XVII. A Theory of the Connexion between Cathode and Röntgen Rays. By J. J. Thomson, M.A., F.R.S., Cavendish Professor of Experimental Physics, Cambridge [Communicated by the Author.]. A moving electrified particle is surrounded by a magnetic field, the lines of magnetic force being circles having the line of motion of the particle for axis. If the particle be suddenly stopped, there will, in consequence of electromagnetic induction, be no instantaneous change in the magnetic field ; the induction gives rise to a magnetic field, which for a moment compensates for that destroyed by the stopping of the particle. The new field thus introduced is not, however, in equilibrium, but moves off through the dielectric as an electric pulse. In this paper we calculate the magnetic force and electric intensity carried by the pulse to any point in the dielectric. The distribution of magnetic force and electric intensity around the moving particle depends greatly on the velocity of the particle, if this velocity is so small that the square of its ratio to the velocity of light can be neglected, then the electric intensity is symmetrically distributed round the particle, and at a distance r from it is equal to e/r2 , where e is the charge on the particle ; the lines of magnetic force are circles with the line of motion of the particle for axis ; the magnitude of the magnetic force at a point P is we sin 6/r 2 , where w is the velocity of the particle, and [theta] the angle a radius from the particle to P makes with the direction of motion. When, however, the velocity of the particle is so great that we can no longer neglect the square of its ratio to the velocity of light, the distribution of electric intensity is no longer uniform, the electric intensity, along with the magnetic force, tends to concentrate in the equatorial plane, that is, the [header] 173 plane through the centre of the particle at right angles to its direction of motion ; this tendency increases with the velocity of the particle until, when this is equal to the velocity of light, both the magnetic force and the electric intensity vanish at all parts of the field except the equatorial plane, and in this plane they are infinite. The pulses started by the stopping of the charged particle are, as might be expected, different when the ratio of the velocity of the particle to that of light is small, and when it is nearly unity. But even when the velocity is small, the pulse started by stopping the particle carries to an external point a disturbance in which the magnetic force is enormously greater than it was at the same point before the particle was stopped. The time the pulse takes to pass over a point P is, if the charged particle be spherical, equal to the time light takes to pass over a distance equal to the diameter of this sphere ; the thickness of this pulse is excessively small compared with the wave-length of visible light. When the velocity of the particle approaches that of fight two pulses are started when it is stopped. One of these is a thin plane sheet whose thickness is equal to the diameter of the charged particle ; this wave is propagated in the direction in which the particle was moving ; there is no corresponding wave propagated backwards : the other is a spherical pulse spreading outwards in all directions, whose thickness is again equal to the diameter of the charged particle, and thus, if this particle is of molecular dimensions, or perhaps even smaller, very small compared with the wave-length of ordinary light. The theory 1 wish to put forward is that the Röntgen rays are these thin pulses of electric and magnetic disturbance which are started when the small negatively charged particles which constitute the cathode rays are stopped. We shall now proceed to calculate the disturbance propagated through the dielectric when a charged particle is suddenly stopped. The components of the magnetic force and the electric intensity all satisfy Poisson's equation [formula redacted] the solution of this equation was shown by Poisson to be [formula redacted] where [phi] is the value of the function at a point P at the time t ; [omega]1 the mean value of [phi] when t = 0 over the surface [footer] 174 [header] of a sphere whose centre is at P, and whose radius is Vt ; [omega] 2 is the mean value of [formula redacted] when t = 0 over the surface of the same sphere. Let t = 0 be the time when the particle is suddenly brought to rest. Take the centre of the particle when it is brought to rest as the origin of coordinates, and the line of motion of the centre of the particle as the axis of z. Then [alpha], [beta], [gamma], the components of the magnetic force when the particle is stopped, are for all points outside the particle given by the equation [citation redacted] [formula redacted] At all points inside the particle, which we shall take to be a sphere of radius a, [alpha] = [beta] = [gamma] = 0. In these equations V is the velocity of light through the dielectric, iv the velocity of the charged sphere before it was stopped, e the charge on the sphere. To get the values of [alpha], [beta], [gamma] at any time after the particle is stopped, we have by Poisson's method to integrate the values just given over the surfaces of certain spheres ; in the general case