# CHAPTER XXI.

## MAGNETIC ACTION ON LIGHT.

806.] The most important step in establishing a relation between electric and magnetic phenomena and those of light must be the discovery of some instance in which the one set of phenomena is affected by the other. In the search for such phenomena we must be guided by any knowledge we may have already obtained with respect to the mathematical or geometrical form of the quantities which we wish to compare. Thus, if we endeavour, as Mrs. Somerville did, to magnetize a needle by means of light, we must remember that the distinction between magnetic north and south is a mere matter of direction, and would be at once reversed if we reverse certain conventions about the use of mathematical signs. There is nothing in magnetism analogous to those phenomena of electrolysis which enable us to distinguish positive from negative electricity, by observing that oxygen appears at one pole of a cell and hydrogen at the other.

Hence we must not expect that if we make light fall on one end of a needle, that end will become a pole of a certain name, for the two poles do not differ as light does from darkness.

We might expect a better result if we caused circularly polarized light to fall on the needle, right-handed light failing on one end and left-handed on the other, for in some respects these kinds of light may be said to be related to each other in the same way as the poles of a magnet. The analogy, however, is faulty even here, for the two rays when combined do not neutralize each other, but produce a plane polarized ray,

Faraday, who was acquainted with the method of studying the strains produced in transparent solids by means of polarized light, made many experiments in hopes of detecting some action on polarized light while passing through a medium in which electrolytic conduction or dielectric induction exists[1]. He was not, however, able to detect any action of this kind, though the experiments were arranged in the way best adapted to discover effects of tension, the electric force or current being at right angles to the direction of the ray, and at an angle of forty-five degrees to the plane of polarization. Faraday varied these experiments in may ways without discovering any action on light due to electrolytic currents or to static electric induction.

He succeeded, however, in establishing a relation between light and magnetism, and the experiments by which he did so are described in the nineteenth series of his Experimental Researches. We shall take Faraday's discovery as our starting point for further investigation into the nature of magnetism, and we shall therefore describe the phenomenon which he observed.

807.] A ray of plane-polarized light is transmitted through a transparent diamagnetic medium, and the plane of its polarization, when it emerges from the medium, is ascertained by observing the position of an analyser when it cuts off the ray. A magnetic force is then made to act so that the direction of the force within the transparent medium coincides with the direction of the ray. The light at once reappears, but if the analyser is turned round through a certain angle, the light is again cut off. This shews that the effect of the magnetic force is to turn the plane of polarization, round the direction of the ray as an axis, through a certain angle, measured by the angle through which the analyser must be turned in order to cut off the light.

808.] The angle through which the plane of polarization is turned is proportional—

(1) To the distance which the ray travels within the medium. Hence the plane of polarization changes continuously from its position at incidence to its position at emergence.

(2) To the intensity of the resolved part of the magnetic force in the direction of the ray.

(3) The amount of the rotation depends on the nature of the medium. No rotation has yet been observed when the medium is air or any other gas.

These three statements are included in the more general one, that the angular rotation is numerically equal to the amount by which the magnetic potential increases, from the point at which the ray enters the medium to that at which it leaves it, multiplied by a coefficient, which, for diamagnetic media, is generally positive.

809.] In diamagnetic substances, the direction in which the plane of polarization is made to rotate is the same as the direction in which a positive current must circulate round the ray in order to produce a magnetic force in the same direction as that which actually exists in the medium.

Verdet, however, discovered that in certain ferromagnetic media, as, for instance, a strong solution of perchloride of iron in wood-spirit or ether, the rotation is in the opposite direction to the current which would produce the magnetic force.

This shews that the difference between ferromagnetic and diamagnetic substances does not arise merely from the 'magnetic permeability' being in the first case greater, and in the second less, than that of air, but that the properties of the two classes of bodies are really opposite.

The power acquired by a substance under the action of magnetic force of rotating the plane of polarization of light is not exactly proportional to its diamagnetic or ferromagnetic magnetizability. Indeed there are exceptions to the rule that the rotation is positive for diamagnetic and negative for ferromagnetic substances, for neutral chromate of potash is diamagnetic, but produces a negative rotation.

