Chapter LXIII: repeats the promise of freedom to the English church (2)

It is to the non-uniformity of the field surrounding a magnet that the apparent attraction between a magnet and a magnetizable body such as iron is ultimately due. This was pointed out by W. Thomson (afterwards Lord Kelvin) in 1847, as the result of a mathematical investigation undertaken to explain Faraday's experimental observations. If the inductively magnetized body lies in a part of the field which happens to be uniform there will be no resulting force tending to move the body, and it will not be "attracted." If however there is a small variation of the force in the space occupied by the body, it can be shown that the body will be urged, not necessarily towards a magnetic pole, but _towards places of stronger magnetic force_. It will not in general move along a line of force, as would an isolated pole, but will follow the direction in which the magnetic force increases most rapidly, and in so doing it may cross the lines of force obliquely or even at right angles.

If a magnetized needle were supported so that it could move freely about its centre of gravity it would not generally settle with its axis in a horizontal position, but would come to rest with its north-seeking pole either higher or lower than its centre. For the practical observation of this phenomenon it is usual to employ a needle which can turn freely in the plane of the magnetic meridian upon a horizontal axis passing through the centre of gravity of the needle. The angle which the magnetic axis makes with the plane of the horizon is called the _inclination_ or _dip._ Along an irregular line encircling the earth in the neighbourhood of the geographical equator the needle takes up a horizontal position, and the dip is zero. At places north of this line, which is called the _magnetic equator_, the north end of the needle points downwards, the inclination generally becoming greater with increased distance from the equator. Within a certain small area in the Arctic Circle (about 97° W. long., 70° N. lat.) the north pole of the needle points vertically downwards, the dip being 90°. South of the magnetic equator the south end of the needle is always inclined downwards, and there is a spot within the Antarctic Circle (148° E. long., 74° S. lat.) where the needle again stands vertically, but with its north end directed upwards. All these observations may be accounted for by the fact first recognized by W. Gilbert in 1600, that the earth itself is a great magnet, having its poles at the two places where the dipping needle is vertical. To be consistent with the terminology adopted in Britain, it is necessary to regard the pole which is geographically north as being the south pole of the terrestrial magnet, and that which is geographically south as the north pole; in practice however the names assigned to the terrestrial magnetic poles correspond with their geographical situations. Within a limited space, such as that contained in a room, the field due to the earth's magnetism is sensibly uniform, the lines of force being parallel straight lines inclined to the horizon at the angle of dip, which at Greenwich in 1910 was about 67°. It is by the horizontal component of the earth's total force that the compass-needle is directed.

The magnets hitherto considered have been assumed to have each two poles, the one north and the other south. It is possible that there may be more than two. If, for example, a knitting needle is stroked with the south pole of a magnet, the strokes being directed from the middle of the needle towards the two extremities alternately, the needle will acquire a north pole at each end and a south pole in the middle. By suitably modifying the manipulation a further number of _consequent poles_, as they are called, may be developed. It is also possible that a magnet may have no poles at all. Let a magnetic pole be drawn several times around a uniform steel ring, so that every part of the ring may be successively subjected to the magnetic force. If the operation has been skilfully performed the ring will have no poles and will not attract iron filings. Yet it will be magnetized; for if it is cut through and the cut ends are drawn apart, each end will be found to exhibit polarity. Again, a steel wire through which an electric current has been passed will be magnetized, but so long as it is free from stress it will give no evidence of magnetization; if, however, the wire is twisted, poles will be developed at the two ends, for reasons which will be explained later. A wire or rod in this condition is said to be _circularly magnetized_; it may be regarded as consisting of an indefinite number of elementary ring-magnets, having their axes coincident with the axis of the wire and their planes at right angles to it. But no magnet can have a single pole; if there is one, there must also be at least a second, of the opposite sign and of exactly equal strength. Let a magnetized knitting needle, having north and south poles at the two ends respectively, be broken in the middle; each half will be found to possess a north and a south pole, the appropriate supplementary poles appearing at the broken ends. One of the fragments may again be broken, and again two bipolar magnets will be produced; and the operation may be repeated, at least in imagination, till we arrive at molecular magnitudes and can go no farther. This experiment proves that the condition of magnetization is not confined to those parts where polar phenomena are exhibited, but exists throughout the whole body of the magnet; it also suggests the idea of _molecular magnetism_, upon which the accepted theory of magnetization is based. According to this theory the molecules of any magnetizable substance are little permanent magnets the axes of which are, under ordinary conditions, disposed in all possible directions indifferently. The process of magnetization consists in turning round the molecules by the application of magnetic force, so that their north poles may all point more or less approximately in the direction of the force; thus the body as a whole becomes a magnet which is merely the resultant of an immense number of molecular magnets.

