Chapter LXIII: repeats the promise of freedom to the English church (4)

_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.

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.

_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).

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:--

+-----------+-----------+---------+------+
| Observer. | Kilos per | lb. per | Date.|
| | sq. cm. | sq. in. | |
+-----------+-----------+---------+------+
| 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,

or

[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.

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.

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
is

P = (B - 2[pi]I)I = 2[pi]I² + HI = (B² - H²) / 8[pi].

The width of the gap may be diminished until it is no greater than the
distance between two neighbouring molecules, when it will cease to be
distinguishable, but, assuming the molecular theory of magnetism to be
true, the above statement will still hold good for the intermolecular
gap. The same pressure P will be exerted across any imaginary section
of a magnetized rod, the stress being sustained by the intermolecular
springs, whatever their physical nature may be, to which the
elasticity of the metal is due. The whole of the rod will therefore be
subject to a compressive longitudinal stress P, the associated
contraction R, expressed as a fraction of the original length, being

R = P/M = (B² - H²) / 8[pi]M,

where M is Young's modulus. This was found to be insufficient to
account for the whole of the retraction exhibited by iron in strong
fields, but it was pointed out by L. T. More[36] that R ought to be
regarded as a "correction" to be applied to the results of
experiments on magnetic change of length, the magnetic stress being no
less an extraneous effect than a stress applied mechanically. Those
who support this view generally speak of the stress as "Maxwell's
stress," and assume its value to be B^2/8[pi]. The stress in question
seems, however, to be quite unconnected with the "stress in the
medium" contemplated by Maxwell, and its value is not exactly
B^2/8[pi] except in the particular case of a permanent ring magnet,
when H = O. Further, Maxwell's stress is a tension along the lines of
force, and is equal to B^2/8[pi] only when B = H, and there is no
magnetization.[37] Some writers have indeed contended that the stress
in magnetized iron is not compressive, but tensile, even when, as in
the case of a ring-magnet, there are no free ends. The point at issue
has an important bearing upon the possible correlation of magnetic
phenomena, but, though it has given rise to much discussion, no
accepted conclusion has yet been reached.[38]

7. EFFECTS OF MECHANICAL STRESS UPON MAGNETIZATION

The effects of traction, compression and torsion in relation to magnetism have formed the subject of much patient investigation, especially at the hands of J. A. Ewing, C. G. Knott and the indefatigable physicists of Tokyo University. The results of their experiments embrace a multiplicity of details of which it is impossible to give an adequate summary. Only a few of the most important can be mentioned here; the reader who wishes for fuller information should consult the original papers.[39]

It was first discovered by E. Villari in 1868 that the magnetic susceptibility of an iron wire was increased by stretching when the magnetization was below a certain value, but diminished when that value was exceeded; this phenomenon has been termed by Lord Kelvin, who discovered it independently, the "Villari reversal," the value of the magnetization for which stretching by a given load produces no effect being known as the "Villari critical point" for that load. The Villari critical point for a given sample of iron is reached with a smaller magnetizing force when the stretching load is great than when it is small; the reversal also occurs with smaller loads and with weaker fields when the iron is soft than when it is hard. The following table shows the values of I and H corresponding to the Villari critical point in some of Ewing's experiments:--

+------------------------------+-----------------------------+
| Soft Iron. | Hard Iron. |
+-----------------+------+-----+-----------------+------+----+
|Kilos per sq. mm.| I. | H. |Kilos per sq. mm.| I. | H. |
+-----------------+------+-----+-----------------+------+----+
| 2.15 | 1220 | 7.3 | 27.6 | 1180 | 34 |
| 4.3 | 1040 | 4.3 | 32.2 | 1150 | 32 |
| 8.6 | 840 | 3.4 | 37.3 | 1110 | 29 |
| 12.9 | 690 | 3.05| 42.5 | 1020 | 25 |
+-----------------+------+-----+-----------------+------+----+