810.] There are other substances, which, independently of the application of magnetic force, cause the plane of polarization to turn to the right or to the left, as the ray travels through the substance. In some of these the property is related to an axis, as in the case of quartz. In others, the property is independent of the direction of the ray within the medium, as in turpentine, solution of sugar, &c. In all these substances, however, if the plane of polarization of any ray is twisted within the medium like a right-handed screw, it will still be twisted like a right-handed screw if the ray is transmitted through the medium in the opposite direction. The direction in which the observer has to turn his analyser in order to extinguish the ray after introducing the medium into its path, is the same with reference to the observer whether the ray comes to him from the north or from the south. The direction of the rotation in space is of course reversed when the direction of the ray is reversed. But when the rotation is produced by magnetic action, its direction in space is the same whether the ray be travelling north or south. The rotation is always in the same direction as that of the electric current which produces, or would produce, the actual magnetic state of the field, if the medium belongs to the positive class, or in the opposite direction if the medium belongs to the negative class.

It follows from this, that if the ray of light, after passing through the medium from north to south, is reflected by a mirror, so as to return through the medium from south to north, the rotation will be doubled when it results from magnetic action. When the rotation depends on the nature of the medium alone, as in turpentine, &c., the ray, when reflected back through the medium, emerges in the same plane as it entered, the rotation during the first passage through the medium having been exactly reversed during the second.

811.] The physical explanation of the phenomenon presents considerable difficulties, which can hardly be said to have been hitherto overcome, either for the magnetic rotation, or for that which certain media exhibit of themselves. We may, however, prepare the way for such an explanation by an analysis of the observed facts.

It is a well-known theorem in kinematics that two uniform circular vibrations, of the same amplitude, having the same periodic time, and in the same plane, but revolving in opposite directions, are equivalent, when compounded together, to a rectilinear vibration. The periodic time of this vibration is equal to that of the circular vibrations, its amplitude is double, and its direction is in the line joining the points at which two particles, describing the circular vibrations in opposite directions round the same circle, would meet. Hence if one of the circular vibrations has its phase accelerated, the direction of the rectilinear vibration will be turned, in the same direction as that of the circular vibration, through an angle equal to half the acceleration of phase.

It can also be proved by direct optical experiment that two rays of light, circularly-polarized in opposite directions, and of the same intensity, become, when united, a plane-polarized ray, and that if by any means the phase of one of the circularly-polarized rays is accelerated, the plane of polarization of the resultant ray is turned round half the angle of acceleration of the phase.

812.] We may therefore express the phenomenon of the rotation of the plane of polarization in the following manner:—A plane-polarized ray falls on the medium. This is equivalent to two circularly-polarized rays, one right-handed, the other left-handed (as regards the observer) . After passing through the medium the ray is still plane-polarized, but the plane of polarization is turned, say, to the right (as regards the observer). Hence, of the two circularly-polarized rays, that which is right-handed must have had its phase accelerated with respect to the other during its passage through the medium.

In other words, the right-handed ray has performed a greater number of vibrations, and therefore has a smaller wave-length, within the medium, than the left-handed ray which has the same periodic time.

This mode of stating what takes place is quite independent of any theory of light, for though we use such terms as wave-length, circular-polarization, &c., which may be associated in our minds with a particular form of the undulatory theory, the reasoning is independent of this association, and depends only on facts proved by experiment.

813.] Let us next consider the configuration of one of these rays at a given instant. Any undulation, the motion of which at each point is circular, may be represented by a helix or screw. If the screw is made to revolve about its axis without any longitudinal motion, each particle will describe a circle, and at the same time the propagation of the undulation will be represented by the apparent longitudinal motion of the similarly situated parts of the thread of the screw. It is easy to see that if the screw is right-handed, and the observer is placed at that end towards which the undulation travels, the motion of the screw will appear to him left-handed, that is to say, in the opposite direction to that of the hands of a watch.
Fig. 67.
Hence such a ray has been called, originally by French writers, but now by the whole scientific world, a left-handed circularly-polarized ray.

A right-handed circularly-polarized ray is represented in like manner by a left-handed helix. In Fig. 67 the right-handed helix ${\displaystyle A}$, on the right-hand of the figure, represents a left-handed ray, and the left-handed helix ${\displaystyle B}$, on the left-hand, represents a right-handed ray.

814.] Let us now consider two such rays which have the same wave-length within the medium. They are geometrically alike in all respects, except that one is the perversion of the other, like its image in a looking-glass. One of them, however, say ${\displaystyle A}$, has a shorter period of rotation than the other. If the motion is entirely due to the forces called into play by the displacement, this shews that greater forces are called into play by the same displacement when the configuration is like A than when it is like ${\displaystyle B}$. Hence in this case the left-handed ray will be accelerated with respect to the right-handed ray, and this will be the case whether the rays are travelling from ${\displaystyle N}$ to ${\displaystyle S}$ or from ${\displaystyle S}$ to ${\displaystyle N}$.