In every magnet the strength of the south pole is exactly equal to that of the north pole, the action of the same magnetic force upon the two poles being equal and oppositely directed. This may be shown by means of the uniform field of force due to the earth's magnetism. A magnet attached to a cork and floated upon water will set itself with its axis in the magnetic meridian, but it will be drawn neither northward nor southward; the forces acting upon the two poles have therefore no horizontal resultant. And again if a piece of steel is weighed in a delicate balance before and after magnetization, no change whatever in its weight can be detected; there is consequently no upward or downward resultant force due to magnetization; the contrary parallel forces acting upon the poles of the magnet are equal, constituting a couple, which may tend to turn the body, but not to propel it.

Iron and its alloys, including the various kinds of steel, though exhibiting magnetic phenomena in a pre-eminent degree, are not the only substances capable of magnetization. Nickel and cobalt are also strongly magnetic, and in 1903 the interesting discovery was made by F. Heusler that an alloy consisting of copper, aluminium and manganese (Heusler's alloy), possesses magnetic qualities comparable with those of iron. Practically the metals iron, nickel and cobalt, and some of their alloys and compounds constitute a class by themselves and are called _ferromagnetic_ substances. But it was discovered by Faraday in 1845 that all substances, including even gases, are either attracted or repelled by a sufficiently powerful magnetic pole. Those substances which are attracted, or rather which tend, like iron, to move from weaker to stronger parts of the magnetic field, are termed _paramagnetic_; those which are repelled, or tend to move from stronger to weaker parts of the field, are termed diamagnetic. Between the ferromagnetics and the paramagnetics there is an enormous gap. The maximum magnetic susceptibility of iron is half a million times greater than that of liquid oxygen, one of the strongest paramagnetic substances known. Bismuth, the strongest of the diamagnetics, has a negative susceptibility which is numerically 20 times less than that of liquid oxygen.

Many of the physical properties of a metal are affected by magnetization. The dimensions of a piece of iron, for example, its elasticity, its thermo-electric power and its electric conductivity are all changed under the influence of magnetism. On the other hand, the magnetic properties of a substance are affected by such causes as mechanical stress and changes of temperature. An account of some of these effects will be found in another section.[2]

2. TERMINOLOGY AND ELEMENTARY PRINCIPLES

In what follows the C.G.S. electromagnetic system of units will be generally adopted, and, unless otherwise stated, magnetic substances will be assumed to be _isotropic_, or to have the same physical properties in all directions.

_Vectors._--Physical quantities such as magnetic force, magnetic
induction and magnetization, which have direction as well as
magnitude, are termed vectors; they are compounded and resolved in the
same manner as mechanical force, which is itself a vector. When the
direction of any vector quantity denoted by a symbol is to be attended
to, it is usual to employ for the symbol either a block letter, as H,
I, B, or a German capital, as [H], [F], [B].[3]

_Magnetic Poles and Magnetic Axis._--A _unit magnetic pole_ is that
which acts on an equal pole at a distance of one centimetre with a
force of one dyne. A pole which points north is reckoned positive, one
which points south negative. The action between any two magnetic poles
is mutual. If m1 and m2 are the strengths of two poles, d the distance
between them expressed in centimetres, and f the force in dynes,

[f] = m1m2/d² (1).

The force is one of attraction or repulsion, according as the sign of
the product m1m2 is negative or positive. The poles at the ends of an
infinitely thin uniform magnet, or _magnetic filament_, would act as
definite centres of force. An actual magnet may generally be regarded
as a bundle of magnetic filaments, and those portions of the surface
of the magnet where the filaments terminate, and so-called "free
magnetism" appears, may be conveniently called poles or polar regions.
A more precise definition is the following: When the magnet is placed
in a uniform field, the parallel forces acting on the positive poles
of the constituent filaments, whether the filaments terminate outside
the magnet or inside, have a resultant, equal to the sum of the forces
and parallel to their direction, acting at a certain point N. The
point N, which is the centre of the parallel forces, is called the
_north_ or _positive pole_ of the magnet. Similarly, the forces acting
in the opposite direction on the negative poles of the filaments have
a resultant at another point S, which is called the _south_ or
_negative pole_. The opposite and parallel forces acting on the poles
are always equal, a fact which is sometimes expressed by the statement
that the total magnetism of a magnet is zero. The line joining the two
poles is called the _axis of the magnet_.

_Magnetic Field._--Any space at every point of which there is a finite
magnetic force is called a _field of magnetic force_, or a _magnetic
field_. The _strength_ or _intensity_ of a magnetic field at any point
is measured by the force in dynes which a unit pole will experience
when placed at that point, the _direction_ of the field being the
direction in which a positive pole is urged. The field-strength at any
point is also called the _magnetic force_ at that point; it is denoted
by H, or, when it is desired to draw attention to the fact that it is
a vector quantity, by the block letter H, or the German character [H].
Magnetic force is sometimes, and perhaps more suitably, termed
_magnetic intensity_; it corresponds to the intensity of gravity g in
the theory of heavy bodies (see Maxwell, _Electricity and Magnetism_,
§ 12 and § 68, footnote). A _line of force_ is a line drawn through a
magnetic field in the direction of the force at each point through
which it passes. A _uniform magnetic field_ is one in which H has
everywhere the same value and the same direction, the lines of force
being, therefore, straight and parallel. A magnetic field is generally
due either to a conductor carrying an electric current or to the poles
of a magnet. The magnetic field due to a long straight wire in which a
current of electricity is flowing is at every point at right angles to
the plane passing through it and through the wire; its strength at any
point distant r centimetres from the wire is

