47. Variation of current with voltage for surface ionization.

The theory of this subject has been worked out independently by Child^[80] and Rutherford^[81]. Let V be the potential difference between two parallel plates at a distance d apart. Suppose that the ionization is confined to a thin layer near the surface of the plate A (see Fig. 1) which is charged positively. When the electric field is acting, there is a distribution of positive ions between the plates A and B.

The current i₁ per square centimetre through the gas is constant for all values of x, and is given by

By Poisson’s equation

Then

Integrating

where A is a constant. Now A is equal to the value of

when x = 0. By making the ionization very intense, the value of

can be made extremely small.

Putting A = 0, we see that

This gives the potential gradient between the plates for different values of x.

Integrating between the limits 0 and d,

or

If i₂ is the value of the current when the electric field is reversed, and K₂ the velocity of the negative ion,

and

The current in the two directions is thus directly proportional to the velocities of the positive and negative ions. The current should vary directly as the square of the potential difference applied, and inversely as the cube of the distance between the plates.

The theoretical condition of surface ionization cannot be fulfilled by the ionization due to active substances, as the ionization extends some centimetres from the active plate. If, however, the distance between the plates is large compared with the distance over which the ionization extends, the results will be in rough agreement with the theory. Using an active preparation of radium, the writer has made some experiments on the variation of current with voltage between parallel plates distant about 10 cms. from each other^[82].

The results showed

(1) That the current through the gas for small voltages increased more rapidly than the potential difference applied, but not as rapidly as the square of that potential difference.

(2) The current through the gas depended on the direction of the electric field; the current was always smaller when the active plate was charged positively on account of the smaller mobility of the positive ion. The difference between i₁ and i₂ was greatest when the gas was dry, which is the condition for the greatest difference between the velocities of the ions.

An interesting result follows from the above theory. For given values of V and d, the current cannot exceed a certain definite value, however much the ionization may be increased. In a similar way, when an active preparation of radium is used as a source of surface ionization, it is found that, for a given voltage and distance between the plates, the current does not increase beyond a certain value however much the activity of the material is increased.

48. Magnetic field produced by an ion in motion. It will be shown later that the two most important kinds of rays emitted by radio-active substances consist of electrified particles, spontaneously projected with great velocity. The easily absorbed rays, known as α rays, are positively electrified atoms of matter; the penetrating rays, known as β rays, carry a negative charge, and have been found to be identical with the cathode rays produced by the electric discharge in a vacuum tube.

A brief account will accordingly be given of the effects produced by a moving charged body, and also of some of the experimental methods which have been used to determine the mass and velocity of the particles of the cathode stream^[83].

Consider an ion of radius a, carrying a charge of electricity e, and moving with a velocity u, small compared with the velocity of light. In consequence of the motion, a magnetic field is set up around the charged ion, which is carried with it. The charged ion in motion constitutes a current element of magnitude eu, and the magnetic field H at any point distant r from the sphere is given by

where θ is the angle the radius vector makes with the direction of motion. The lines of magnetic force are circles around the axis of motion. When the ion is moving with a velocity small compared with the velocity of light, the lines of electric force are nearly radial, but as the speed of light is approached, they tend to leave the axis of motion and to bend towards the equator. When the speed of the body is very close to that of light, the magnetic and electric field is concentrated to a large extent in the equatorial plane.

The presence of a magnetic field around the moving body implies that magnetic energy is stored up in the medium surrounding it. The amount of this energy can be calculated very simply for slow speeds.