This therefore is the explanation of the phenomenon as it is produced by turpentine, &c. In these media the displacement caused by a circularly-polarized ray calls into play greater forces of restitution when the configuration is like ${\displaystyle A}$ than when it is like ${\displaystyle B}$. The forces thus depend on the configuration alone, not on the direction of the motion.

But in a diamagnetic medium acted on by magnetism in the direction ${\displaystyle SN}$, of the two screws ${\displaystyle A}$ and ${\displaystyle B}$, that one always rotates with the greatest velocity whose motion, as seen by an eye looking from ${\displaystyle S}$ to ${\displaystyle N}$, appears like that of a watch. Hence for rays from ${\displaystyle S}$ to ${\displaystyle N}$ the right-handed ray ${\displaystyle B}$ will travel quickest, but for rays from ${\displaystyle N}$ to ${\displaystyle S}$ the left-handed ray ${\displaystyle A}$ will travel quickest.

815.] Confining our attention to one ray only, the helix ${\displaystyle B}$ has exactly the same configuration, whether it represents a ray from ${\displaystyle S}$ to ${\displaystyle N}$ or one from ${\displaystyle N}$ to ${\displaystyle S}$. But in the first instance the ray travels faster, and therefore the helix rotates more rapidly. Hence greater forces are called into play when the helix is going round one way than when it is going round the other way. The forces, therefore, do not depend solely on the configuration of the ray, but also on the direction of the motion of its individual parts.

816.] The disturbance which constitutes light, whatever its physical nature may be, is of the nature of a vector, perpendicular to the direction of the ray. This is proved from the fact of the interference of two rays of light, which under certain conditions produces darkness, combined with the fact of the non-interference of two rays polarized in planes perpendicular to each other. For since the interference depends on the angular position of the planes of polarization, the disturbance must be a directed quantity or vector, and since the interference ceases when the planes of polarization are at right angles, the vector representing the disturbance must be perpendicular to the line of intersection of these planes, that is, to the direction of the ray.

817.] The disturbance, being a vector, can be resolved into components parallel to ${\displaystyle x}$ and ${\displaystyle y}$, the axis of ${\displaystyle z}$ being parallel to the direction of the ray. Let ${\displaystyle \xi }$ and ${\displaystyle \eta }$ be these components, then, in the case of a ray of homogeneous circularly-polarized light,
 ${\displaystyle \xi =r\cos \theta }$,⁠${\displaystyle \eta =r\sin \theta }$, (1)
 where ${\displaystyle \theta =nt-qz+\alpha }$. (2)

In these expressions, ${\displaystyle r}$ denotes the magnitude of the vector, and ${\displaystyle \theta }$ the angle which it makes with the direction of the axis of ${\displaystyle x}$.

The periodic time, ${\displaystyle \tau }$, of the disturbance is such that
 ${\displaystyle n\tau =2\pi }$. (3)

The wave-length, ${\displaystyle \lambda }$, of the disturbance is such that
 ${\displaystyle q\lambda =2\pi }$. (4)

The velocity of propagation is ${\displaystyle {\frac {n}{q}}}$.

The phase of the disturbance when ${\displaystyle t}$ and ${\displaystyle z}$ are both zero is ${\displaystyle \alpha }$.

The circularly-polarized light is right-handed or left-handed according as ${\displaystyle q}$ is negative or positive.

Its vibrations are in the positive or the negative direction of rotation in the plane of (${\displaystyle x}$, ${\displaystyle y}$), according as ${\displaystyle n}$ is positive or negative.

The light is propagated in the positive or the negative direction of the axis of ${\displaystyle z}$, according as ${\displaystyle n}$ and ${\displaystyle q}$ are of the same or of opposite signs.

In all media ${\displaystyle n}$ varies when ${\displaystyle q}$ varies, and ${\displaystyle {\frac {dn}{dq}}}$ is always of the same sign with ${\displaystyle {\frac {n}{q}}}$.

Hence, if for a given numerical value of ${\displaystyle n}$ the value of ${\displaystyle {\frac {n}{q}}}$ is greater when ${\displaystyle n}$ is positive than when ${\displaystyle n}$ is negative, it follows that for a value of ${\displaystyle q}$, given both in magnitude and sign, the positive value of ${\displaystyle n}$ will be greater than the negative value.

