 56,

602); and by L. Holborn (_Berl. Ber._, 1897, p. 95, and _Wied. Ann._,

1897, 61, 281). The latter gives values of the constants a and b for

different samples of iron and steel, some of which are shown in the

following table:--

[kappa] = a + bH

Metal. a b

English tungsten steel 8.90 0.264

Tungsten steel, hardened 2.23 0.032

Silver steel 8.66 0.384

Tool steel 8.30 0.400

Refined steel 11.28 1.92

Cast iron 3.16 0.236

Soft iron 16.6 18.6

Hard drawn iron 5.88 1.76

For most samples of steel the straight-line law was found to hold

approximately up to H = 3; in the case of iron and of soft steel the

approximation was less close.

The behaviour of nickel in weak fields has been observed by Ewing

(_Phil. Trans._, 1888, 179A, 325), who found that the initial value of

[kappa] was 1.7, and that it remained sensibly constant until H had

reached a value of about five units. While therefore the initial

susceptibility of nickel is less than that of iron and steel, the range

of magnetic force within which it is approximately constant is about one

hundred times greater. Ewing has also made a careful study (_Proc. Roy.

Soc._, 1889, 46, 269) of "magnetic viscosity" under small forces--the

cause of the magnetometer "drift" referred to by Rayleigh. On the

application of a small magnetizing force to a bar of soft annealed iron,

a certain intensity of magnetization is instantly produced; this,

however, does not remain constant, but slowly increases for some seconds

or even minutes, and may ultimately attain a value nearly twice as great

as that observed immediately after the force was applied.[30] When the

magnetizing current is broken, the magnetization at once undergoes

considerable diminution, then gradually falls to zero, and a similar

sudden change followed by a slow one is observed when a feeble current

is reversed. Ewing draws attention to a curious consequence of this

time-lag. By the alternate application and withdrawal of a small

magnetizing force a cyclic condition may be established in an iron rod.

If now the alternations are performed so rapidly that time is not

allowed for more than the first sudden change in the magnetization,

there will be no hysteresis loss, the magnetization exactly following

the magnetizing force. Further, if the alternations take place so slowly

that the full maximum and minimum values of the magnetization are

reached in the intervals between the reversals, there will again be no

dissipation of energy. But at any intermediate frequency the ascending

and descending curves of magnetization will enclose a space, and energy

will be dissipated. It is remarkable that the phenomena of magnetic

viscosity are much more evident in a thick rod than in a thin wire, or

even in a large bundle of thin wires. In hardened iron and steel the

effect can scarcely be detected, and in weak fields these metals exhibit

no magnetic hysteresis of any kind.

6. CHANGES OF DIMENSIONS ATTENDING MAGNETIZATION

It is well known that the form of a piece of ferromagnetic metal is in

general slightly changed by magnetization. The phenomenon was first

noticed by J. P. Joule, who in 1842 and 1847 described some experiments

which he had made upon bars of iron and steel. His observations, were

for the most part confirmed by a number of subsequent workers, notably

by A. M. Mayer; but with the single exception of the discovery by W. F.

Barrett in 1882 that a nickel bar contracts when magnetized, nothing of

importance was added by Joule's results for nearly forty years. Later

researches have however thrown much new light upon a class of phenomena

which cannot fail to have an important bearing upon the complete theory

of molecular magnetism.[31] According to Joule's observations, the

length of a bar of iron or soft steel was increased by magnetization,

the elongation being proportional up to a certain point to the square of

the intensity of magnetization; but when the "saturation point" was

approached the elongation was less than this law would require, and a

stage was finally reached at which further increase of the magnetizing

force produced little or no effect upon the length. From data contained

in Joule's paper it may be calculated that the strongest external field

H0 produced by his coil was about 126 C.G.S. units, but since the

dimensional ratio of his bars was comparatively small, the actual

magnetizing force H must have been materially below that value. In 1885

it was shown by Bidwell, in the first of a series of papers on the

subject, that if the magnetizing force is pushed beyond the point at

which Joule discontinued his experiments, the extension of the bar does

not remain unchanged, but becomes gradually less and less, until the

bar, after first returning to its original length, ultimately becomes

actually shorter than when in the unmagnetized condition. The elongation

is generally found to reach a maximum under a magnetizing force of 50 to

120 units, and to vanish under a force of 200 to 400, retraction

occurring when still higher forces are applied. In order to meet the

objection that the phenomenon might be due to electromagnetic action

between the coil and the rod, Bidwell made some experiments with iron

rings, and found that the length of their diameters varied under

magnetization in precisely the same manner as the length of a straight

rod. Experiments were afterwards made with rods of iron, nickel, and

cobalt, the external field being carried up to the high value of 1500

units. The results are indicated in Fig. 24. It appears that the

contraction which followed the initial extension of the iron reached a

limit in fields of 1000 or 1100. Nickel exhibited retraction from the

very beginning (as observed by Barrett), its greatest change of length

considerably exceeding that undergone by iron; in a field of 800 the

original length was diminished by as much as 1/40,000 part, but stronger

forces failed to produce any further effect. The curve for cobalt is a

very remarkable one. Little or no change of length was observed until

the strength of the field H0 reached about 50; then the rod began to

contract, and after passing a minimum at H0 = 400, recovered its

original length at H0 = 750; beyond this point there was extension, the

amount of which was still increasing fast when the experiment was

stopped at H0 = 1400. Similar results were obtained with three different

samples of the metal. Roughly speaking, therefore, cobalt behaves

oppositely to iron.