H = 2i/r, (2)

i being the current in C.G.S. units.[4] The lines of force are
evidently circles concentric with the wire and at right angles to it;
their direction is related to that of the current in the same manner
as the rotation of a corkscrew is related to its thrust. The field at
the centre of a circular conductor of radius r through which current
is passing is

H = 2[pi]i/r, (3)

the direction of the force being along the axis and related to the
direction of the current as the thrust of a corkscrew to its rotation.
The field strength in the interior of a long uniformly wound coil
containing n turns of wire and having a length of l centimetres is
(except near the ends)

H = 4[pi]in/l. (4)

In the middle portion of the coil the strength of the field is very
nearly uniform, but towards the end it diminishes, and at the ends is
reduced to one-half. The direction of the force is parallel to the
axis of the coil, and related to the direction of the current as the
thrust of a corkscrew to its rotation. If the coil has the form of a
ring of mean radius r, the length will be 2[pi]r, and the field inside
the coil may be expressed as

H = 2ni/r. (5)

The uniformity of the field is not in this case disturbed by the
influence of ends, but its strength at any point varies inversely as
the distance from the axis of the ring. When therefore sensible
uniformity is desired, the radius of the ring should be large in
relation to that of the convolutions, or the ring should have the form
of a short cylinder with thin walls. The strongest magnetic fields
employed for experimental purposes are obtained by the use of
electromagnets. For many experiments the field due to the earth's
magnetism is sufficient; this is practically quite uniform throughout
considerable spaces, but its total intensity is less than half a unit.

_Magnetic Moment and Magnetization._--The moment, M, M or [M], of a
uniformly and longitudinally magnetized bar-magnet is the product of
its length into the strength of one of its poles; it is the moment of
the couple acting on the magnet when placed in a field of unit
intensity with its axis perpendicular to the direction of the field.
If l is the length of the magnet, M = ml. The action of a magnet at a
distance which is great compared with the length of the magnet depends
solely upon its moment; so also does the action which the magnet
experiences when placed in a uniform field. The moment of a small
magnet may be resolved like a force. The _intensity of magnetization_,
or, more shortly, the _magnetization_ of a uniformly magnetized body
is defined as the magnetic moment per unit of volume, and is denoted
by I, I, or [I]. Hence

I = M/v = ml/v = m/a,

v being the volume and a the sectional area. If the magnet is not
uniform, the magnetization at any point is the ratio of the moment of
an element of volume at that point to the volume itself, or I =
m·ds/dv. where ds is the length of the element. The direction of the
magnetization is that of the magnetic axis of the element; in
isotropic substances it coincides with the direction of the magnetic
force at the point. If the direction of the magnetization at the
surface of a magnet makes an angle [epsilon] with the normal, the
normal component of the magnetization, I cos [epsilon], is called the
_surface density_ of the magnetism, and is generally denoted by
[sigma].

_Potential and Magnetic Force._--The _magnetic potential_ at any point
in a magnetic field is the work which would be done against the
magnetic forces in bringing a unit pole to that point from the
boundary of the field. The line through the given point along which
the potential decreases most rapidly is the direction of the resultant
magnetic force, and the rate of decrease of the potential in any
direction is equal to the component of the force in that direction. If
V denote the potential, F the resultant force, X, Y, Z, its components
parallel to the co-ordinate axes and n the line along which the force
is directed, then

[delta]V [delta]V [delta]V/ [delta]V
- -------- = F, - -------- = X, - --------- = Y, - -------- = Z. (6)
[delta]n [delta]x [delta]y [delta]z

Surfaces for which the potential is constant are called _equipotential
surfaces_. The resultant magnetic force at every point of such a
surface is in the direction of the normal (n) to the surface; every
line of force therefore cuts the equipotential surfaces at right
angles. The potential due to a single pole of strength m at the
distance r from the pole is

V = m/r, (7)

the equipotential surfaces being spheres of which the pole is the
centre and the lines of force radii. The potential due to a thin
magnet at a point whose distance from the two poles respectively is r
and r´ is

V = m(l/r = l/r´). (8)

When V is constant, this equation represents an equipotential surface.

