James Clerk Maxwell.

# The scientific papers of James Clerk Maxwell (Volume 1) online

. (page 43 of 50)
Online LibraryJames Clerk MaxwellThe scientific papers of James Clerk Maxwell (Volume 1) → online text (page 43 of 50)
Font size whose strength is unity and length unity placed perpendicularly to the plane
of the ring, is a unit current; and F units, of electricity, measured statically,

ON PHYSICAL LINES OF FORCE. 499

traverse the section of this current in one second, — these units being such that
any two of them, placed at unit of distance, repel each other with unit of force.

We may suppose either that E units of positive electricity move in the
positive direction through the wire, or that E units of negative electricity move
in the negative direction, or, tliirdly, that \E units of positive electricity move
in the positive direction, while ^E units of negative electricity move in the
negative direction at the same time.

The last is the supposition on which MM. Weber and Kohlrausch* proceed,
who have found

^^=155,370,000,000 (130),

the unit of length being the millimetre, and that of time being one second,
whence

j5' = 310,740,000,000 (131).

Prop. XVI. — To find the rate of propagation of transverse vibrations
through the elastic medium of which the cells are composed, on the suppo-
sition that its elasticity is due entirely to forces acting between pairs of particles.

By the ordinary method of investigation we know that

y=J^, (132),

where m is the coefiScient of transverse elasticity, and p is the density. By
referring to the equations of Part I., it will be seen that if /) is the density
of the matter of the vortices, and /x is the " coefficient of magnetic induction,"

l^ = TTp (133);

whence 7rm= F'/i (134);

and by (108), E= V\fjx (135).

In air or vacuum /x = 1, and therefore

V=E 1

= 310,740,000,000 millimetres per second | (136).

= 193,088 miles per second J

* Abhandlungen der K&nig. Sdchnachen Geaellschaftf Vol. iii. (1857), p. 260.

63—2

500 ON PHYSICAL LINES OF FORCE.

The velocity of light in air, as determined by M. Fizeau*, ia 70,843 leagues
per second (25 leagues to a degree) which gives

7=314,858,000,000 millimetres
= 195,647 miles per second (137).

The velocity of transverse undulations in our hypothetical medium, calculated
from the electro-magnetic experiments of MM. Kohlrausch and Weber, agrees so
exactly with the velocity of light calculated from the optical experiments of
M. Fizeau, that we can scarcely avoid the inference that light consists in the
transverse undulations of the same medium which is the cause of electric and
nmgnetic phenomena.

Prop. XVII. — To find the electric capacity of a Leyden jar composed of
any given dielectric placed between two conducting surfaces.

Let the electric tensions or potentials of the two surfaces be "^^ and Sl'.j.
Let S be the area of each surface, and 6 the distance between them, and let
e and - e be the quantities of electricity on each surface ; then the capacity

^=^ <i'^«)-

Within the dielectric we have the variation of "^ perpendicular to the surface

Beyond either surface this variation is zero.

Hence by (115) applied at the surface, the electricity on unit of area is

OT ('''^■'

and we deduce the whole capacity of the apparatus,

^-l&e (^^°)'

so that the quantity of electricity required to bring the one surface to a

* Comptes Rmdus, Vol. xxix. (1849), p. 90. In Galbraitu and Haughton's Manual of Astronomy,
M. Fizeau's result is stated at 169,944 geographical miles of 1000 fathoms, which gives 193,118
statute miles; the value deduced from aberration is 192,000 miles.

ON PHYSICAL LINES OF FORCE. 501

given tension varies directly as the surface, inversely as the thickness, and
inversely as the square of E.

Now the coefl&cient of induction of dielectrics is deduced from the capacity
of induction-apparatus formed of them ; so that if D is that coefficient, D varies
inversely as E", and is unity for air. Hence

^=Yr^ <"•)•

where V and V^ are the velocities of light in air and in the medium. Now

V
if i is the. index of refraction, -jir = i, and

Z> = - (142);

so that the inductive power of a dielectric varies directly as the square of the
index of refraction, and inversely as the magnetic inductive power.

In dense media, however, the optical, electric, and magnetic phenomena
may be modified in different degrees by the particles of gross matter ; and their
mode of arrangement may influence these phenomena differently in different
directions. The axes of optical, electric, and magnetic properties will probably
coincide ; but on account of the unknown and probably complicated nature of
the reactions of the heavy particles on the setherial medium, it may be im-
possible to discover any general numerical relations between the optical, electric,
and magnetic ratios of these axes.