[Illustration: FIG. 24]

Joule and others experimented with hardened steel, but failed to find a

key to the results they obtained, which are rather complex, and have

been thought to be inconsistent. The truth appears to be that a hardened

steel rod generally behaves like one of iron or soft steel in first

undergoing extension under increasing magnetizing force, and recovering

its original length when the force has reached a certain critical value,

beyond which there is contraction. But this "critical value" of the

force is found to depend in an unexpected manner upon the hardness of

the steel; the critical value diminishes as the hardness becomes greater

_up to a certain point_, corresponding to a yellow temper, after which

it increases and with the hardest steel becomes very high. For steel

which has been made red-hot, suddenly cooled, and then let down to a

yellow temper, the critical value of the magnetizing force is smaller

than for steel which is either softer or harder; it is indeed so small

that the metal contracts like nickel even under weak magnetizing forces,

without undergoing any preliminary extension that can be detected.

[Illustration: FIG. 25.]

Joule also made experiments upon iron wires under tension, and drew the

erroneous inference (which has been often quoted as if it were a

demonstrated fact) that under a certain critical tension (differing for

different specimens of iron but independent of the magnetizing force)

magnetization would produce no effect whatever upon the dimensions of

the wire. What actually happens when an iron wire is loaded with various

weights is clearly shown in Fig. 25. Increased tension merely has the

effect of diminishing the maximum elongation and hastening the

contraction; with the two greatest loads used in the experiment there

was indeed no preliminary extension at all.[32] The effects of tension

upon the behaviour of a nickel wire are of a less simple character. In

weak fields the magnetic contraction is always diminished by pulling

stress; in strong fields the contraction increases under a small load

and diminishes under a heavy one. Cobalt, curiously enough, was found to

be quite unaffected by tensile stress.

Certain experiments by C. G. Knott on magnetic twist, which will be

referred to later, led him to form the conclusion that in an iron wire

carrying an electric current the magnetic elongation would be increased.

This forecast was shown by Bidwell to be well founded. The effect

produced by a current is exactly opposite to that of tension, raising

the elongation curve instead of depressing it. In the case of a wire

0.75 mm. in diameter the maximum elongation was nearly doubled when a

current of two amperes was passing through the iron, while the "critical

value" of the field was increased from 130 to 200. Yet notwithstanding

this enormous effect in iron, the action of a current upon nickel and

cobalt turned out to be almost inappreciable.

Some experiments were next undertaken with the view of ascertaining how

far magnetic changes of length in iron were dependent upon the hardness

of the metal, and the unexpected result was arrived at that softening

produces the same effect as tensile stress; it depresses the elongation

curve, diminishing the maximum extension, and reducing the "critical

value" of the magnetizing force. A thoroughly well annealed ring of soft

iron indeed showed no extension at all, beginning to contract, like

nickel, under the smallest magnetizing forces. The experiments were not

sufficiently numerous to indicate whether, as is possible, there is a

critical degree of hardness for which the height of the elongation curve

is a maximum.

Finally, experiments were made to ascertain the effect of magnetization

upon the dimensions of iron rings in directions perpendicular to the

magnetization, and upon the volume of the rings.[33] It was found that

the curve showing the relation of transverse changes of dimensions to

magnetizing force was similar in general character to the familiar

elongation curves, but the signs were reversed; the curve was inverted,

indicating at first retraction, which, after passing a maximum and

vanishing in a critical field, was succeeded by elongation. The curve

showing the circumferential (or longitudinal) changes was also plotted,

and from the two curves thus obtained it was easy, on the assumption

that the metal was isotropic in directions at right angles to the

magnetization, to calculate changes of volume; for if circumferential

elongation be denoted by l1, and transverse elongation by l2, then the

cubical dilatation (+ or -) = l1 + 2l2 approximately. If l1 were exactly

equal to -2l2 for all values of the magnetizing force, it is clear that

the volume of the ring would be unaffected by magnetization. In the case

of the ring in question, the circumferential changes were in weak fields

less than twice as great as the transverse ones, while in strong fields

they were more than twice as great; under increasing magnetic force

therefore the volume of the ring was first diminished, then it regained

its original value (for H = 90), and ultimately increased. It was also

shown that annealing, which has such a large effect upon circumferential

(or longitudinal) changes, has almost none upon transverse ones. Hence

the changes of volume undergone by a given sample of wrought iron under

increasing magnetization must depend largely upon the state of the metal

as regards hardness; there may be always contraction, or always

expansion, or first one and then the other.