Now this is what is observed in a diamagnetic medium, acted on by a magnetic force, ${\displaystyle \gamma }$, in the direction of ${\displaystyle z}$. Of the two circularly-polarized rays of a given period, that is accelerated of which the direction of rotation in the plane of (${\displaystyle x}$, ${\displaystyle y}$) is positive. Hence, of two circularly-polarized rays, both left-handed, whose wave-length within the medium is the same, that has the shortest period whose direction of rotation in the plane of ${\displaystyle xy}$ is positive, that is, the ray which is propagated in the positive direction of ${\displaystyle z}$ from south to north. We have therefore to account for the fact, that when in the equations of the system ${\displaystyle q}$ and ${\displaystyle r}$ are given, two values of ${\displaystyle n}$ will satisfy the equations, one positive and the other negative, the positive value being numerically greater than the negative.

818.] We may obtain the equations of motion from a consideration of the potential and kinetic energies of the medium. The potential energy, ${\displaystyle V}$, of the system depends on its configuration, that is, on the relative position of its parts. In so far as it depends on the disturbance due to circularly-polarized light, it must be a function of ${\displaystyle r}$, the amplitude, and ${\displaystyle q}$, the coefficient of torsion, only. It may be different for positive and negative values of ${\displaystyle q}$ of equal numerical value, and it probably is so in the case of media which of themselves rotate the plane of polarization.

The kinetic energy, ${\displaystyle T}$, of the system is a homogeneous function of the second degree of the velocities of the system, the coefficients of the different terms being functions of the coordinates.

819.] Let us consider the dynamical condition that the ray may be of constant intensity, that is, that ${\displaystyle r}$ may be constant.

Lagrange's equation for the force in ${\displaystyle r}$ becomes
 ${\displaystyle {\frac {d}{dt}}{\frac {dT}{d{\dot {r}}}}-{\frac {dT}{dr}}+{\frac {dV}{dr}}=0}$. (5)
Since ${\displaystyle r}$ is constant, the first term vanishes. We have therefore the equation
 ${\displaystyle -{\frac {dT}{dr}}+{\frac {dV}{dr}}=0}$, (6)

in which ${\displaystyle q}$ is supposed to be given, and we are to determine the value of the angular velocity ${\displaystyle {\dot {\theta }}}$, which we may denote by its actual value, ${\displaystyle n}$.

The kinetic energy, ${\displaystyle T}$, contains one term involving ${\displaystyle n^{2}}$; other terms may contain products of ${\displaystyle n}$ with other velocities, and the rest of the terms are independent of ${\displaystyle n}$. The potential energy, ${\displaystyle V}$, is entirely independent of ${\displaystyle n}$. The equation is therefore of the form
 ${\displaystyle An^{2}+Bn+C=0}$. (7)
This being a quadratic equation, gives two values of ${\displaystyle n}$. It appears from experiment that both values are real, that one is positive and the other negative, and that the positive value is numerically the greater. Hence, if ${\displaystyle A}$ is positive, both ${\displaystyle B}$ and ${\displaystyle C}$ are negative, for, if ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ are the roots of the equation,
 ${\displaystyle A(n_{1}+n_{2})+B=0}$. (8)

The coefficient, ${\displaystyle B}$, therefore, is not zero, at least when magnetic force acts on the medium. We have therefore to consider the expression ${\displaystyle Bn}$, which is the part of the kinetic energy involving the first power of ${\displaystyle n}$, the angular velocity of the disturbance.

820.] Every term of ${\displaystyle T}$ is of two dimensions as regards velocity. Hence the terms involving ${\displaystyle n}$ must involve some other velocity. This velocity cannot be ${\displaystyle {\dot {r}}}$ or ${\displaystyle {\dot {q}}}$, because, in the case we consider, ${\displaystyle r}$ and ${\displaystyle q}$ are constant. Hence it is a velocity which exists in the medium independently of that motion which constitutes light. It must also be a velocity related to ${\displaystyle n}$ in such a way that when it is multiplied by ${\displaystyle n}$ the result is a scalar quantity, for only scalar quantities can occur as terms in the value of ${\displaystyle T}$, which is itself scalar. Hence this velocity must be in the same direction as ${\displaystyle n}$, or in the opposite direction, that is, it must be an angular velocity about the axis of ${\displaystyle z}$.\

Again, this velocity cannot be independent of the magnetic force, for if it were related to a direction fixed in the medium, the phenomenon would be different if we turned the medium end for end, which is not the case.

We are therefore led to the conclusion that this velocity is an invariable accompaniment of the magnetic force in those media which exhibit the magnetic rotation of the plane of polarization.

821.] We have been hitherto obliged to use language which is perhaps too suggestive of the ordinary hypothesis of motion in the undulatory theory. It is easy, however, to state our result in a form free from this hypothesis.