It seems probable, however, that the value of E, for any given axis,
depends upon the velocity of light whose vibrations are parallel to that axis,
or whose plane of polarization is perpendicular to that axis.

In a uniaxal crystal, the axial value of E will depend on the velocity of
the extraordinary ray, and the equatorial value will depend on that of the
ordinary ray.

In "positive" crystals, the axial value of E will be the least and in
negative the greatest.

The value of D„ which varies inversely as E\ will, cwteris panbus, be greatest
for the axial direction in positive crystals, and for the equatorial direction in
negative crystals, such as Iceland spar. If a spherical portion of a crystal,
radius =a, be suspended in a field of electric force which would act on unit of

502 ON PHYSICAL LINES OF FORCE.

electricity with force =1, and if A and D, be the coefficients of dielectric
induction along the two axes in the plane of rotation, then if 6 be the incli-
nation of the axis to the electric force, the moment tending to turn the sphere
will be

3 ^J^L;L?^_^^^7Vsin2^ (143),

^ (2A+1)(2A + 1)

and the axis of greatest dielectric induction (D^ will tend to become parallel to

the lines of electric force.

PART IV.

THE THEORY OF MOLECULAR VORTICES APPLIED TO THE ACTION OF MAGNETISM

ON POLARIZED LIGHT.

The connexion between the distribution of lines of magnetic force and that
of electric currents may be completely expressed by saying that the work done
on a unit of imaginary magnetic matter, when carried round any closed curve,
is proportional to the quantity of electricity which passes through the closed
curve. The mathematical form of this law may be expressed as in equations (9)*,
wliich I here repeat, where a, /8, y are the rectangular components of magnetic
intensity, and p, q, r are the rectangular components of steady electric cuiTents,

^ 47r \dy dz

1 /da
^ = ii[d-z-

-1)

(9).

The same mathematical connexion is found between other sets of phenomena
in physical science.

(1) If a, /S, y represent displacements, velocities, or forces, then p, q, r
will be rotatory displacements, velocities of rotation, or moments of couples pro-
ducing rotation, in the elementary portions of the mass.

* Phil Mag. March, 1861 [p. 462 of this vol].

ON PHYSICAL LINES OF FORCE. 503

(2) If a, /8, y represent rotatory displacements in a uniform and continuous
substance, then p, q, r represent the relative linear displacement of a particle
with respect to those in its immediate neighbourhood. See a paper by Prof. W.
Thomson "On a Mechanical Representation of Electric, Magnetic, and Galvanic
Forces," Camh. and Dublin Math. Journal, Jan. 1847.

(3) If a, j8, y represent the rotatory velocities of vortices whose centres
are fixed, then p, q, r represent the velocities with which loose particles placed
between them would be carried along. See the second part of this paper (Phil.
Mag. April, 1861) [p. 469].

It appears from all these instances that the connexion between magnetism
and electricity has the same mathematical form as that between certain
pairs of phenomena, of which one has a linear and the other a rotatory
character. Professor Challis* conceives magnetism to consist in currents of a
fluid whose direction corresponds with that of the lines of magnetic force ; and
electric currents, on this theory, are accompanied by, if not dependent on, a
rotatory motion of the fluid about the axis of the current. Professor Helmholtzf
has investigated the motion of an incompressible fluid, and has conceived lines
drawn so as to correspond at every point with the instantaneous axis of
rotation of the fluid there. He has pointed out that the lines of fluid motion
are arranged according to the same laws with respect to the lines of rotation,
as those by which the lines of magnetic force are arranged with respect to
electric currents. On the other hand, in this paper I have regarded magnetism
as a phenomenon of rotation, and electric currents as consisting of the actual
translation of particles, thus assuming the inverse of the relation between the
two sets of phenomena.

Now it seems natural to suppose that all the direct efiects of any cause
which is itself of a longitudinal character, must be themselves longitudinal, and
that the dii^ect eflects of a rotatory cause must be themselves rotatory\ A
motion of translation along an axis cannot produce a rotation about that axis
unless it meets with some special mechanism, like that of a screw, which
connects a motion in a given direction along the axis with a rotation in a given
direction round it; and a motion of rotation, though it may produce tension
along the axis, cannot of itself produce a current in one direction along the axis
rather than the other.

