Rays of Positive Electricity and Their Application to Chemical Analyses/Rectilinear Propagation of the Positive Rays

This can be shown by placing a solid obstacle in the path of a pencil of positive rays: this casts a shadow on the part of the tube which was phosphorescing under the impact of these rays. Comparing the shape of the shadow with that of the obstacle, it is found that the shadow is very approximately the projection of the outside of the solid on the walls of the tube by lines passing through the hole in the cathode through which the pencil of positive rays emerges.

On The Nature of the Positive Rays, Their Deflection by Electric and Magnetic Forces
As cathode rays were proved to be negatively electrified particles by the study of the deflections they experience when acted on by magnetic and electric forces, and as these deflections gave the means of finding the mass and velocity of the cathode particles, it was natural to attempt to apply the same methods to the positive rays. It was not, however, until twelve years had elapsed since the discovery of the rays that any effect of a magnetic field on them was detected. A small permanent magnet held near a bundle of cathode rays produces a very appreciable effect; it has, however, no apparent action on the positive rays : as a matter of fact the deflection due to a magnetic field on the positive rays is at most about 2 per cent of the deflection of the cathode rays, the deflections being measured at equal distances from the cathode. In 1898, however, Wien, by the use of very powerful magnetic fields, proved that the positive rays were deflected by magnetic forces.$1$

Before discussing Wien's experiments it will be convenient to consider the theory of the deflection of a moving electrified particle by a magnetic field. The force acting on the moving particle is at right angles to the magnetic force, at right angles to the direction of motion of the particle and equal to eHvsinΦ, where H is the magnetic force at the particle, v the velocity of the particle, Φ the angle between H and v, and e the charge on the particle. Since this force is always at right to the direction of motion of the particle it will not alter the speed of the particle but only the direction in which it is moving. Suppose that the particle is originally projected with a velocity v parallel to the axis of x, and that it is moving in a field arranged so as to be very approximately in the direction of the axis of z, the direction of the force along the particle will be parallel to the axis of y and this will be the direction in which it will be deflected. If y is the deflection this direction at the time t, m the mass of the particle, H the magnetic force parallel to the axis of z, and e the charge carried by the particle, the equation of motion of the particle is


 * $$m{\operatorname{d^2}y\over\operatorname{d}t^2} = eH{\operatorname{d}x\over\operatorname{d}t}$$,

Integrating: this equation we get


 * $$m{\operatorname{d}y\over\operatorname{d}t} = \int\limits_{0}^{x} eH {\operatorname{d}x\over\operatorname{d}t} \, dt = \int\limits_{0}^{t} eH \, dx$$      (I)

if the origin of co-ordinates is taken at the point of projection; for since the particle was projected parallel to the axis of x, dy/dt=0 when x=0. Now if the deflection of the particle is small dx/dt will, neglecting the squares of small quantities, be equal to v, and dy/dt to v dy/dx. On this assumption equation (I) may be written


 * $$mv{\operatorname{d}y\over\operatorname{d}x} = \int\limits_{0}^{x} eH  \, dx $$  ;

hence if y is the deflection when x=l


 * $$mvy = \int\limits_{0}^{l}  \bigg\{   \int\limits_{0}^{x} eH dx \bigg\} dx $$.

integrating by parts, we get


 * $$mvy = l \int\limits_{0}^{l} eH  \, dx  -  \int\limits_{0}^{l} xeH  \, dx $$


 * $$\qquad = e \int\limits_{0}^{l} (l-x) \, H \, dx $$

or writing A for


 * $$\int\limits_{0}^{l} (l-x) \, H \, dx $$


 * $$y = \frac{e}{mv} A$$.

Where A depends merely upon the strength of the magnetic field and the distance from the point of projection at which the deflection is measured; it is quite independent of the charge mass, or velocity of the particle.

If the magnetic field is that between two poles of an electromagnet placed close together and reaching up to the point of projection of the particle, then if a is the breadth of the pole pieces, H is approximately constant from x=0 to x=a and vanishes from x=a to x=L. Substituting this value for H in the expression for A we find


 * $$A = a \Big( l - \frac{a}{2}  \Big) $$

when H is the magnetic force between the poles. When this approximation is not sufficiently accurate and we have to take into account the stray magnetic field beyond the poles as well as the variation of the magnetic force between the poles, A may be conveniently determined by the following method,$2$ Wind a coil of triangular section DEF, the base DF being equal to l, the angle EDF a right angle, and DE small compared with the depth of the pole pieces of the electromagnet. Place the coil so that DF is along the direction in which the particle is projected. D D being at the point of projection and F at the distance at which the deflection is measured, connect up the coil with a ballistic galvanometer, or what is more convenient with a Grassot flux meter, and determine the number of lines of magnetic force which pass through this coil when the electromagnet is made or broken; from this number we can easily determine the value of A. For if N is this number, then we see from Fig. 4 that


 * $$N =  \int\limits_{0}^{l} H  \, \times \, PN .  \, dx $$

and from the figure


 * $$\frac{PN}{DE} = \frac{FN}{FD} =  \frac{l-x}{l} $$


 * hence $$N =  \int\limits_{0}^{l} H  .\, \frac{DE}{l} (l-x)  \, dx $$


 * $$ = \frac{DE}{l}  \int\limits_{0}^{l} H (l-x) dx $$


 * $$ = \frac{DE}{l}  .A .$$

Thus when N is known A can be at once determined.

$1$W. Wien, "Verh. d. phys. Gesell.," 17,1898.

$2$ J. J. Thomson, " Phil. Mag.," VI, xvm, p. 844.