Whatever light is, at each point of space there is something going on, whether displacement, or rotation, or something not yet imagined, but which is certainly of the nature of a vector or directed quantity, the direction of which is normal to the direction of the ray. This is completely proved by the phenomena of interference.

In the case of circularly-polarized light, the magnitude of this vector remains always the same, but its direction rotates round the direction of the ray so as to complete a revolution in the periodic time of the wave. The uncertainty which exists as to whether this vector is in the plane of polarization or perpendicular to it, does not extend to our knowledge of the direction in which it rotates in right-handed and in left-handed circularly-polarized light respectively. The direction and the angular velocity of this vector are perfectly known, though the physical nature of the vector and its absolute direction at a given instant are uncertain.

When a ray of circularly-polarized light falls on a medium under the action of magnetic force, its propagation within the medium is affected by the relation of the direction of rotation of the light to the direction of the magnetic force. From this we conclude, by the reasoning of Art. 821, that in the medium, when under the action of magnetic force, some rotatory motion is going on, the axis of rotation being in the direction of the magnetic forces; and that the rate of propagation of circularly-polarized light, when the direction of its vibratory rotation and the direction of the magnetic rotation of the medium are the same, is different from the rate of propagation when these directions are opposite.

The only resemblance which we can trace between a medium through which circularly-polarized light is propagated, and a medium through which lines of magnetic force pass, is that in both there is a motion of rotation about an axis. But here the resemblance stops, for the rotation in the optical phenomenon is that of the vector which represents the disturbance. This vector is always perpendicular to the direction of the ray, and rotates about it a known number of times in a second. In the magnetic phenomenon, that which rotates has no properties by which its sides can be distinguished, so that we cannot determine how many times it rotates in a second.

There is nothing, therefore, in the magnetic phenomenon which corresponds to the wave-length and the wave-propagation in the optical phenomenon. A medium in which a constant magnetic force is acting is not, in consequence of that force, filled with waves travelling in one direction, as when light is propagated through it. The only resemblance between the optical and the magnetic phenomenon is, that at each point of the medium something exists of the nature of an angular velocity about an axis in the direction of the magnetic force.

On the Hypothesis of Molecular Vortices.

822.] The consideration of the action of magnetism on polarized light leads, as we have seen, to the conclusion that in a medium under the action of magnetic force something belonging to the same mathematical class as an angular velocity, whose axis is in the direction of the magnetic force, forms a part of the phenomenon.

This angular velocity cannot be that of any portion of the medium of sensible dimensions rotating as a whole. We must therefore conceive the rotation to be that of very small portions of the medium, each rotating on its own axis. This is the hypothesis of molecular vortices.

The motion of these vortices, though, as we have shewn (Art. 575), 824.] MOLECULAR VOKTICES. 409

it does not sensibly affect the visible motions of large bodies, may be such as to affect that vibratory motion on which the propagation of light, according to the undulatory theory, depends. The dis placements of the medium, during the propagation of light, will produce a disturbance of the vortices, and the vortices when so dis turbed may react on the medium so as to affect the mode of propa gation of the ray.

823.] It is impossible, in our present state of ignorance as to the nature of the vortices, to assign the form of the law which connects the displacement of the medium with the variation of the vortices. We shall therefore assume that the variation of the vortices caused by the displacement of the medium is subject to the same conditions which Helmholtz, in his great memoir on Vortex-motion"*, has shewn to regulate the variation of the vortices of a perfect liquid,

Helmholtz s law may be stated as follows : Let P and Q be two neighbouring particles in the axis of a vortex, then, if in conse quence of the motion of the fluid these particles arrive at the points P Q , the line P Q will represent the new direction of the axis of the vortex, and its strength will be altered in the ratio of P Q to PQ.

Hence if a, /3, y denote the components of the strength of a vor tex, and if f, 77, ( denote the displacements of the medium, the value of a will become

dx dy dz dn

(1)

We now assume that the same condition is satisfied during the small displacements of a medium in which a, (3, y represent, not the components of the strength of an ordinary vortex, but the components of magnetic force.

824.] The components of the angular velocity of an element of

the medium are &&gt;, = \&gt; ( C -H) .

��o&gt;., = % i~ ~\, cU \dz dx

• CreUe s Journal, vol. lv. (1858). Translated by Tait, Phil. May., July, 1867.

��(2)

�� � 410 MAGNETIC ACTION ON LIGHT. [825.