The equipotential surfaces are two series of ovoids surrounding the
two poles respectively, and separated by a plane at zero potential
passing perpendicularly through the middle of the axis. If r and r´
make angles [theta] and [theta]´ with the axis, it is easily shown
that the equation to a line of force is

cos [theta] - cos [theta]´ = constant. (9)

At the point where a line of force intersects the perpendicular
bisector of the axis r = r´ = r0, say, and cos [theta] - cos [theta]´
obviously = l/r0, l being the distance between the poles; l/r0 is
therefore the value of the constant in (9) for the line in question.
Fig. 2 shows the lines of force and the plane sections of the
equipotential surfaces for a thin magnet with poles concentrated at
its ends. The potential due to a small magnet of moment M, at a point
whose distance from the centre of the magnet is r, is

V = M cos [theta]/r², (10)

where [theta] is the angle between r and the axis of the magnet.
Denoting the force at P (see fig. 3) by F, and its components parallel
to the co-ordinate axes by X and Y, we have

[delta]V M
X = - -------- = --- (3 cos² [theta] - 1),
[delta]x r³

[delta]V M
Y = - -------- = --- (3 sin [theta] cos [theta]). (11)
[delta]y r³

If F_r is the force along r and F_t that along t at right angles to r,

M
F_r = X cos [theta] + Y sin [theta] = --- 2 cos [theta], (12)
r³

M
F_t = -X sin [theta] + Y cos [theta] = --- sin [theta], (13)
r³
For the resultant force at P,

M
F = [root](F_r² + F_t²) = --- [root](3 cos² [theta] + 1). (14)
r³

The direction of F is given by the following construction: Trisect OP
at C, so that OC = OP/3; draw CD at right angles to OP, to cut the
axis produced in D; then DP will be the direction of the force at P.
For a point in the axis OX, [theta] = 0; therefore cos [theta] = 1,
and the point D coincides with C; the magnitude of the force is, from
(14),

F_x = 2M/r³, (15)

its direction being along the axis OX. For a point in the line OY
bisecting the magnet perpendicularly, [theta] = [pi]/2 therefore cos
[theta] = 0, and the point D is at an infinite distance. The magnitude
of the force is in this case

F_y = M/r³, (16)

and its direction is parallel to the axis of the magnet. Although the
above useful formulae, (10) to (15), are true only for an infinitely
small magnet, they may be practically applied whenever the distance r
is considerable compared with the length of the magnet.

_Couples and Forces between Magnets._--If a small magnet of moment M
is placed in the sensibly uniform field H due to a distant magnet, the
couple tending to turn the small magnet upon an axis at right angles
to the magnet and to the force is

MH sin [theta], (17)

where [theta] is the angle between the axis of the magnet and the
direction of the force. In fig. 4 S´N´ is a small magnet of moment M´,
and SN a distant fixed magnet of moment M; the axes of SN and S´N´
make angles of [theta] and [phi] respectively with the line through
their middle points. It can be deduced from (17), (12) and (13) that
the couple on S´N´ due to SN, and tending to increase [phi], is

MM´(sin [theta] cos [phi] - 2 sin [phi] cos [theta])/r³. (18)

This vanishes if sin [theta] cos [phi] = 2 sin [phi] cos [theta],
i.e. if tan [phi] = ½ tan [theta], S´N´ being then along a line of
force, a result which explains the construction given above for
finding the direction of the force F in (14). If the axis of SN
produced passes through the centre of S´N´, [theta] = 0, and the
couple becomes

2MM´ sin [phi]/r³, (19)

tending to diminish [phi]; this is called the "end on" position. If
the centre of S´N´ is on the perpendicular bisector of SN, [theta] =
½[pi], and the couple will be

MM´ cos [phi]/r³, (20)

tending to increase [phi]; this is the "broadside on" position. These
two positions are sometimes called the first and second (or A and B)
principal positions of Gauss. The components X, Y, parallel and
perpendicular to r, of the force between the two magnets SN and S´N´
are

X = 3MM´(sin [theta] sin [phi] - 2 cos [theta] cos [phi])/r^4, (21)
Y = 3MM´(sin [theta] cos [phi] + sin [phi] cos [theta])/r^4. (22)

It will be seen that, whereas the couple varies inversely as the cube
of the distance, the force varies inversely as the fourth power.

_Distributions of Magnetism._--A magnet may be regarded as consisting
of an infinite number of elementary magnets, each having a pair of
poles and a definite magnetic moment. If a series of such elements,
all equally and longitudinally magnetized, were placed end to end with
their unlike poles in contact, the external action of the filament
thus formed would be reduced to that of the two extreme poles. The
same would be the case if the magnetization of the filament varied
inversely as the area of its cross-section a in different parts. Such
a filament is called a _simple magnetic solenoid_, and the product aI
is called the _strength_ of the solenoid. A magnet which consists
entirely of such solenoids, having their ends either upon the surface
or closed upon themselves, is called a _solenoidal magnet_, and the
magnetism is said to be distributed solenoidally; there is no free
magnetism in its interior. If the constituent solenoids are parallel
and of equal strength, the magnet is also uniformly magnetized. A thin
sheet of magnetic matter magnetized normally to its surface in such a
manner that the magnetization at any place is inversely proportional
to the thickness h of the sheet at that place is called a _magnetic
shell_; the constant product hI is the _strength_ of the shell and is
generally denoted by [Phi] or [phi]. The potential at any point due to
a magnetic shell is the product of its strength into the solid angle
[omega] subtended by its edge at the given point, or V = [Phi][omega].
For a given strength, therefore, the potential depends solely upon the
boundary of the shell, and the potential outside a closed shell is
everywhere zero. A magnet which can be divided into simple magnetic
shells, either closed or having their edges on the surface of the
magnet, is called a _lamellar magnet_, and the magnetism is said to be
distributed lamellarly. A magnet consisting of a series of plane
shells of equal strength arranged at right angles to the direction of
magnetization will be uniformly magnetized.

