[1] _Annals of Electricity_, 1839, 3, p. 288.
[2] Humphrey Lloyd, _Proc. Roy. Irish Acad._, 1848, 4, p. 57.
MAGNETO-OPTICS. The first relation between magnetism and light was
discovered by Faraday,[1] who proved that the plane of polarization of a
ray of light was rotated when the ray travelled through certain
substances parallel to the lines of magnetic force. This power of
rotating the plane of polarization in a magnetic field has been shown to
be possessed by all refracting substances, whether they are in the
solid, liquid or gaseous state. The rotation by gases was established
independently by H. Becquerel,[2] and Kundt and Röntgen,[3] while
Kundt[4] found that films of the magnetic metals, iron, cobalt, nickel,
thin enough to be transparent, produced enormous rotations, these being
in iron and cobalt magnetized to saturation at the rate of 200,000° per
cm. of thickness, and in nickel about 89,000°. The direction of rotation
is not the same in all bodies. If we call the rotation positive when it
is related to the direction of the magnetic force, like rotation and
translation in a right-handed screw, or, what is equivalent, when it is
in the direction of the electric currents which would produce a magnetic
field in the same direction as that which produces the rotation, then
most substances produce positive rotation. Among those that produce
negative rotation are ferrous and ferric salts, ferricyanide of
potassium, the salts of lanthanum, cerium and didymium, and chloride of
titanium.[5]
The magnetic metals iron, nickel, cobalt, the salts of nickel and
cobalt, and oxygen (the most magnetic gas) produce positive rotation.
For slightly magnetizable substances the amount of rotation in a space
PQ is proportional to the difference between the magnetic potential at
P and Q; or if [theta] is the rotation in PQ, [Omega]_P, [Omega]_Q,
the magnetic potential at P and Q, then [theta] = R([Omega]_P -
[Omega]_Q), where R is a constant, called Verdet's constant, which
depends upon the refracting substance, the wave length of the light,
and the temperature. The following are the values of R (when the
rotation is expressed in circular measure) for the D line and a
temperature of 18° C.:--
Substance. R × 10^5. Observer.
Carbon bisulphide / 1.222 Lord Rayleigh[6] and Köpsel.[7]
\ 1.225 Rodger and Watson.[8]
Water / .377 Arons.[9]
\ .3808 Rodger and Watson.[8]
Alcohol .330 Du Bois.[10]
Ether .315 Du Bois.[10]
Oxygen (at 1 atmosphere) .000179 Kundt and Röntgen (_loc. cit._)
Faraday's heavy glass 1.738
The variation of Verdet's constant with temperature has been
determined for carbon bisulphide and water by Rodger and Watson (_loc.
cit._). They find if R_t, R0 are the values of Verdet's constant at
t°C and 0°C. respectively, then for carbon bisulphide R_t = R0 (1 -
.0016961), and for water R_t = R0 (1 - .0000305t - .00000305t²).
For the magnetic metals Kundt found that the rotation did not increase
so rapidly as the magnetic force, but that as this force was increased
the rotation reached a maximum value. This suggests that the rotation
is proportional to the intensity of magnetization, and not to the
magnetic force.
The amount of rotation in a given field depends greatly upon the wave
length of the light; the shorter the wave length the greater the
rotation, the rotation varying a little more rapidly than the inverse
square of the wave length. Verdet[11] has compared in the cases of
carbon bisulphide and creosote the rotation given by the formula
c² / di \
[theta] = mc[gamma] --------- ( c - [lamda] --------- )
[lambda]² \ d[lambda]/
with those actually observed; in this formula [theta] is the angular
rotation of the plane of polarization, m a constant depending on the
medium, [lambda] the wave length of the light in air, and i its index
of refraction in the medium. Verdet found that, though the agreement
is fair, the differences are greater than can be explained by errors
of experiment.