The next step in our hypothesis is the assumption that the kinetic energy of the medium contains a term of the form

2C (aa&gt; 1 + j8a&gt; 2 + yfi&gt; 3 ). (3)

This is equivalent to supposing that the angular velocity acquired by the element of the medium during the propagation^ of light is a quantity which may enter into combination with that motion by which magnetic phenomena are explained.

In order to form the equations of motion of the medium, we must express its kinetic energy in terms of the velocity of its parts, the components of which are ,77, f We therefore integrate by parts, and find

2 C 1 1 I (aco 1 + /3a) 2 -I ya&gt; 3 ) das dy dz

-av) dxdy

��The double integrals refer to the bounding surface, which may be supposed at an infinite distance. We may, therefore, while in vestigating what takes place in the interior of the medium, confine our attention to the triple integral.

825.] The part of the kinetic energy in unit of volume, expressed by this triple integral, may be written

iv Cfa+W + faf ( 5 )

where u, v, w are the components of the electric current as given in equations (E), Art. 607.

It appears from this that our hypothesis is equivalent to the assumption that the velocity of a particle of the medium whose components are f, r/, (f, is a quantity which may enter into com bination with the electric current whose components are u, v, w.

826.] Returning to the expression under the sign of triple inte gration in (4), substituting for the values of a, /3, y, those of a , /3", y , as given by equations (1), and writing

d d d d

ji for a -=- + 0-7- H-y-v- ; (6)

dh dx dy dz

the expression under the sign of integration becomes

L( d A_ d ^\, d ( d _ d c\+i d (**-

d& \d,y dz&gt; + ^ dh (dz dx) + C dh (dx In the case of waves in planes normal to the axis of z the displace-

�� � 828.] DYNAMICAL THEORY. 411

ments are functions of z and t only, so that -jj = y -^ and this expression is reduced to

• 2i-So-

The kinetic energy per unit of volume, so far as it depends on the velocities of displacement, may now be written

��where p is the density of the medium.

827.] The components, X and Y, of the impressed force, referred to unit of volume, may be deduced from this by Lagrange s equa tions, Art. 564.

"

��-

These forces arise from the action of the remainder of the medium on the element under consideration, and must in the case of an isotropic medium be of the form indicated by Cauchy,

��828.] If we now take the case of a circularly-polarized ray for which = rcos(ntqz), r? = r sin (nt qz), (14)

we find for the kinetic energy in unit of volume

T = pr 2 n*-Cyr 2 q 2 n; (15)

and for the potential energy in unit of volume 7= /*(4 ) 02-42* + &c.)

= i*Q, (16)

where Q is a function of q 2 .

The condition of free propagation of the ray given in Art. 820, equation (6), is dT d y

Tr=^ } (17)

which gives P n 2 -2Cyq 2 n = Q, (18)

whence the value of n may be found in terms of q.

But in the case of a ray of given wave-period, acted on by

�� � 412 MAGNETIC ACTION ON LIGHT. [829.

magnetic force, what we want to determine is the value of ---, when n

d y

is constant, in terms of -~ , when y is constant. Differentiating (1 8) aw/

^ 0. (19)

We thus find = - -L - (20)

ay pn

��829.] If A is the wave-length in air, and i the corresponding index of refraction in the medium,

q\ = 2-ni, n\ = 2nv. (21)

The change in the value of q, due to magnetic action, is in every case an exceedingly small fraction of its own value, so that we may

write g

(22)

��where q Q is the value of q when the magnetic force is zero. The angle, 0, through which the plane of polarization is turned in passing through a thickness c of the medium, is half the sum of the positive and negative values of qc, the sign of the result being changed, because the sign of q is negative in equations (14). We thus obtain

• =- Cr | (23)

\isC i 2 ,. di^ 1

--^T^O^sx) - (24)

1 27r(?y -

vp\

The second term of the denominator of this fraction is approx imately equal to the angle of rotation of the plane of polarization during its passage through a thickness of the medium equal to half a wave-length. It is therefore in all actual cases a quantity which we may neglect in comparison with unity.

Writing ~ = m, (25)

we may call m the coefficient of magnetic rotation for the medium, a quantity whose value must be determined by observation. It is found to be positive for most diamagnetic, and negative for some paramagnetic media. We have therefore as the final result of our theory ^2 .7;

i-K, (26)

��where 6 is the angular rotation of the plane of polarization, m a

�� � 830.] FORMULA FOR THE ROTATION. 413

constant determined by observation of the medium, y the intensity of the magnetic force resolved in the direction of the ray, c the length of the ray within the medium, A. the wave-length of the light in air, and i its index of refraction in the medium.