Most of the experiments described above have been repeated and the

results confirmed by other workers, some of whom have added fresh

observations. The complicated hysteresis effects which attend magnetic

elongation and retraction have been studied by H. Nagaoka, who also, in

conjunction with K. Honda, measured the changes of length of various

metals shaped in the form of ovoids instead of cylindrical rods, and

determined the magnetization curves for the same specimens; a higher

degree of accuracy was thus attained, and satisfactory data were

provided for testing theories. Among other things, it was found that the

behaviour of cast cobalt was entirely changed by annealing; the sinuous

curve shown in Fig. 24 was converted into an almost perfectly straight

line passing through the origin, and lying below the horizontal axis;

while the permeability of the metal was greatly diminished by the

operation. They also tested several varieties of nickel-steel in the

form of both ovoids and wires. With a sample containing 25% of nickel no

appreciable change was detected; others containing larger percentages,

and tested in fields up to 2000, all exhibited elongation, which tended

to an asymptotic value as the field was increased. The influence of

temperature varying between wide limits has formed the subject of a

research by K. Honda and S. Shimizu. For soft iron, tungsten-steel and

nickel little difference appeared to result from lowering the

temperature down to -186° C. (the temperature of liquid air); at

sufficiently high temperatures, 600° to 1000° or more, it was remarked

that the changes of length in iron, steel and cobalt tended in every

case to become proportional to the magnetic force, the curves being

nearly straight lines entirely above the axis. The retraction of nickel

was diminished by rising temperature, and at 400° had almost vanished.

The influence of high temperature on cobalt was very remarkable,

completely altering the character of the change of length: the curves

for annealed cobalt show that at 450° this metal behaves just like iron

at ordinary temperatures, lengthening in fields up to about 300 and

contracting in stronger ones. The same physicists have made some

additional experiments upon the effect of tension on magnetic change of

length. Bidwell's results for iron and nickel were confirmed, and it was

further shown that the elongation of nickel-steel was very greatly

diminished by tension; when magnetized under very heavy loads, the wire

was indeed found to undergo slight contraction. Honda subjected tubes of

iron, steel and nickel to the simultaneous action of circular and

longitudinal fields, and observed the changes of length when one of the

fields was varied while the other remained constant at different

successive values from zero upwards. The experimental results agreed in

sign though not in magnitude with those calculated from the changes

produced by simple longitudinal magnetization, discrepancies being

partly accounted for by the fact that the metals employed were not

actually isotropic. Heusler's alloy has been tested for change of length

by L. Austin, who found continuous elongation with increasing fields,

the curves obtained bearing some resemblance to curves of magnetization.

As regards the effect of magnetization upon volume there are some

discrepancies. Nagaoka and Honda, who employed a fluid dilatometer,

found that the volume of several specimens of iron, steel and nickel was

always slightly increased, no diminution being indicated in low fields;

cobalt, on the other hand, was diminished in volume, and the amount of

the change, though still very small, was greater than that shown by the

other metals. Various nickel-steels all expanded under magnetization,

the increase being generally considerable and proportional to the field;

in the case of an alloy containing 29% of nickel the change was nearly

40 times greater than in soft iron. C. G. Knott, who made an exhaustive

series of experiments upon various metals in the form of tubes,

concluded that in iron there was always a slight increase of volume, and

in nickel and cobalt a slight decrease. It is uncertain how far these

various results are dependent upon the physical condition of the metals.

Attempts have been made to explain magnetic deformation by various

theories of magnetic stress,[34] notably that elaborated by G. R.

Kirchhoff (_Wied. Ann._, 1885, 24, 52, and 1885, 25, 601), but so far

with imperfect success. E. Taylor Jones showed in 1897 that only a

small proportion of the contraction exhibited by a nickel wire when

magnetized could be accounted for on Kirchhoff's theory from the

observed effects of pulling stress upon magnetization; and in a more

extended series of observations Nagaoka and Honda found wide

quantitative divergences between the results of experiment and

calculation, though in nearly all cases there was agreement as to

quality. They consider, however, that Kirchhoff's theory, which

assumes change of magnetization to be simply proportional to strain,

is still in its infancy, the present stage of its evolution being

perhaps comparable with that reached by the theory of magnetization at

the time when the ratio I/H was supposed to be constant. In the light

of future researches further development may reasonably be expected.

It has been suggested[35] that an iron rod under magnetization may be

in the same condition as if under a mechanically applied longitudinal

stress tending to shorten the iron. If a long magnetized rod is

divided transversely and the cut ends placed nearly in contact, the

magnetic force inside the narrow air gap will be B = H + 4[pi]I. The

force acting on the magnetism of one of the faces, and urging this

face towards the other, will be less than B by 2[pi]I, the part of the

total force due to the first face itself; hence the force per unit of

area with which the faces would press against each other if in contact

P = (B - 2[pi]I)I = 2[pi]I² + HI = (B² - H²) / 8[pi].

Encyclopaedia Britannica, 11th Edition, "McKinley, William" to "Magnetism, Terrestrial" Volume 17, Slice 3 · 第 41 / 45 章

_Directing Couple acting on an Elongated Body._--In a non-uniform