this integration leads to complicated elliptic integrals. We shall get a clearer idea of the physical nature of the disturbance if we consider two special cases, (1) when we can neglect the square and higher powers of w/V; (2) when w/V is very nearly unity. In the first case, when we neglect w2/Y2, [alpha], [beta], d[alpha]/dt, d[beta]/dt when t = 0 all satisfy Laplace's equation, hence the mean value of any of these quantities over the surface of a sphere which does not enclose the origin, nor cut through any part of the electrified sphere, is equal to the value of this quantity at the centre of the sphere ; we can easily see, too, that when the sphere entirely surrounds the electrified sphere the mean value of any of these quantities over its surface is zero. Thus we have, by Poisson's solution, the following values for the components of the magnetic force after a time t from the stoppage of the electrified sphere, [header] 175 [formula redacted] As we are neglecting w2 we may leave out the second terms in these equations. These values hold from t to t=(r-a)/V. When t>(r + a)/V, [alpha]=[beta]=0 We must now allow for the absence of magnetic force inside the sphere of radius a ; the easiest way to do this is to suppose that the expressions (1) hold right up to the centre of this sphere, and superpose on the distribution represented by (1) a distribution inside the sphere given by [formula redacted] where [formula redacted]; while outside the sphere we have for this distribution [alpha]=[beta]=0 If we superpose this distribution we may suppose that at any time [formula redacted] where [alpha]1, [beta]1 are the values given by equations (2) which may be now supposed to hold from t = 0 to t = r/V, while, when t > r/V, [alpha]1, [beta]1 both vanish ; [alpha]2 , [beta]2 are the magnetic forces arising from the disturbance given initially by (3). This disturbance will begin to be felt at a point P at a time [formula redacted], and will cease after a time [formula redacted]; is the centre of the charged sphere. Thus the thickness of the pulse due to this distribution is equal to the diameter of the sphere. We can easily show that [formula redacted] taken over the part of a sphere whose centre is at P and radius is Vt, which is within the sphere whose radius is a, is equal to [formula redacted] where x is the x coordinate of P. [footer] 176 [header] Hence [omega]1, the mean value of wex/r 3 , is [formula redacted] Thus [formula redacted] Hence [formula redacted] from [formula redacted] to [formula redacted]. If the sphere is small, Vt/a is large compared with unity, and Vt is approximately equal to OP ; hence [formula redacted] [formula redacted] [alpha]2 and [beta] 2 are, when a is small, very large compared with [alpha]1 and [beta]1. We have now the complete solution of the problem, and we see that after the sphere is stopped, the magnetic force at a point P remains unaltered until t=(r — a)/V, when a very thin pulse of intense negative magnetic force arrives, the intensity of the field being ew sin [formula redacted]. OP, where [theta] is the angle between OP and the axis of z ; the magnetic force previously at P was in the opposite direction, and equal to [formula redacted] . This very intense pulse only lasts for a very short time ; and the view I wish to put forward is that this pulse constitutes one kind of Röntgen radiation. The reasons for this view will be given after we have considered the case of a sphere moving with the velocity of light. We may, however, point out that since the state represented by [alpha]1/[beta]1 lasts for the time r/V, while that for the state [alpha]2, [beta]2 only for the time 2a / V, [formula redacted] This must evidently be the case, for the line integral of the magnetic force round a circuit is equal to 4[pi] times the current through the circuit ; in this case the currents are dielectric currents, and equal the rate of increase of the electric displacement through the circuit, so that the time integral of [header] 177 the line integral is equal to the change in the displacement ; but when we neglect w2/ V2 , the distribution of the displacement is the same when the sphere is moving as when in the steady state at rest : thus the time integral must vanish. Let us now consider the case when the velocity of the particles is nearly equal to that of light. In the limiting case when w=V we see, from the expressions given for a, ft, that they vanish unless z=0, when they become infinite ; in this case the original magnetic field is confined to a plane through the centre of the sphere at right angles to its direction of motion. When w is nearly but not quite equal to V, the disturbance is practically confined between the two cones whose semi-vertical angles are [formula redacted] and [formula redacted], where $ is a small angle. To simplify the analysis and yet retain the essential physical features of the case, we shall suppose that the initial disturbance, instead of being confined between these two cones, is confined between the planes [formula redacted] and [formula redacted], where d is a small quantity; and that both the magnetic force and the electric intensity are parallel to the planes, the lines of electric intensity being radial at right angles to the axis of z, and the lines of