In a magnetic field of strength H, the magnetic energy stored up in unit volume of the medium of unit permeability is given by

Integrating the value of this expression over the region exterior to a sphere of radius a, the total magnetic energy due to the motion of the charged body is given by

The magnetic energy, due to the motion, is analogous to kinetic energy, for it depends upon the square of the velocity of the body. In consequence of the charge carried by the ion, additional kinetic energy is associated with it. If the velocity of the ion is changed, electric and magnetic forces are set up tending to stop the change of motion, and more work is done during the change than if the ion were uncharged. The ordinary kinetic energy of the body is

In consequence of its charge, the kinetic energy associated with it is increased by

It thus behaves as if it possessed a mass m + m₁ where m₁ is the electrical mass, with the value

We have so far only considered the electrical mass of a charged ion moving with a velocity small compared with that of light. As the speed of light is approached, the magnetic energy can no longer be expressed by the equation already given. The general values of the electrical mass of a charged body for speed were first worked out by J. J. Thomson^[84] in 1887. A more complete examination was made in 1889 by Heaviside^[85], while Searle^[86] worked out the case for a charged ellipsoid. Recently, the question was again attacked by Abraham^[87]. Slightly different expressions for the variation of electrical mass with speed have been obtained, depending upon the conditions assumed for the distribution of the electricity on the sphere. The expression found by Abraham, which has been utilized by Kaufmann to show that the mass of the electron is electromagnetic in origin, is given later in section 82.

All the calculations agree in showing that the electrical mass is practically constant for slow speeds, but increases as the speed of light is approached, and is theoretically infinite when the speed of light is reached. The nearer the velocity of light is approached, the greater is the resisting force to a change of motion. An infinite force would be required to make an electron actually attain the velocity of light, so that, according to the present theory, it would be impossible for an electron to move faster than light, i.e. faster than an electromagnetic disturbance travels in the ether.

The importance of these deductions lies in the fact that an electric charge in motion, quite independently of any material nucleus, possesses an apparent mass in virtue of its motion, and that this mass is a function of the speed. Indeed, we shall see later (see section 82) that the apparent mass of the particles constituting the cathode stream can be explained in virtue of their charge, without the necessity of assuming a material body in which the charge is distributed. This has led to the suggestion that all mass may be electrical in origin, and due purely to electricity in motion.

49. Action of a magnetic field on a moving ion. Let us consider the case of an ion of mass m carrying a charge e and moving freely with a velocity u. If u is small compared with the velocity of light, the ion in motion corresponds to a current element of magnitude eu. If the ion moves in an external magnetic field of strength H, it is acted on by a force at right angles both to the direction of motion, and to that of the magnetic force and equal in magnitude to Heu sin θ, where θ is the angle between the direction of the magnetic force and the direction of motion. Since the force due to the magnetic field is always perpendicular to the direction of motion, it has no effect upon the velocity of the particle, but can only alter the direction of its path.

If ρ is the radius of curvature of the path of the ion, the force along the normal is equal to

and this is balanced by the force Heu sin θ.

If

i.e. if the ion is moving at right angles to the direction of the magnetic field

or

Since u is constant, ρ is also constant, i.e. the particle describes a circular orbit of radius ρ. The radius of the circular orbit is thus directly proportional to u, and inversely proportional to H.

If the ion is moving at an angle θ with the direction of the magnetic field, it describes a curve which is compounded of a motion of a particle of velocity u sin θ perpendicular to the field and u cos θ in the direction of the field. The former describes a circular orbit of radius ρ, given by

the latter is unaffected by the magnetic field and moves uniformly in the direction of the magnetic field with a velocity u cos θ. The motion of the particle is in consequence a helix, traced on a cylinder of radius

whose axis is in the direction of the magnetic field. Thus an ion projected obliquely to the direction of a uniform magnetic field always moves in a helix whose axis is parallel to the lines of magnetic force^[88].

50. Determination of e/m for the cathode stream. The cathode rays, first observed by Varley, were investigated in detail by Crookes. These rays are projected from the cathode in a vacuum tube at low pressure. They travel in straight lines, and are readily deflected by a magnet, and produce strong luminosity in a variety of substances placed in their path. The rays are deflected by a magnetic field in the same direction as would be expected for a negatively charged particle projected from the cathode. In order to explain the peculiar properties of these rays Crookes supposed that they consisted of negatively electrified particles, moving with great velocity and constituting, as he appropriately termed it, “a new or fourth state of matter.” The nature of these rays was for twenty years a subject of much controversy, for while some upheld their material character, others considered that they were a special form of wave motion in the ether.