Verdet[12] has shown that the rotation of a salt solution is the sum of
the rotations due to the salt and the solvent; thus, by mixing a salt
which produces negative rotation with water which produces positive
rotation, it is possible to get a solution which does not exhibit any
rotation. Such solutions are not in general magnetically neutral. By
mixing diamagnetic and paramagnetic substances we can get magnetically
neutral solutions, which, however, produce a finite rotation of the
plane of polarization. The relation of the magnetic rotation to chemical
constitution has been studied in great detail by Perkin,[13]
Wachsmuth,[14] Jahn[15] and Schönrock.[16]
The rotation of the plane of polarization may conveniently be regarded
as denoting that the velocity of propagation of circular-polarized light
travelling along the lines of magnetic force depends upon the direction
of rotation of the ray, the velocity when the rotation is related to the
direction of the magnetic force, like rotation and translation on a
right-handed screw being different from that for a left-handed rotation.
A plane-polarized ray may be regarded as compounded of two oppositely
circularly-polarized rays, and as these travel along the lines of
magnetic force with different velocities, the one will gain or lose in
phase on the other, so that when they are again compounded they will
correspond to a plane-polarized ray, but in consequence of the change of
phase the plane of polarization will not coincide with its original
position.
_Reflection from a Magnet._--Kerr[17] in 1877 found that when
plane-polarized light is incident on the pole of an electromagnet,
polished so as to act like a mirror, the plane of polarization of the
reflected light is rotated by the magnet. Further experiments on this
phenomenon have been made by Righi,[18] Kundt,[19] Du Bois,[20]
Sissingh,[21] Hall,[22] Hurion,[23] Kaz[24] and Zeeman.[25] The simplest
case is when the incident plane-polarized light falls normally on the
pole of an electromagnet. When the magnet is not excited the reflected
ray is plane-polarized; when the magnet is excited the plane of
polarization is rotated through a small angle, the direction of rotation
being opposite to that of the currents exciting the pole. Righi found
that the reflected light was slightly elliptically polarized, the axes
of the ellipse being of very unequal magnitude. A piece of gold-leaf
placed over the pole entirely stops the rotation, showing that it is not
produced in the air near the pole. Rotation takes place from magnetized
nickel and cobalt as well as from iron, and is in the same direction
(Hall). Righi has shown that the rotation at reflection is greater for
long waves than for short, whereas, as we have seen, the Faraday
rotation is greater for short waves than for long. The rotation for
different coloured light from iron, nickel, cobalt and magnetite has
been measured by Du Bois; in magnetite the direction of rotation is
opposite to that of the other metals. When the light is incident
obliquely and not normally on the polished pole of an electromagnet, it
is elliptically polarized after reflection, even when the plane of
polarization is parallel or at right angles to the plane of incidence.
According to Righi, the amount of rotation when the plane of
polarization of the incident light is perpendicular to the plane of
incidence reaches a maximum when the angle of incidence is between 44°
and 68°, while when the light is polarized in the plane of incidence the
rotation steadily decreases as the angle of incidence is increased. The
rotation when the light is polarized in the plane of incidence is always
less than when it is polarized at right angles to that plane, except
when the incidence is normal, when the two rotations are of course
equal.
_Reflection from Tangentially Magnetized Iron._--In this case Kerr[26]
found: (1) When the plane of incidence is perpendicular to the lines of
magnetic force, no rotation of the reflected light is produced by
magnetization; (2) no rotation is produced when the light is incident
normally; (3) when the incidence is oblique, the lines of magnetic force
being in the plane of incidence, the reflected light is elliptically
polarized after reflection, and the axes of the ellipse are not in and
at right angles to the plane of incidence. When the light is polarized
in the plane of incidence, the rotation is at all angles of incidence in
the opposite direction to that of the currents which would produce a
magnetic field of the same sign as the magnet. When the light is
polarized at right angles to the plane of incidence, the rotation is in
the same direction as these currents when the angle of incidence is
between 0° and 75° according to Kerr, between 0° and 80° according to
Kundt, and between 0° and 78° 54´ according to Righi. When the incidence
is more oblique than this, the rotation of the plane of polarization is
in the opposite direction to the electric currents which would produce a
magnetic field of the same sign.