The effects of pulling stress may be observed either when the wire is stretched by a constant load while the magnetizing force is varied, or when the magnetizing force is kept constant while the load is varied. In the latter case the first application of stress is always attended by an increase--often a very great one--of the magnetization, whether the field is weak or strong, but after a load has been put on and taken off several times the changes of magnetization become cyclic. From experiments of both classes it appears that for a given field there is a certain value of the load for which the magnetization is a maximum, the maximum occurring at a smaller load the stronger the field. In very strong fields the maximum may even disappear altogether, the effect of the smallest stress being to diminish the magnetization; on the other hand, with very weak fields the maximum may not have been reached with the greatest load that the wire can support without permanent deformation. When the load on a hardened wire is gradually increased, the maximum value of I is found to correspond with a greater stress than when the load is gradually diminished, this being an effect of hysteresis. Analogous changes are observed in the residual magnetization which remains after the wire has been subjected to fields of different strength. The effects of longitudinal pressure are opposite to those of traction; when the cyclic condition has been reached, pressure reduces the magnetization of iron in weak fields and increases it in strong fields (Ewing, _Magnetic Induction_, 1900, 223).

The influence of traction in diminishing the susceptibility of nickel was first noticed by Kelvin (W. Thomson), and was subsequently investigated by Ewing and Cowan. The latter found the effect to be enormous, not only upon the induced magnetization, but in a still greater degree upon the residual. Even under so "moderate" a load as 33 kilogrammes per square mm., the induced magnetization of a hard-drawn nickel wire in a field of 60 fell from 386 to 72 units, while the residual was reduced from about 280 to 10. Ewing has also examined the effects produced by longitudinal compression upon the susceptibility and retentiveness of nickel, and found, as was to be expected, that both were greatly increased by pressure. The maximum susceptibility of one of his bars rose from 5.6 to 29 under a stress of 19.8 kilos per square mm. There were reasons for believing that no Villari reversal would be found in nickel. Ewing and Cowan looked carefully for it, especially in weak fields, but failed to discover anything of the kind.[40] Some experiments by A. Heydweiller,[41] which appeared to indicate a reversal in weak fields (corresponding to I = 5, or thereabouts), have been shown by Honda and Shimizu to be vitiated by the fact that his specimen was not initially in a magnetically neutral state; they found that when the applied field had the same direction as that of the permanent magnetization, Heydweiller's fallacious results were easily obtained; but if the field were applied in the direction opposite to that of the permanent magnetization, or if, as should rightly be the case, there were no permanent magnetization at all, then there was no indication of any Villari reversal. Thus a very important question, which has given rise to some controversy, appears to be now definitely settled.

The effects of longitudinal pressure upon the magnetization of cast cobalt have been examined by C. Chree,[42] and also by J. A. Ewing.[43] Chree's experiments were undertaken at the suggestion of J. J. Thomson, who, from the results of Bidwell's observations on the magnetic deformation of cobalt, was led to expect that that metal would exhibit a reversal opposite in character to the effect observed in iron. The anticipated reversal was duly found by Chree, the critical point corresponding, under the moderate stress employed, to a field of about 120 units. Ewing's independent experiments showed that the magnetization curve for a cobalt rod under a load of 16.2 kilogrammes per square mm. crossed the curve for the same rod when not loaded at H = 53. Both observers noticed analogous effects in the residual magnetization. The effect of tension was subsequently studied by Nagaoka and Honda, who in 1902 confirmed, _mutatis mutandis_, the results obtained by Chree and Ewing for cast cobalt, while for annealed cobalt it turned out that tension always caused diminution of magnetization, the diminution increasing with increasing fields. They also investigated the magnetic behaviour of various nickel-steels under tension, and found that there was always increase of magnetization. Thus it has been proved that in annealed cobalt and in nickel-steel there is no Villari reversal.

It has been pointed out by J. J. Thomson (_Applications of Dynamics to Physics and Chemistry_, 47) that on dynamical principles there must be a reciprocal relation between the changes of dimensions produced by magnetization and the changes of magnetization attending mechanical strain. Since, for example, stretching diminishes the magnetization of nickel, it follows from theory that the length of a nickel rod should be diminished by magnetization and conversely. So, too, the Villari reversals in iron and cobalt might have been predicted--as indeed that in cobalt actually was--from a knowledge of the changes of length which those metals exhibit when magnetized.

The complete reciprocity of the effects of magnetization upon length and of stretching upon magnetization is shown by the following parallel statements:--

_Iron._