The nature of these rays was successfully demonstrated by J. J. Thomson^[89] in 1897. If the rays consisted of negatively electrified particles, they should be deflected in their passage through an electric as well as through a magnetic field. Such an experiment had been tried by Hertz, but with negative results. J. J. Thomson, however, found that the rays were deflected by an electric field in the direction to be expected for a negatively charged particle, and showed that the failure of Hertz to detect the same was due to the masking of the electric field by the strong ionization produced in the gas by the cathode stream. This effect was got rid of by reducing the pressure of the gas in the tube.

The experimental arrangement used for the electric deflection of the rays is shown in Fig. 10.

The cathode rays are generated at the cathode C, and a narrow pencil of rays is obtained by passing the rays through a perforated disc AB. The rays then passed midway between two parallel insulated plates D and E, d centimetres apart, and maintained at a constant difference of potential V. The point of incidence of the pencil of rays was marked by a luminous patch produced on a phosphorescent screen placed at PP´.

The particle carrying a negative charge e in passing between the charged plates, is acted on by a force Xe directed towards the positive plate, where X, the strength of the electric field, is given by

The application of the electric field thus causes the luminous patch to move in the direction of the positive plate. If now a uniform magnetic field is applied at the plates D and E, perpendicular to the pencil of rays, and parallel to the plane of the plates, and in such a direction that the electric and magnetic forces are opposed to one another, the patch of light can be brought back to its undisturbed position by adjusting the strength of the magnetic field. If H is the strength of the magnetic field, the force on the particle due to the magnetic field is Heu, and when a balance is obtained

Now if the magnetic field H is acting alone, the curvature ρ of the path of the rays between the plates can be deduced from the deflection of the luminous patch. But we have seen that

From equations (1) and (2), the value of u and e/m for the particle can be determined.

The velocity u is not constant, but depends upon the potential difference between the electrodes, and this in turn depends upon the pressure and nature of the residual gas in the tube.

By altering these factors, the cathode particles may be made to acquire velocities varying between about 10⁹ and 10¹⁰ cms. per second. This velocity is enormous compared with that which can be impressed ordinarily upon matter by mechanical means. On the other hand, the value of e/m for the particles is sensibly constant for different velocities.

As a result of a series of experiments the mean value e/m = 7·7 × 10⁶ was obtained. The value of e/m is independent of the nature or pressure of the gas in the vacuum tube and independent of the metal used as cathode. A similar value of e/m was obtained by Lenard^[90] and others.

Kaufmann^[91] and Simon^[92] used a different method to determine the value of e/m. The potential difference V between the terminals of the tube was measured. The work done on the charged particle in moving from one end of the tube to the other is Ve, and this must be equal to the kinetic energy

acquired by the moving particle. Thus

By combination of this equation with (2) obtained by measurement of the magnetic deflexion, both u and e/m can be determined.

Simon found by this method that

It will be seen later (section 82) that a similar value was deduced by Kaufmann for the electrons projected from radium.

These results, which have been based on the effect of a magnetic and electric field on a moving ion, were confirmed by Weichert, who determined by a direct method the time required for the particle to traverse a known distance.

The particles which make up the cathode stream were termed “corpuscles” by J. J. Thomson. The name “electron,” first employed by Johnstone Stoney, has also been applied to them and has come into general use^[93].

The methods above described do not give the mass of the electron, but only the ratio of the charge to the mass. A direct comparison can, however, be made between the ratio e/m for the electron and the corresponding value for the hydrogen atoms set free in the electrolysis of water. Each of the hydrogen atoms is supposed to carry a charge e, and it is known that 96,000 coulombs of electricity, or, in round numbers, 10⁴ electromagnetic units of quantity are required to liberate one gram of hydrogen. If N is the number of atoms in one gram of hydrogen, then Ne = 10⁴. But if m is the mass of a hydrogen atom, then Nm = 1. Dividing one by the other e/m = 10⁴. We have seen already that a gaseous ion carries the same charge as a hydrogen atom, while indirect evidence shows that the electron carries the same charge as an ion, and consequently the same charge as the atom of hydrogen. Hence we may conclude that the apparent mass of the electron is only about ¹⁄₁₀₀₀ of the mass of the hydrogen atom. The electron thus behaves as the smallest body known to science.