It can be shown that uniform magnetization is possible only when the
form of the body is ellipsoidal. (Maxwell, _Electricity and
Magnetism_, II., § 437). The cases of greatest practical importance
are those of a sphere (which is an ellipsoid with three equal axes)
and an ovoid or prolate ellipsoid of revolution. The potential due to
a uniformly magnetized sphere of radius a for an external point at a
distance r from the centre is

V = (4/3)[pi] a³I cos [theta]/r², (23)

[theta] being the inclination of r to the magnetic axis. Since
(4/3)[pi]a³I is the moment of the sphere (= volume × magnetization),
it appears from (10) that the magnetized sphere produces the same
external effect as a very small magnet of equal moment placed at its
centre and magnetized in the same direction; the resultant force
therefore is the same as in (14). The force in the interior is
uniform, opposite to the direction of magnetization, and equal to
(4/3)[pi]I. When it is desired to have a uniform magnet with
definitely situated poles, it it usual to employ one having the form
of an ovoid, or elongated ellipsoid of revolution, instead of a
rectangular or cylindrical bar. If the magnetization is parallel to
the major axis, and the lengths of the major and minor axes are 2a and
2c, the poles are situated at a distance equal to (2/3)a from the
centre, and the magnet will behave externally like a simple solenoid
of length (4/3)a. The internal force F is opposite to the direction of
the magnetization, and equal to NI, where N is a coefficient depending
only on the ratio of the axes. The moment = (4/3)[pi] ac²I =
-(4/3)[pi] ac²FN.

The distribution of magnetism and the position of the poles in magnets
of other shapes, such as cylindrical or rectangular bars, cannot be
specified by any general statement, though approximate determinations
may be obtained experimentally in individual cases.[5] According to F.
W. G. Kohlrausch[6] the distance between the poles of a cylindrical
magnet the length of which is from 10 to 30 times the diameter, is
sensibly equal to five-sixths of the length of the bar. This
statement, however, is only approximately correct, the distance
between the poles depending upon the intensity of the
magnetization.[7] In general, the greater the ratio of length to
section, the more nearly will the poles approach the end of the bar,
and the more nearly uniform will be the magnetization. For most
practical purpose a knowledge of the exact position of the poles is of
no importance; the magnetic moment, and therefore the mean
magnetization, can always be determined with accuracy.

_Magnetic Induction or Magnetic Flux._--When magnetic force acts on
any medium, whether magnetic, diamagnetic or neutral, it produces
within it a phenomenon of the nature of a flux or flow called
_magnetic induction_ (Maxwell, _loc. cit._, § 428). Magnetic
induction, like other fluxes such as electrical, thermal or fluid
currents, is defined with reference to an area; it satisfies the same
conditions of continuity as the electric current does, and in
isotropic media it depends on the magnetic force just as the electric
current depends on the electromotive force. The magnitude of the flux
produced by a given magnetic force differs in different media. In a
uniform magnetic field of unit intensity formed in empty space the
induction or magnetic flux across an area of 1 square centimetre
normal to the direction of the field is arbitrarily taken as the unit
of induction. Hence if the induction per square centimetre at any
point is denoted by B, then in empty space B is numerically equal to
H; moreover in isotropic media both have the same direction, and for
these reasons it is often said that in empty space (and practically in
air and other non-magnetic substances) B and H are identical. Inside a
magnetized body, B is the force that would be exerted on a unit pole
if placed in a narrow crevasse cut in the body, the walls of the
crevasse being perpendicular to the direction of the magnetization
(Maxwell, § § 399, 604); and its numerical value, being partly due to
the free magnetism on the walls, is generally very different from that
of H. In the case of a straight uniformly magnetized bar the direction
of the magnetic force due to the poles of the magnet is from the north
to the south pole outside the magnet, and from the south to the north
inside. The magnetic flux per square centimetre at any point (B, B, or
[B]) is briefly called the _induction_, or, especially by electrical
engineers, the _flux-density_. The direction of magnetic induction may
be indicated by _lines of induction_; a line of induction is always a
closed curve, though it may possibly extend to and return from
infinity. Lines of induction drawn through every point in the contour
of a small surface form a re-entrant tube bounded by lines of
induction; such a tube is called a _tube of induction_. The
cross-section of a tube of induction may vary in different parts, but
the total induction across any section is everywhere the same. A
special meaning has been assigned to the term "lines of induction."
Suppose the whole space in which induction exists to be divided up
into _unit tubes_, such that the surface integral of the induction
over any cross-section of a tube is equal to unity, and along the axis
of each tube let a line of induction be drawn. These axial lines
constitute the system of lines of induction which are so often
referred to in the specification of a field. Where the induction is
high the lines will be crowded together; where it is weak they will be
widely separated, the number per square centimetre crossing a normal
surface at any point being always equal to the numerical value of B.
The induction may therefore be specified as B lines per square
centimetre. The direction of the induction is also of course indicated
by the direction of the lines, which thus serve to map out space in a
convenient manner. Lines of induction are frequently but inaccurately
spoken of as lines of force.