magnetic force circles with their centres on the axis of z. Let E be the electric intensity at a point distant p from this axis ; then the total normal induction over the surface of a cylinder passing through this point and with the axis of z for its axis is equal to [formula redacted] this must equal 4[pi]e ; hence [formula redacted] Hence if [alpha], [beta] are the components of the magnetic force just after the particle is stopped, [formula redacted] Both d[alpha]/dt and d[beta]/dt are zero except when [formula redacted], when they are infinite. These equations give the initial state of the field outside the charged particle ; inside this particle, which we shall take to 178 [header] be a sphere of radius d, we shall suppose that the electric intensity and the magnetic force both vanish. Thus the original distribution of the field is confined between two parallel planes; and from this space we must exclude that inside the sphere as this is free from magnetic force. Let us now consider how this distribution will spread through space. Consider what will happen at a point P. There will be no effect at P until a sphere of radius Vt and centre P cuts the space between the planes. This will not happen until [formula redacted], where c is the distance of P from the plane through the centre of the sphere perpendicular to the direction in which the sphere was moving before it was stopped. When t is greater than this value, the sphere will cut the space between the planes ; and to apply Poisson's solution we have to find the mean value of the magnetic force over the surface of this sphere. Take the plane of xz to pass through P. Let Q be a point on the surface of the sphere, dS an element of the area of this surface, [phi] the angle the plane through Q and the axis of z makes with the plane of xz, p the distance of Q from the axis of z, and [theta] the angle between p and the normal to tbe sphere at Q ; then the element of the surface included between z and z + dz, [phi] and [phi] + d[phi] is given by the equation [formula redacted] Now initially [formula redacted] so that [formula redacted] Now if a is written for Vt the radius of the sphere, and if the x coordinate of P is b, then we may easily prove that [formula redacted] hence [formula redacted] [formula redacted] The limits of [phi] are [formula redacted] [header] 179 In finding the mean value of [beta] over the sphere we must double this value, for to each value of [beta] and z there correspond two elements of the surface of the sphere which contribute equally to the integral ; hence [formula redacted] Now the limits of z depend upon whether the sphere does not or does cut right through the slab between the two parallel planes ; in the former case Vt is less than c + d, and the limits of z are c-Vt and d ; then [formula redacted]; in the latter case Vt is greater than c + d, and the limits of z are — d and + d ; hence in this case [formula redacted]. Hence [omega]1, the mean value of the initial value of [beta] over the surface of this sphere, is [formula redacted] in the first case, and [formula redacted] in the second ; hence [formula redacted] according as Vt < or > c + d. This value of [formula redacted] is the same whether the point P is in front or behind the plane. We now proceed to find the value of [formula redacted] Now d[beta]/dz is zero except at the surface of the plane ; hence 180 [header] when the sphere cuts z = d and not z = — d, we have [formula redacted] When the sphere cuts both z = d and z=—d, then [formula redacted] Thus [omega]2, the mean value of the initial value of d[beta]/dt over the surface of the sphere, is given by the equation [formula redacted], when the sphere cuts z = d and not z = —d, [formula redacted] when the sphere cuts z = — d and not z = d, 0 = when the sphere cuts both. Hence by Poisson's formula [formula redacted] when the sphere cuts z = d and not z=—d, = 0 when the sphere cuts z= —d and not z — d, = 0 when the sphere cuts z = d and also z= —d. Thus the distribution of magnetic force between the planes z = + d is propagated forwards unchanged with the velocity V, there is no corresponding pulse propagated in the negative direction. In addition to the plane pulse there will also, as in the previous case, be a spherical one, whose thickness is 2d ; we can calculate the magnetic force at any point in this pulse as follows : — Let H be the magnetic force at a point in this pulse at a distance b from the axis of z, then the line integral of this magnetic force round the circle whose radius is b and whose axis is the axis of z is [formula redacted] ; the magnetic force lasts for a time 2d/V, so that the time integral of the line integral is [formula redacted]. At any point in front of the particle the time integral of the magnetic force due to the plane pulse round the same circuit is [formula redacted] Hence the time integral of the whole magnetic force round [header] 181 this circuit is equal to [formula redacted]. This is equal to 4[pi] times the change in the electrostatic polarization through the same circuit : now when the particle was stopped, this polarization was zero, and when the field has reached a steady state, the electric intensity is uniformly distributed, so that the polarization through the circle is [formula