In later experiments J. J. Thomson showed that the negative ions set free at low pressures by an incandescent carbon filament, and also the negative ions liberated from a zinc plate exposed to the action of ultra-violet light, had the same value for e/m as the electrons produced in a vacuum tube. It thus seemed probable that the electron was a constituent of all matter. This view received strong support from measurements of quite a different character. Zeeman in 1897 found that the lines of the spectrum from a source of light exposed in a strong magnetic field were displaced and doubled. Later work has shown that the lines in some cases are trebled, in others sextupled, while, in a few cases, the multiplication is still greater. These results received a general explanation on the radiation theories previously advanced by Lorenz and Larmor. The radiation, emitted from any source, was supposed to result from the orbital or oscillatory motion of the charged parts constituting the atom. Since a moving ion is acted on by an external magnetic field, the motion of the charged ions is disturbed when the source of light is exposed between the poles of a strong magnet. There results a small change in the period of the emitted light, and a bright line in the spectrum is, in consequence, displaced by the action of the magnetic field. According to theory, the small change in the wave-length of the emitted light depends upon the strength of the magnetic field and on the ratio e/m of the charge carried by the ion to its mass. By comparison of the theory with the experimental results, it was deduced that the moving ion carried a negative charge, and that the value of e/m was about 10⁷. The charged ion, responsible for the radiation from a luminous body, is thus identical with the electron set free in a vacuum tube.

It thus seems reasonable to suppose that the atoms of all bodies are complex and are built up, in part at least, of electrons, whose apparent mass is very small compared with that of the hydrogen atom. The properties of such disembodied charges has been examined mathematically among others by Larmor, who sees in this conception the ultimate basis of a theory of matter. J. J. Thomson and Lord Kelvin have investigated mathematically certain arrangements of a number of electrons which are stable for small disturbances. This question will be discussed more in detail in section 270.

51. Canal rays. If a discharge is passed through a vacuum tube provided with a perforated cathode, within certain limits of pressure, luminous streams are observed to pass through the holes and to emerge on the side of the cathode remote from the anode. These rays were first observed by Goldstein^[94] and were called by him the “Canal-strahlen.” These rays travel in straight lines and produce phosphorescence in various substances.

Wien^[95] showed that the canal rays were deflected by strong magnetic and electric fields, but the amount of deflection was very small compared with that of the cathode rays under similar conditions. The deflection was found to be opposite in direction to the cathode rays, and this indicates that the canal rays consist of positive ions. Wien determined their velocity and the ratio e/m, by measuring the amount of their magnetic and electric deflection. The value of e/m was found to be variable, depending upon the gas in the tube, but the maximum value observed was 10⁴. This shows that the positive ion, in no case, has a mass less than that of the hydrogen atom. It seems probable that the canal rays consist of positive ions, derived either from the gas or the electrodes, which travel towards the cathode, and have sufficient velocity to pass through the holes of the cathode and to appear in the gas beyond.

It is remarkable that, so far, no case has been observed where the carrier of a positive charge has an apparent mass less than that of the hydrogen atom. Positive electricity always appears to be associated with bodies atomic in size. We have seen that the process of ionization in gases is supposed to consist of the expulsion of an electron from the atom. The corresponding positive charge remains behind on the atom and travels with it. This difference between positive and negative electricity appears to be fundamental, and no explanation of it has, as yet, been forthcoming.

52. Radiation of energy. If an electron moves uniformly in a straight line with constant velocity, the magnetic field, which travels with it, remains constant, and there is no loss of energy from it by radiation. If, however, its motion is hastened or retarded, the magnetic field is altered, and there results a loss of energy from the electron in the form of electromagnetic radiation. The rate of loss of energy from an accelerated electron was first calculated by Larmor^[96] and shown to be

where e is the charge on the electron in electromagnetic units, and V the velocity of light.