When induction or magnetic flux takes place in a ferromagnetic metal,
the metal becomes magnetized, but the magnetization at any point is
proportional not to B, but to B - H. The factor of proportionality
will be 1-4[pi], so that

I = (B - H)/4[pi], (24)

or

B = H + 4[pi] I. (25)

Unless the path of the induction is entirely inside the metal, free
magnetic poles are developed at those parts of the metal where
induction enters and leaves, the polarity being south at the entry and
north at the exit of the flux. These free poles produce a magnetic
field which is superposed upon that arising from other sources. The
_resultant magnetic field_, therefore, is compounded of two fields,
the one being due to the poles, and the other to the external causes
which would be operative in the absence of the magnetized metal. The
intensity (at any point) of the field due to the magnetization may be
denoted by H_i, that of the external field by H0, and that of the
resultant field by H. In certain cases, as, for instance, in an iron
ring wrapped uniformly round with a coil of wire through which a
current is passing, the induction is entirely within the metal; there
are, consequently, no free poles, and the ring, though magnetized,
constitutes a poleless magnet. Magnetization is usually regarded as
the direct effect of the resultant magnetic force, which is therefore
often termed the _magnetizing force_.

_Permeability and Susceptibility._--The ratio B/H is called the
_permeability_ of the medium in which the induction is taking place,
and is denoted by µ. The ratio I/H is called the _susceptibility_ of
the magnetized substance, and is denoted by [kappa]. Hence

B = µH and I = [kappa]H. (26)

Also

B H + 4[pi]I
µ = --- = ---------- = 1 + 4[pi][kappa], (27)
H H

and

µ - 1
[kappa] = ----- (28)
4[pi]

Since in empty space B has been assumed to be numerically equal to H,
it follows that the permeability of a vacuum is equal to 1. The
permeability of most material substances differs very slightly from
unity, being a little greater than 1 in paramagnetic and a little less
in diamagnetic substances. In the case of the ferromagnetic metals and
some of their alloys and compounds, the permeability has generally a
much higher value. Moreover, it is not constant, being an apparently
arbitrary function of H or of B; in the same specimen its value may,
under different conditions, vary from less than 2 to upwards of 5000.
The magnetic susceptibility [kappa] expresses the numerical relation
of the magnetization to the magnetizing force. From the equation
[kappa] = (µ - 1)/4[pi], it follows that the magnetic susceptibility
of a vacuum (where µ = l) is 0, that of a diamagnetic substance (where
µ < l) has a negative value, while the susceptibility of paramagnetic
and ferromagnetic substances (for which µ > 1) is positive. No
substance has yet been discovered having a negative susceptibility
sufficiently great to render the permeability (= 1 + 4[pi][kappa])
negative.

_Magnetic Circuit._--The circulation of magnetic induction or flux
through magnetic and non-magnetic substances, such as iron and air, is
in many respects analogous to that of an electric current through good
and bad conductors. Just as the lines of flow of an electric current
all pass in closed curves through the battery or other generator, so
do all the lines of induction pass in closed curves through the magnet
or magnetizing coil. The total magnetic induction or flux corresponds
to the current of electricity (practically measured in amperes); the
induction or flux density B to the density of the current (number of
amperes to the square centimetre of section); the magnetic
permeability to the specific electric conductivity; and the line
integral of the magnetic force, sometimes called the magneto-motive
force, to the electromotive force in the circuit. The principal points
of difference are that (1) the magnetic permeability, unlike the
electric conductivity, which is independent of the strength of the
current, is not in general constant; (2) there is no perfect insulator
for magnetic induction, which will pass more or less freely through
all known substances. Nevertheless, many important problems relating
to the distribution of magnetic induction may be solved by methods
similar to those employed for the solution of analogous problems in
electricity. For the elementary theory of the magnetic circuit see
ELECTRO-MAGNETISM.

_Hysteresis, Coercive Force, Retentiveness._--It is found that when a
piece of ferromagnetic metal, such as iron, is subjected to a magnetic
field of changing intensity, the changes which take place in the
induced magnetization of the iron exhibit a tendency to lag behind
those which occur in the intensity of the field--a phenomenon to which
J. A. Ewing (_Phil. Trans._ clxxvi. 524) has given the name of
_hysteresis_ (Gr. [Greek: hystereô], to lag behind). Thus it happens
that there is no definite relation between the magnetization of a
piece of metal which has been previously magnetized and the strength
of the field in which it is placed. Much depends upon its antecedent
magnetic condition, and indeed upon its whole magnetic history. A
well-known example of hysteresis is presented by the case of permanent
magnets. If a bar of hard steel is placed in a strong magnetic field,
a certain intensity of magnetization is induced in the bar; but when
the strength of the field is afterwards reduced to zero, the
magnetization does not entirely disappear. That portion which is
permanently retained, and which may amount to considerably more than
one-half, is called the _residual magnetization_. The ratio of the
residual magnetization to its previous maximum value measures the
_retentiveness_, or _retentivity_, of the metal.[8] Steel, which is
well suited for the construction of permanent magnets, is said to
possess great "coercive force." To this term, which had long been used
in a loose and indefinite manner, J. Hopkinson supplied a precise
meaning (_Phil. Trans._ clxxvi. 460). The _coercive force_, or
_coercivity_, of a material is that reversed magnetic force which,
while it is acting, just suffices to reduce the residual induction to
nothing after the material has been temporarily submitted to any great
magnetizing force. A metal which has great retentiveness may at the
same time have small coercive force, and it is the latter quality
which is of chief importance in permanent magnets.