830.] The only test to which this theory has hitherto been sub jected, is that of comparing the values of 6 for different kinds of light passing through the same medium and acted on by the same magnetic force.

This has been done for a considerable number of media by M. Verdet *, who has arrived at the following results :

(1) The magnetic rotations of the planes of polarization of the rays of different colours follow approximately the law of the inverse square of the wave-length.

(2) The exact law of the phenomena is always such that the pro duct of the rotation by the square of the wave-length increases from the least refrangible to the most refrangible end of the spectrum.

(3) The substances for which this increase is most sensible are also those which have the greatest dispersive power.

He also found that in the solution of tartaric acid, which of itself produces a rotation of the plane of polarization, the magnetic rotation is by no means proportional to the natural rotation.

In an addition to the same memoir f Verdet has given the results of very careful experiments on bisulphide of carbon and on creosote, two substances in which the departure from the law of the inverse square of the wave-length was very apparent. He has also com pared these results with the numbers given by three different for- mula3&gt; ; 2 , fli ^

(I) e, (ii) e =

A" x l*A.

(Ill) e = mcy ( -X^).

The first of these formulae, (I), is that which we have already ob tained in Art. 829, equation (26). The second, (II), is that which results from substituting in the equations of motion, Art. 826, equa-

d z n r/ 3 "

tions (10), (11), terms of the form -^ and ~-, instead of

��dz*dt

��* Recherches sur les propri^t^s optiques developpdes dans les corps transparents par Faction du magn^tisme, 4 rae partie. Comptes Rendu*, t. Ivi. p. 630 (6 April, 1863). t Comptes Bendus. Ivii. p. 670 (19 Oct., 1863).

�� � 414 MAGNETIC ACTION ON LIGHT. [830.

d 3 and -- , o ?, I am n ^ aware that this form of the equations has

��been suggested by any physical theory. The third formula, (III), results from the physical theory of M. C. Neumann 56 , jn which the

equations of motion contain terms of the form -=? and --- =f t.

dt dt

It is evident that the values of given by the formula (III) are not even approximately proportional to the inverse square of the wave-length. Those given by the formulae (I) and (II) satisfy this condition, and give values of which agree tolerably well with the observed values for media of moderate dispersive power. For bisul phide of carbon and creosote, however, the values given by (II) differ very much from those observed. Those given by (I) agree better with observation, but, though the agreement is somewhat close for bisulphide of carbon, the numbers for creosote still differ by quan tities much greater than can be accounted for by any errors of observation.

Magnetic Rotation of the Plane of Polarization (from Verdet}.

Bisulphide of Carbon at 24. 9 C. Lines of the spectrum Observed rotation Calculated by I.

II. III.

��Creosote at 24. 3 C.

Lines of the spectrum C D E F

Observed rotation 573 758 1000 1241 1723

Calculated by I. 617 780 1000 1210 1603

II. 623 789 1000 1200 1565

III. 976 993 1000 1017 1041 Eotation of the ray E = 21. 58 .

We are so little acquainted with the details of the molecular

• Explicare tentatur quomodo fiat ut lucis planum polarizationis per vires elec-

tricas vel magneticas declinetur. Halis Saxonum, 1858.

t These three forms of the equations of motion were first suggested by Sir G. B. Airy (Phil. Mag., June 1846) as a means of analysing the phenomenon then recently discovered by Faraday. Mac Cullagh had previously suggested equations containing

,33

terms of the form - in order to represent mathematically the phenomena of quartz. dz s

These equations were offered by Mac Cullagh and Airy, not as giving a mechanical explanation of the phenomena, but as shewing that the phenomena may be explained by equations, which equations appear to be such as might possibly be deduced from some plausible mechanical assumption, although no such assumption has yet been made.

��i C D

�E

�P

�G

�592 768

�1000

�1234

�1704

�589 760

�1000

�1234

�1713

�606 772

�1000

�1216

�1640

�943 967

�1000

�1034

�1091

�Rotation of the ray

�E = 25. 28 .

� � �� � 831.] ARGUMENT OF THOMSON. 415

constitution of bodies, that it is not probable that any satisfactory theory can be formed relating- to a particular phenomenon, such as that of the magnetic action on light, until, by an induction founded on a number of different cases in which visible phenomena are found to depend upon actions in which the molecules are concerned, we learn something more definite about the properties which must be attributed to a molecule in order to satisfy the conditions of ob served facts.