redacted], where [theta] is the acute angle between OP and the axis of z, P being a point on the circumference of the circle ; hence [formula redacted] or [formula redacted] where r is the distance OP. The minus sign denoting that the magnetic force in the spherical pulse is in the opposite direction to that in the plane pulse. At a point behind the charged particle there is no plane pulse, so that [formula redacted], where [theta]' is the acute angle between OP and the axis of z ; thus [formula redacted] hence if [theta] is the angle between OP and the positive direction of the axis of z, the magnetic force at any point in the spherical wave is given by [formula redacted] Thus we see that the stoppage of a charged particle will give rise to very thin pulses of intense magnetic force and electric intensity; when the velocity of the particle is small there will be one spherical pulse; when the velocity is nearly equal to that of light there will in addition to the spherical pulse be a plane one propagated only in the direction in 182 [header] which the particle was originally moving. It is these pulses which I believe constitute the Röntgen rays. As they consist of electric and magnetic disturbances, they might be expected to produce some effects analogous to those of light. If they were so thin that the time taken by them to pass over a molecule of a substance were small compared with the time of vibration of the molecule, there would be no refraction, and the thinness of the pulse would also account for the absence of diffraction. In the preceding investigation we have supposed that the stoppage of the particle is instantaneous ; if the impact lasts for a finite time T the negative pulse will be broadened out, so that its thickness, instead of being 2a, will be 2a + VT, where V is the velocity of light. The intensity of the magnetic force in the pulse will vary inversely as the thickness of the pulse, so that when the collision lasts for the time T, the magnetic force in the negative pulse will be 2a/(2a + VT) of the value given above. The more sudden the collision, the thinner the pulse and the greater the magnetic force and the energy in the pulse ; the pulse will, however, possess the properties of the Röntgen rays until T is comparable to one of the times of vibration of a substance through which it has to pass. In the case of the cathode rays all the circumstances seem favourable to a very sudden collision, as the mass of the moving particles is very small and their velocity exceedingly great. In some experiments which I described in the Philosophical Magazine for Oct. 1897 on cathode rays, the velocity of the negative particles was about one third of that of light, and in some more recent experiments made on the Lenard rays, with the apparatus described by Des Coudres, considerably higher velocities were found. A change in the time of the collision will alter the thickness of the pulse and so change the nature of the ray. If we suppose that part of the absorption of the rays is due to the communication of energy to charged ions in their path, we find that the thicker the pulse the greater the absorption. For suppose that E is the electric intensity in the pulse, m the mass, and e the charge on an ion ; then if u is the velocity communicated to the ion when the pulse passes over it, t the time taken by the pulse to pass over it, [formula redacted] or if d is the thickness of the pulse [formula redacted]; [header] 183 thus the energy 1/2mu2 communicated to the ion is equal to [formula redacted] Now the energy in the pulse is proportional to E2d/V2 , so that the ratio of the energy communicated to the ion to the energy in the pulse is proportional to d. Thus the broader the pulse, the greater the absorption and the less the penetrating power. The energy in the pulse is inversely proportional to its thickness. If we return to the expression for the intensity of the magnetic force in case (1), we see that it is proportional to sin [theta], so that the disturbance is greatest at right angles to the cathode rays : thus, if the cathode particles are stopped at their first encounter, the Röntgen rays would be brightest at right angles to the cathode rays; if, however, as would seem most probable, the cathode particles had to make several encounters before they were reduced to rest, changing their direction between each encounter, the distribution of the cathode rays would be much more uniform. Experiments on the distribution of Röntgen rays produced by the impact of the cathode particles directly against the walls of the discharge-tube are, as Sir George Stokes has pointed out, affected by the much greater absorption of the oblique rays produced by the greater thickness of glass traversed by them. Experiments on rays produced by focus-tubes would give results more easily interpreted. The result to which we have been led from the consideration of the effects produced by the sudden stoppage of an electrified particle, viz. : that the Röntgen effects are produced by a very thin pulse of intense electromagnetic disturbance, is in agreement with the view expressed by Sir George Stokes in the Wilde Lecture ([formula redacted]), that the Röntgen rays are not waves of very short wave-length, but impulses. Cambridge, Dec. 16, 1897.