_Demagnetizing Force._--It has already been mentioned that when a
ferromagnetic body is placed in a magnetic field, the resultant
magnetic force H, at a point within the body, is compounded of the
force H0, due to the external field, and of another force, H_i,
arising from the induced magnetization of the body. Since H_i
generally tends to oppose the external force, thus making H less than
H0, it may be called the _demagnetizing force_. Except in the few
special cases when a uniform external field produces uniform
magnetization, the value of the demagnetizing force cannot be
calculated, and an exact determination of the actual magnetic force
within the body is therefore impossible. An important instance in
which the calculation can be made is that of an elongated _ellipsoid
of revolution_ placed in a uniform field H0, with its axis of
revolution parallel to the lines of force. The magnetization at any
point inside the ellipsoid will then be

[kappa]H0
I = ------------ (29)
1 + [kappa]N

where

/ 1 \ / 1 1 + e \
N = 4[pi]( --- - 1 ) ( --- log ----- ),
\ e2 / \ 2e 1 - e /

e being the eccentricity (see Maxwell's _Treatise_, § 438). Since I =
[kappa]H, we have

[kappa]H + [kappa]NI = [kappa]H0, (30)

or

H = H0 - NI,

NI being the demagnetizing force H_i. N may be called, after H. du
Bois (_Magnetic Circuit_, p. 33), the _demagnetizing factor_, and the
ratio of the length of the ellipsoid 2c to its equatorial diameter 2a
(= c/a), the _dimensional ratio_, denoted by the symbol [m].

Since e = [root](1 - a²/c²) = [root](1 - 1/[m]²),

the above expression for N may be written

4[pi] / [m] [m] + [root]([m]² - 1) \
N = -------- ( ------------------ log ---------------------- - 1 )
[m]² - 1 \ 2 [root]([m]² - 1) [m] - [root]([m]² - 1) /
_ _
4[pi] | [m] / \ |
= -------- | ---------------- log ( [m] + [root]([m]² - 1) ) - 1 |,
[m]² - 1 |_ [root]([m]² - 1) \ / _|

from which the value of N for a given dimensional ratio can be
calculated. When the ellipsoid is so much elongated that 1 is
negligible in relation to [m]², the expression approximates to the
simpler form

4[pi]
N = ----- (log 2[m] - 1). (31)
[m]²

In the case of a _sphere_, e = O and N = (4/3)[pi]; therefore from
(29)

[kappa]H0 3[kappa]
I = [kappa]H = -------------------- = ---------------- H0, (32)
1 + (4/3)[pi][kappa] 3 + 4[pi][kappa]

Whence

3 3
H = ---------------- H0 = ----- H0, (33)
3 + 4[pi][kappa] µ + 2

and

3µ
B = µH = ----- H0. (34)
µ + 2

Equations (33) and (34) show that when, as is generally the case with
ferromagnetic substances, the value of µ is considerable, the
resultant magnetic force is only a small fraction of the external
force, while the numerical value of the induction is approximately
three times that of the external force, and nearly independent of the
permeability. The demagnetizing force inside a _cylindrical rod_
placed longitudinally in a uniform field H0 is not uniform, being
greatest at the ends and least in the middle part. Denoting its mean
value by [H]_i, and that of the demagnetizing factor by [N], we have

H = H0 - [H]_i = H0 - [N]I. (35)

Du Bois has shown that when the dimensional ratio [m] (=
length/diameter) exceeds 100, [N][m]² = constant = 45, and hence for
long thin rods

[N] = 45/[m]². (36)

From an analysis of a number of experiments made with rods of
different dimensions H. du Bois has deduced the corresponding mean
demagnetizing factors. These, together with values of [m]²[N] for
cylindrical rods, and of N and [m]²N for ellipsoids of revolution, are
given in the following useful table (_loc. cit._ p. 41):--