* Phil. Mag. December, 1860, January and February, 18G1.
t Crelle, Journal, Vol. LV. (1858), p. 25.

504 ON PHYSICAL LINES OF FORCE.

Electric currents are known to produce effects of transference in tlie direc-
tion of the current. They transfer the electrical state from one body to another,
and they transfer the elements of electrolytes in opposite directions, but they
do not* cause the plane of polarization of light to rotate when the light tra-
verses the axis of the current.

On the other hand, the magnetic state is not characterized by any strictly
longitudinal phenomenon. The north and south poles differ only in their names,
and these names might be exchanged without altering the statement of any
magnetic phenomenon ; whereas the positive and negative poles of a battery are
completely distinguished by the different elements of water which are evolved
there. The magnetic state, however, is characterized by a well-marked rotatory
phenomenon discovered by Faraday f — the rotation of the plane of polarized light
when transmitted along the lines of magnetic force.

When a transparent diamagnetic substance has a ray of plane-polarized light
passed through it, and if lines of magnetic force are then produced in the
substance by the action of a magnet or of an electric current, the plane of
polarization of the transmitted light is found to be changed, and to be turned
through an angle depending on the intensity of the magnetizing force within
the substance.

The direction of this rotation in diamagnetic substances is the same as that
in which positive electricity must circulate round the substance in order to
produce the actual magnetizing force within it; or if we suppose the horizontal
part of terrestrial magnetism to be the magnetizing force acting on the sub-
stance, the plane of polarization would be turned in the direction of the earth's
true rotation, that is, from west upwards to east.

In paramagnetic substances, M. VerdetJ has found that the plane of polari-
zation is turned in the opposite direction, that is, in the direction in which
negative electricity would flow if the magnetization were effected by a helix
surrounding the substance.

In both cases the absolute direction of the rotation is the same, whether
the light passes from north to south or from south to north, — a fact which dis-
tinguishes this phenomenon from the rotation produced by quartz, turpentine, &c.,

♦ Faraday, Experimental Eesearclies, 951—954, and 2216—2220.

t Ibid., Series xix.

X Comptes Rendus, Vol. XLiii. p. 529; Vol. XLiv, p. 1209.

ON PHYSICAL LINES OF FORCE. 505

in which the absolute direction of rotation is reversed when that of the light
is reversed. The rotation in the latter case, whether related to an axLs, as in
quartz, or not so related, as in fluids, indicates a relation between the direction
of the ray and the direction of rotation, which is similar in its formal expression
to that between the longitudinal and rotatory motions of a right-handed or a
left-handed screw; and it indicates some property of the substance the mathe-
matical form of which exhibits right-handed or left-handed relations, such as are
known to appear in the external forms of crystals having these properties. In
the magnetic rotation no such relation appears, but the direction of rotation is
directly connected with that of the magnetic lines, in a way which seems to
indicate that magnetism is really a phenomenon of rotation.

The transference of electrolytes in fixed directions by the electric current,
and the rotation of polarized light in fixed directions by magnetic force, are
the facts the consideration of which has induced me to regard magnetism as a
phenomenon of rotation, and electric currents as phenomena of translation, instead
of following out the analogy pointed out by Helmholtz, or adopting the theory
propounded by Professor Challis.

The theory that electric currents are linear, and magnetic forces rotator}'
phenomena, agrees so far with that of Ampere and Weber ; and the hypothesis
that the magnetic rotations exist wherever magnetic force extends, that the
centrifugal force of these rotations accounts for magnetic attractions, and that
the inertia of the vortices accounts for induced currents, is supported by the
opinion of Professor W. Thomson*. In fact the whole theory of molecular vor-
tices developed in this paper has been suggested to me by observing the
direction in which those investigators who study the action of media are looking
for the explanation of electro-magnetic phenomena.

Professor Thomson has pointed out that the cause of the magnetic action
on light must be a real rotation going on in the magnetic field. A right-handed
circularly polarized ray of light is found to travel with a different velocity
according as it passes from north to south, or from south to north, along a
line of magnetic force. Now, whatever theory we adopt about the direction of
vibrations in plane-polarized light, the geometrical arrangement of the parts of
the medium during the passage of a right-handed circularly polarized ray is
exactly the same whether the ray is moving north or south. The only difference

* See Nichol's Cyclopcedia, art "Magnetism, Dynamical Relations of," edition 1860; Proceedings
of Royal Society, June 1856 and June 1861 ; and Phil. Mag. 1857.

