the co-ordinates of an ion, R1 the coefficient of resistance to the
motion of the ions, and α 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 εl(pt−qz) we find by substitution that q is given by the equation
| q2 − K0p2 − | 4πne2p2P | = ± | 4πne3Hp3 | , |
| P2 − H2e2p2 | P2 − H2e2p2 |
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 = ιY0 and to q2 X0 = −ιY0. 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 | + | 4πne3Hp | , |
| v12 | v2 | m2 (s2 − p2)2 |
| 1 | = | 1 | − | 4πne3Hp | . |
| v22 | v2 | m2 (s2 − p2)2 |
The rotation r of the plane of polarization per unit length
| = 12p ( | 1 | − | 1 | ) = | 2πne3Hp2v | . |
| v1 | v2 | m2 (s2 − p2)2 |
Since 1/v2 = K0 + 4πne2/m (s2 − p2), we have if µ is the refractive index for light of frequency p, and v0 the velocity of light in vacuo.
| µ2 − 1 = 4πne2v02/m (s2 − p2). | (1) |
So that we may put
| r = (µ2 − 1)2 p2H / sπµnev03. | (2) |
Becquerel (Comptes rendus, 125, p. 683) gives for r the expression
| 12 | e | H | dµ | , | ||
| m | v0 | dλ |
where λ is the wave length. This is equivalent to (2) if µ is given by (1). He has shown that this expression is in good agreement with experiment. The sign of r depends on the sign of e, hence the rotation due to negative ions would be opposite to that for positive. For the great majority of substances the direction of rotation is that corresponding to the negation ion. We see from the equations that the rotation is very large for such a value of p as makes P = 0; this value corresponds to a free period of the ions, so that the rotation ought to be very large in the neighbourhood of an absorption band. This has been verified for sodium vapour by Macaluso and Corbino.43
If plane-polarized light falls normally on a plane face of the medium containing the ions, then if the electric force in the incident wave is parallel to x and is equal to the real part of Aεl(pt−qz), if the reflected beam in which the electric force is parallel to x is represented by Bεl(pt+qz) and the reflected beam in which the electric force is parallel to the axis of y by Cεl(pt+qz), then the conditions that the magnetic force parallel to the surface is continuous, and that the electric forces parallel to the surface in the air are continuous with Y0, X0 in the medium, give
| A | = | B | = | ιC |
| (q + q1) (q + q2) | (q2 − q1q2) | q (q2 − q1) |
or approximately, since q1 and q2 are nearly equal,
| ιC | = | q (q2 − q1) | = | (µ2 − 1) pH | . |
| B | q2 − q12 | 4πµneV02 |
Thus in transparent bodies for which µ is real, C and B differ in phase by π/2, and the reflected light is elliptically polarized, the major axis of the ellipse being in the plane of polarization of the incident light, so that in this case there is no rotation, but only elliptic polarization; when there is strong absorption so that µ contains an imaginary term, C/B will contain a real part so that the reflected light will be elliptically polarized, but the major axis is no longer in the plane of polarization of the incident light; we should thus have a rotation of the plane of polarization superposed on the elliptic polarization.
Zeeman’s Effect.—Faraday, after discovering the effect of a magnetic field on the plane of polarization of light, made numerous experiments to see if such a field influenced the nature of the light emitted by a luminous body, but without success. In 1885 Fievez,44 a Belgian physicist, noticed that the spectrum of a sodium flame was changed slightly in appearance by a magnetic field; but his observation does not seem to have attracted much attention, and was probably ascribed to secondary effects. In 1896 Zeeman45 saw a distinct broadening of the lines of lithium and sodium when the flames containing salts of these metals were between the poles of a powerful electromagnet; following up this observation, he obtained some exceedingly remarkable and interesting results, of which those observed with the blue-green cadmium line may be taken as typical. He found that in a strong magnetic field, when the lines of force are parallel to the direction of propagation of the light, the line is split up into a doublet, the constituents of which are on opposite sides of the undisturbed position of the line, and that the light in the constituents of this doublet is circularly polarized, the rotation in the two lines being in opposite directions. When the magnetic force is at right angles to the direction of propagation of the light, the line is resolved into a triplet, of which the middle line occupies the same position as the undisturbed line; all the constituents of this triplet are plane-polarized, the plane of polarization of the middle line being at right angles to the magnetic force, while the outside lines are polarized on a plane parallel to the lines of magnetic force. A great deal of light is thrown on this phenomenon by the following considerations due to H. A. Lorentz.46
Let us consider an ion attracted to a centre of force by a force proportional to the distance, and acted on by a magnetic force parallel to the axis of z: then if m is the mass of the particle and e its charge, the equations of motion are
| m | d 2x | + αx = − He | dy | ; |
| dt2 | dt |
| m | d 2y | + αy = He | dx | ; |
| dt2 | dt |
| m | d 2z | + ax = 0. |
| dt2 |
The solution of these equations is
| x = A cos (p1t + β) + B cos (p2t + β1) |
| y = A sin (p1t + β) − B sin (p2t + β1) |
| z = C cos (pt + γ) |
or approximately
| p1 = p + 12 | He | , p2 = p − 12 | He | . |
| m | m |
Thus the motion of the ion on the xy plane may be regarded as made up of two circular motions in opposite directions described with frequencies p1 and p2 respectively, while the motion along z has the period p, which is the frequency for all the vibrations when H = 0. Now suppose that the cadmium line is due to the motion of such an ion; then if the magnetic force is along the direction of propagation, the vibration in this direction has its period unaltered, but since the direction of vibration is perpendicular to the wave front, it does not give rise to light. Thus we are left with the two circular motions in the wave front with frequencies p1 and p2 giving the circularly polarized constituents of the doublet. Now suppose the magnetic force is at right angles to the direction of propagation of the light; then the vibration parallel to the magnetic force being in the wave front produces luminous effects and gives rise to a plane-polarized ray of undisturbed period (the middle line of the triplet), the plane of polarization being at right angles to the magnetic force. The components in the wave-front of the circular orbits at right angles to the magnetic force will be rectilinear motions of frequency p1 and p2 at right angles to the magnetic force—so that they will produce plane-polarized light, the plane of polarization being parallel to the magnetic force; these are the outer lines of the triplet.
If Zeeman’s observations are interpreted from this point of view, the directions of rotation of the circularly-polarized light in the doublet observed along the lines of magnetic force show that the ions which produce the luminous vibrations are negatively electrified, while the measurement of the charge of frequency due to the magnetic field shows that e/m is of the order 107. This result is of great interest, as this is the order of the value of e/m in the negatively electrified particles which constitute the Cathode Rays (see Conduction, Electric III. Through Gases). Thus we infer that the “cathode particles” are found in bodies, even where not subject to the action of intense electrical fields, and are in fact an ordinary constituent of the molecule. Similar particles are found near an incandescent wire, and also near a metal plate illuminated by ultra-violet light. The value of e/m deduced from the Zeeman effect ranges from 107 to 3.4 × 107, the value of e/m for the particle in the cathode rays is 1.7 × 107. The majority of the determinations of e/m from the Zeeman effect give numbers larger than this, the maximum being about twice this value.