_Demagnetizing Factors._

+------+--------------------+-------------------+
| | Cylinder. | Ellipsoid. |
| [m]. +----------+---------+----------+--------+
| | [N]. | [m]²[N].| N. | [m]²N. |
+------+----------+---------+----------+--------+
| 0 | 12.5664 | 0 | 12.5664 | 0 |
| 0.5 | -- | -- | 6.5864 | -- |
| 1 | -- | -- | 4.1888 | -- |
| 5 | -- | -- | 0.7015 | -- |
| 10 | 0.2160 | 21.6 | 0.2549 | 25.5 |
| 15 | 0.1206 | 27.1 | 0.1350 | 30.5 |
| 20 | 0.0775 | 31.0 | 0.0848 | 34.0 |
| 25 | 0.0533 | 33.4 | 0.0579 | 36.2 |
| 30 | 0.0393 | 35.4 | 0.0432 | 38.8 |
| 40 | 0.0238 | 38.7 | 0.0266 | 42.5 |
| 50 | 0.0162 | 40.5 | 0.0181 | 45.3 |
| 60 | 0.0118 | 42.4 | 0.0132 | 47.5 |
| 70 | 0.0089 | 43.7 | 0.0101 | 49.5 |
| 80 | 0.0069 | 44.4 | 0.0080 | 51.2 |
| 90 | 0.0055 | 44.8 | 0.0065 | 52.5 |
| 100 | 0.0045 | 45.0 | 0.0054 | 54.0 |
| 150 | 0.0020 | 45.0 | 0.0026 | 58.3 |
| 200 | 0.0011 | 45.0 | 0.0016 | 64.0 |
| 300 | 0.00050 | 45.0 | 0.00075 | 67.5 |
| 400 | 0.00028 | 45.0 | 0.00045 | 72.0 |
| 500 | 0.00018 | 45.0 | 0.00030 | 75.0 |
| 1000 | 0.00005 | 45.0 | 0.00008 | 80.0 |
+------+--------------------+-------------------+

In the middle part of a rod which has a length of 400 or 500 diameters
the effect of the ends is insensible; but for many experiments the
condition of endlessness may be best secured by giving the metal the
shape of a ring of uniform section, the magnetic field being produced
by an electric current through a coil of wire evenly wound round the
ring. In such cases H_i = 0 and H = H0.

The residual magnetization I_r retained by a bar of ferromagnetic
metal after it has been removed from the influence of an external
field produces a demagnetizing force [N]I_r, which is greater the
smaller the dimensional ratio. Hence the difficulty of imparting any
considerable permanent magnetization to a short thick bar not
possessed of great coercive force. The magnetization retained by a
long thin rod, even when its coercive force is small, is sometimes
little less than that which was produced by the direct action of the
field.

_Demagnetization by Reversals._--In the course of an experiment it is
often desired to eliminate the effects of previous magnetization, and,
as far as possible, wipe out the magnetic history of a specimen. In
order to attain this result it was formerly the practice to raise the
metal to a bright red heat, and allow it to cool while carefully
guarded from magnetic influence. This operation, besides being very
troublesome, was open to the objection that it was almost sure to
produce a material but uncertain change in the physical constitution
of the metal, so that, in fact, the results of experiments made before
and after the treatment were not comparable. Ewing introduced the
method (_Phil. Trans._ clxxvi. 539) of demagnetizing a specimen by
subjecting it to a succession of magnetic forces which alternated in
direction and gradually diminished in strength from a high value to
zero. By means of a simple arrangement, which will be described
farther on, this process can be carried out in a few seconds, and the
metal can be brought as often as desired to a definite condition,
which, if not quite identical with the virgin state, at least closely
approximates to it.

_Forces acting on a Small Body in the Magnetic Field._--If a small
magnet of length ds and pole-strength m is brought into a magnetic
field such that the values of the magnetic potential at the negative
and positive poles respectively are V1 and V2, the work done upon the
magnet, and therefore its potential energy, will be

W = m(V2 - V1) = m dV,

which may be written

dV dV
W = m ds -- = M -- = -MH0 = -vIH0,
ds ds

where M is the moment of the magnet, v the volume, I the
magnetization, and H0 the magnetic force along _ds_. The small magnet
may be a sphere rigidly magnetized in the direction of H0; if this is
replaced by an isotropic sphere inductively magnetized by the field,
then, for a displacement so small that the magnetization of the sphere
may be regarded as unchanged, we shall have

[kappa]
dW = -vI dH0 = -v -------------------- H0 dH0;
1 + (4/3)[pi][kappa]

whence

v [kappa]
W = - --- -------------------- H^2_0. (37)
2 1 + (4/3)[pi][kappa]

The mechanical force acting on the sphere in the direction of
displacement x is

dW [kappa] dH^2_0
F = - -- = v -------------------- ------. (38)
dx 1 + (4/3)[pi][kappa] dx

If H0 is constant, the force will be zero; if H0 is variable, the
sphere will tend to move in the direction in which H0 varies most
rapidly. The coefficient [kappa]/(1 + (4/3)[pi][kappa]) is positive
for ferromagnetic and paramagnetic substances, which will therefore
tend to move from weaker to stronger parts of the field; for all known
diamagnetic substances it is negative, and these will tend to move
from stronger to weaker parts. For small bodies other than spheres the
coefficient will be different, but its sign will always be negative
for diamagnetic substances and positive for others;[9] hence the
forces acting on any small body will be in the same directions as in
the case of a sphere.[10]

_Directing Couple acting on an Elongated Body._--In a non-uniform
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

H0
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,

H0
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)