VOL. I. 64

506 ON PHYSICAL LINES OF FORCE,

is, that the particles describe their circles in opposite directions. Since, therefore,
the configuration is the same in the two cases, the forces acting between par-
ticles must be the same in both, and the motions due to these forces must be
equal in velocity if the medium was originally at rest; but if the medium be
in a state of rotation, either as a whole or in molecular vortices, the circular
vibrations of light may differ in velocity according as their direction is similar
or contrary to that of the vortices.

We have now to investigate whether the hjrpothesis developed in this
paper — that magnetic force is due to the centrifugal force of small vortices, and
that these vortices consist of the same matter the vibrations of which constitute
liglit — leads to any conclusions as to the effect of magnetism on polarized light.
We suppose transverse vibrations to be transmitted through a magnetized
medium. How will the propagation of these vibrations be affected by the
circumstance that portions of that medium are in a state of rotation ?

In the following investigation, I have found that the only effect which the
rotation of the vortices will have on the light will be to make the plane of
polarization rotate in the same direction as the vortices, through an angle
proportional —

{A) to the thickness of the substance,

(B) to the resolved part of the magnetic force parallel to the ray,

(C) to the index of refraction of the ray,

{D) inversely to the square of the wave-length in air,
[E) to the mean radius of the vortices,
{F) to the capacity for magnetic induction.

A and B have been fully investigated by M. Verdet'"*, who has shewn that
the rotation is strictly proportional to the thickness and to the magnetizing
force, and that, when the ray is inclined to the magnetizmg force, the rotation
is as the cosine of that inclination. D has been supposed to give the true
relation between the rotation of different rays; but it is probable that C must
be taken into account in an accurate statement of the phenomena. The rotation
varies, not exactly inversely as the square of the wave length, but a little faster;
so that for the highly refrangible rays the rotation is greater than that given
by this law, but more nearly as the index of refraction divided by the square
of the wave-lfength.

♦ Annates de Chimie et de Physique, s6r. 3, Vol. XLi. p. 370; Vol. xliii. p. 37.

ON PHYSICAL LINES OF FORCE. 507

The relation (E) between the amount of rotation and the size of the
vortices shews that different substances may differ in rotating power inde-
pendently of any observable difference in other respects. We know nothing
of the absolute size of the vortices ; and on our hypothesis the optical phenomena
are probably the only data for determining their relative size in different sub-
stances.

On our theory, the direction of the rotation of the plane of polarization
depends on that of the mean moment of momenta, or angular momentum, of the
molecular vortices ; and since M. Verdet has discovered that magnetic substances
have an effect on light opposite to that of diamagnetic substances, it follows that
the molecular rotation must be opposite in the two classes of substances.

We can no longer, therefore, consider diamagnetic bodies as being those
whose coeflBcient of magnetic indu-ction is less than that of space empty of
gross matter. We must admit the diamagnetic state to be the opposite of the
paramagnetic ; and that the vortices, or at least the influential majority of them,
in diamagnetic substances, revolve in the direction in which positive electricity
revolves in the magnetizing bobbin, while in paramagnetic substances they
revolve in the opposite direction.

This result agrees so far with that part of the theory of M. Weber*
which refers to the paramagnetic and diamagnetic conditions. M. Weber sup-
poses the electricity in paramagnetic bodies to revolve the same way as the
surrounding helix, while in diamagnetic bodies it revolves the opposite way.
Now if we regard negative or resinous electricity as a substance the absence
of which constitutes positive or vitreous electricity, the results will be those
actually observed. This will be true independently of any other hypothesis
than that of M. Weber about magnetism and diamagnetism, and does not
require us to admit either M. Weber's theory of the mutual action of electric
particles in motion, or our theory of cells and cell-walls.

I am inclined to believe that iron differs from other substances in the
manner of its action as well as in the intensity of its magnetism; and I think
its behaviour may be explained on our hypothesis of molecular vortices, by
supposing that the particles of the iron itself are set in rotation by the tan-
gential action of the vortices, in an opposite direction to their own. These
large heavy particles would thus be revolving exactly as we have supposed the

* Taylor's Scientific Memoirs, Vol. v. p. 477.

64—2

508 ON PHYSICAL LINES OF FORCE.

infinitely small particles constituting electricity to revolve, but without being
free like them to change their place and form currents.