The theory proposed in the preceding pages is evidently of a provisional kind, resting as it does on unproved hypotheses relating to the nature of molecular vortices, and the mode in which they are affected by the displacement of the medium. We must therefore regard any coincidence with observed facts as of much less scientific value in the theory of the magnetic rotation of the plane of polari zation than in the electromagnetic theory of light, which, though it involves hypotheses about the electric properties of media, does not speculate as to the constitution of their molecules.

831.] NOTE. The whole of this chapter may be regarded as an expansion of the exceedingly important remark of Sir William Thomson in the Proceedings of the Royal Society, June 1856 : The magnetic influence on light discovered by Faraday depends on the direction of motion of moving particles. For instance, in a medium possessing it, particles in a straight line parallel to the lines of magnetic force, displaced to a helix round this line as axis, and then projected tangentially with such velocities as to describe circles, will have different velocities according as their motions are round in one direction (the same as the nominal direction of the galvanic current in the magnetizing coil), or in the contrary direction. But the elastic reaction of the medium must be the same for the same displacements, whatever be the velocities and directions of the par ticles ; that is to say, the forces which are balanced by centrifugal force of the circular motions are equal, while the lumiriiferous motions are unequal. The absolute circular motions being there fore either equal or such as to transmit equal centrifugal forces to the particles initially considered, it follows that the luminiferous motions are only components of the whole motion ; and that a less luminiferous component in one direction, compounded with a mo tion existing in the medium when transmitting no light, gives an equal resultant to that of a greater luminiferous motion in the con trary direction compounded with the same non -luminous motion. I think it is not only impossible to conceive any other than this

�� � 41(5 MAGNETIC ACTION ON LIGHT. [831.

dynamical explanation of the fact that circularly-polarized light transmitted through magnetized glass parallel to the lines of mag netizing force, with the same quality, right-handed always, or left- handed always, is propagated at different rates according as its course is in the direction or is contrary to the direction in which a north magnetic pole is drawn ; but I believe it can be demonstrated that no other explanation of that fact is possible. Hence it appears that Faraday s optical discovery affords a demonstration of the re ality of Ampere s explanation of the ultimate nature of magnetism ; and gives a definition of magnetization in the dynamical theory of heat. The introduction of the principle of moments of momenta (" the conservation of areas") into the mechanical treatment of Mr. Rankine s hypothesis of " molecular vortices," appears to indi cate a line perpendicular to the plane of resultant rotatory mo mentum ("the invariable plane") of the thermal motions as the magnetic axis of a magnetized body, and suggests the resultant moment of momenta of these motions as the definite measure of the "magnetic moment." The explanation of all phenomena of electromagnetic attraction or repulsion, and of electromagnetic in duction, is to be looked for simply in the inertia and pressure of the matter of which the motions constitute heat. Whether this matter is or is not electricity, whether it is a continuous fluid inter- permeating the spaces between molecular nuclei, or is itself mole- cularly grouped ; or whether all matter is continuous, and molecular heterogeneousness consists in finite vortical or other relative mo tions of contiguous parts of a body ; it is impossible to decide, and perhaps in vain to speculate, in the present state of science.

A theory of molecular vortices, which I worked out at consider able length, was published in the Phil. Mag. for March, April, and May, 1861, Jan. and Feb. 1862.

I think we have good evidence for the opinion that some pheno menon of rotation is going on in the magnetic field, that this rota tion is performed by a great number of very small portions of matter, each rotating on its own axis, this axis being parallel to the direction of the magnetic force, and that the rotations of these dif ferent vortices are made to depend on one another b}^ means of some kind of mechanism connecting them.

The attempt which I then made to imagine a working model of this mechanism must be taken for no more than it really is, a de monstration that mechanism may be imagined capable of producing a connexion mechanically equivalent to the actual connexion of the

�� � 831.] THEORY OF MOLECULAK VORTICES. 417

parts of the electromagnetic field. The problem of determining the mechanism required to establish a given species of connexion be tween the motions of the parts of a system always admits of an infinite number of solutions. Of these, some may be more clumsy or more complex than others, but all must satisfy the conditions of mechanism in general.

The folio wing results of the theory, however, are of higher value :

(1) Magnetic force is the effect of the centrifugal force of the vortices.

(2) Electromagnetic induction of currents is the effect of the forces called into play when the velocity of the vortices is changing.

(3) Electromotive force arises from the stress on the connecting mechanism.

(4) Electric displacement arises from the elastic yielding of the connecting mechanism.

��VOL. II.

�� �

1. Experimental Researches, 951–954 and 2216–2220.