The theory of the phenomena just described has been dealt with by
Airy,[27] C. Neumann,[28] Maxwell,[29] Fitzgerald,[30] Rowland,[31] H.
A. Lorentz,[32] Voight,[33] Ketteler,[34] van Loghem,[35] Potier,[36]
Basset,[37] Goldhammer,[38] Drude,[39] J. J. Thomson,[40] and
Leatham;[41] for a critical discussion of many of these theories we
refer the reader to Larmor's[42] British Association Report. Most of
these theories have proceeded on the plan of adding to the expression
for the electromotive force terms indicating a force similar in
character to that discovered by Hall (see MAGNETISM) in metallic
conductors carrying a current in a magnetic field, i.e. an electromotive
force at right angles to the plane containing the magnetic force and the
electric current, and proportional to the sine of the angle between
these vectors. The introduction of a term of this kind gives rotation of
the plane of polarization by transmission through all refracting
substance, and by reflection from magnetized metals, and shows a fair
agreement between the theoretical and experimental results. The simplest
way of treating the questions seems, however, to be to go to the
equations which represent the propagation of a wave travelling through a
medium containing ions. A moving ion in a magnetic field will be acted
upon by a mechanical force which is at right angles to its direction of
motion, and also to the magnetic force, and is equal per unit charge to
the product of these two vectors and the sine of the angle between them.
For the sake of brevity we will take the special case of a wave
travelling parallel to the magnetic force in the direction of the axis
of z.
Then supposing that all the ions are of the same kind, and that there
are _n_ of these each with mass _m_ and charge _e_ per unit volume,
the equations representing the field are (see ELECTRIC WAVES):--
dX0 d[xi] d[beta]
K0 --- + 4[pi]ne ----- = -------;
dt dt dz
dX[0] d[beta]
----- = -------;
dz dt
dY0 d[eta] d[alpha]
K0 --- + 4[pi]ne ------ = - --------
dt dt dz
dY0 d[alpha]
--- = - --------;
dz dt
d²[xi] d[xi] / 4[pi] \ d[eta]
m ------ + R1 ----- + a[xi] = ( X0 + ----- ne[xi] ) e + He ------
dt² dt \ 3 / dt
d²[eta] d[eta] / 4[pi] \ d[xi]
m ------- + R1 ------ + a[eta] = ( Y0 + ----- ne[eta] ) e - He -----;
dt² dt \ 3 / dt
where H is the external magnetic field, X0, Y0 the components of the
part of the electric force in the wave not due to the charges on the
atoms, [alpha] and [beta] the components of the magnetic force, [xi]
and [eta] the co-ordinates of an ion, R1 the coefficient of resistance
to the motion of the ions, and [alpha] the force at unit distance
tending to bring the ion back to its position of equilibrium, K0 the
specific inductive capacity of a vacuum. If the variables are
proportional to [epsilon]^[l(pt - qz)] we find by substitution that q
is given by the equation
4[pi]ne²p²P 4[pi]ne³Hp³
q² - K0p² - ----------- = ± -----------,
P² - H²e²p² P² - H²e²p²
where
P = (a - (4/3)[pi]ne²) + R1[iota]p - mp²,
or, by neglecting R, P = m(s² - p²), where s is the period of the free
ions. If, q1², q2² are the roots of this equation, then corresponding
to q1 we have X0 = [iota]Y0 and to q2 X0 = -[iota]Y0. We thus get two
oppositely circular-polarized rays travelling with the velocities p/q1
and p/q2 respectively. Hence if v1, v2 are these velocities, and v the
velocity when there is no magnetic field, we obtain, if we neglect
terms in H²,
1 1 4[pi]ne³Hp
--- = -- + ------------,
v1² v² m²(s² - p²)²
1 1 4[pi]ne³Hp
--- = -- - ------------.
v2² v² m²(s² - p²)²