The whole energy of rotation of the magnetized field would thus be greatly
increased, as we know it to be ; but the angular momentum of the iron
particles would be opposite to that of the aethereal cells and immensely greater,
so that the total angular momentum of the substance will be in the direction
of rotation of the iron, or the reverse of that of the vortices. Since, however,
the angular momentum depends on the absolute size of the revolving portions
of the substance, it may depend on the state of aggregation or chemical
arrangement of the elements, as well as on the ultimate nature of the com-
ponents of the substance. Other phenomena in nature seem to lead to the
conclusion that all substances are made up of a number of parts, finite in size,
the particles composing these parts being themselves capable of internal motion.

Prop. XVIII. — To find the angular momentum of a vortex.

The angular momentum of any material system about an axis is the sum
of the products of the mass, dm, of each particle multipHed by twice the area
it describes about that axis in unit of time ; or if ^ is the angular momentum

As we do not know the distribution of density within the vortex, we shall
determine the relation between the angular momentum and the energy of the
vortex which was found in Prop. VI.

Since the time of revolution is the same throughout the vortex, the mean

angular velocity o> will be uniform and =-, where a is the velocity at the

circumference, and )• the radius. Then

A = ^dmroi,
and the energy E = \^dmr^(a^ = ^Ao>,

= -!^/xa-Fby Prop. VI.*

OTT

whence A = -—[xraV (144)

* Phil. Mag. April 1861 [p. 472 of this vol.].

ON PHYSICAL LINES OF FORCE. 50^>

for the axis of x, with similar expressions for the other axes, V bein^r tlie
volume, and r the radius of the vortex.

Prop. XIX. — To determine the conditions of undulatory motion in a medium
containing vortices, the vibrations being perpendicular to the direction of pro-
pagation.

Let the waves be plane-waves propagated in the direction of 2, and let
the axis of x and y be taken in the directions of greatest and least elasticity
in the plane xy. Let x and y represent the displacement parallel to these axes,
which will be the same throughout the same wave-surface, and therefore we
shall have x and y functions of z and t only.

Let X be the tangential stress on unit of area parallel to xy, tending to
move the part next the origin in the direction of x.

Let Y be the corresponding tangential stress in the
direction of y.

Let ^1 and k^ be the coefficients of elasticity with respect
to these two kinds of tangential stress ; then, if the medium
is at rest,

Now let us suppose vortices in the medium whose velocities are represented
as usual by the symbols a, ^, y, and let us suppose that the value of a is

increasing at the rate -j- , on account of the action of the tangential stresses

alone, there being no electromotive force in the field. The angular momentum
in the stratum whose area is unity, and thickness dz, is therefore increasing

at the rate — fir -j- dz; and if the part of the force Y which produces this effect

is Y', then the moment of Y' is - Y'dz, so that Y' = - — fir ^- .

' Air'^ at

The complete value of Y when the vortices are in a state of varied motion

(145).

dy 1 da
■dz~4n'^'^ dt

o- 1 1 ^ J dx 1 d^

Similarly. Z=*,^ + j^M*"^ ,

510

ON PHYSICAL LINES OF FORCE.

The whole force acting upon a stratum whose thickness is dz and area
unity, is ,— dz in the direction of x, and -p dz in direction of y. The mass
of the stratum is pdz, so that we have as the equations of motion,

d'x dX J d'x , d I d^
'di

P df-~~d^~^'dz^'^dzi^'^'^

d'y _dY _i d-y d 1

df dz

dz'

da.
dz Att'^ di

(146).

Now the changes of velocity -j- and -^ are produced by the motion of

the medium contaiimig the vortices, which distorts and twists every element
of its mass; so that we must refer to Prop. X.* to determine these quantities
in terms of the motion. We find there at equation (68),

da = a-T- Sx-\-fi -J- Bx + y — Bx

(68).

dx dy '' dz

Since Bx and By are functions of z and t only, we may write this equation

and in like manner,

so that if we now put
equations of motion

da _ d"x
dt ~^ dzdt

d^_ d?y_
dt ^ dzdt

.(147),

I /xr

a'/), h^ = Tfp, and — — y = c', we may write the

d'x .d'x d^y
^"^ dz'^"" dz^dt

df

df~ dz:' dz'dtl
These equations may be satisfied by the values

(148).

provided
and

X = A cos (nt — mz + a)\ h49)

Online LibraryJames Clerk MaxwellThe scientific papers of James Clerk Maxwell (Volume 1) → online text (page 43 of 50)