Any alteration in the velocity of a moving charge is thus always accompanied by a radiation of energy from it. Since the electron, set free in a vacuum tube, increases in velocity in passing through the electric field, energy must be radiated from it during its passage from cathode to anode. It can, however, readily be calculated that, in ordinary cases, this loss of energy is small compared with the kinetic energy acquired by the electron in passing through the electric field.

An electron moving in a circular orbit is a powerful radiator of energy, since it is constantly accelerated towards the centre. An electron moving in an orbit of radius equal to the radius of an atom (about 10^-8 cms.) would lose most of its kinetic energy of motion in a small fraction of a second, even though its velocity was originally nearly equal to the velocity of light. If, however, a number of electrons are arranged at equal angular intervals on the circumference of a circle and move with constant velocity round the ring, the radiation of energy is much less than for a single electron, and rapidly diminishes with an increase in the number of electrons round the ring. This result, obtained by J. J. Thomson, will be discussed in more detail later when the stability of systems composed of rotating electrons is under consideration.

Since the radiation of energy is proportional to the square of the acceleration, the proportion of the total energy radiated depends upon the suddenness with which an electron is started or stopped. Now some of the cathode ray particles are stopped abruptly when they impinge on the metal cathode, and, in consequence, give up a fraction of their kinetic energy in the form of electromagnetic radiation. Stokes and Weichert suggested that this radiation constituted the X rays, which are known to have their origin at the surface on which the cathode rays impinge. The mathematical theory has been worked out by J. J. Thomson^[97]. If the motion of an electron is suddenly arrested, a thin spherical pulse in which the magnetic and electric forces are very intense travels out from the point of impact with the velocity of light. The more suddenly the electron is stopped, the thinner and more intense is the pulse. On this view the X rays are not corpuscular like the cathode rays, which produce them, but consist of transverse disturbances in the ether, akin in some respects to light waves of short wave-length. The rays are thus made up of a number of pulses, which are non-periodic in character, and which follow one another at irregular intervals.

On this theory of the nature of the X rays, the absence of direct deflection, refraction, or polarization is to be expected, if the thickness of the pulse is small compared with the diameter of an atom. It also explains the non-deflection of the path of the rays by a magnetic or electric field. The intensity of the electric and magnetic force in the pulse is so great that it is able to cause a removal of an electron from some of the atoms of the gas, over which the pulse passes, and thus causes the ionization observed.

The cathode rays produce X rays, and these in turn give rise to a secondary radiation whenever they impinge on a solid body. This secondary radiation is emitted equally in all directions, and consists partly of a radiation of the X ray type and also of electrons projected with considerable velocity. This secondary radiation gives rise to a tertiary radiation and so on.

Barkla^[98] has shown that the secondary radiation emitted from a gas through which the rays pass consists in part of scattered X rays of about the same penetrating power as the primary rays as well as some easily absorbed rays.

Part of the cathode rays is diffusely reflected on striking the cathode. These scattered rays consist in part of electrons of the same speed as in the primary beam, but also include some others of much less velocity. The amount of diffuse reflection depends upon the nature of the cathode and the angle of incidence of the rays.

We shall see later (chapter IV.) that similar effects are produced when the rays from radio-active substances impinge upon solid bodies.

In this chapter an account of the ionization theory of gases has been given to the extent that is necessary for the interpretation of the measurements of radio-activity by the electric method. It would be out of place here to discuss the development of that theory in detail, to explain the passage of electricity through flames and vapours, the discharge of electricity from hot bodies, and the very complicated phenomena observed in the passage of electricity through a vacuum tube.

For further information on this important subject, the reader is referred to J. J. Thomson’s Conduction of Electricity through Gases, in which the whole subject is treated in a full and complete manner. A simple account of the effect of moving charges and the electronic theory of matter was given by the same author in the Silliman Lectures of Yale University and published under the title Electricity and Matter (Scribner, New York, 1904).