# Rays of Positive Electricity and Their Application to Chemical Analyses/Electrostatic Deflection of the Particle

Let us suppose as before that the particle is projected with a velocity v parallel to the axis of x: let the electric force acting on the particle be parallel to the axis of z and equal to Z, then the equation of motion of the particle under the electric force is
    $m{\operatorname {d^{2}} z \over \operatorname {d} t^{2}}=eZ$
When the deflection is small,
    ${\operatorname {d^{2}} z \over \operatorname {d} t^{2}}=v^{2}{\operatorname {d^{2}} z \over \operatorname {d} x^{2}}$
approximately, and hence
    $mv^{2}{\operatorname {d^{2}} z \over \operatorname {d} x^{2}}=eZ$
    or
    $z={\frac {e}{mv^{2}}}\int \limits _{0}^{l}{\bigg\int \limits _{0}^{x}Z\,dx{\bigg )}\,dx$
    $\qquad ={\frac {e}{mv^{2}}}B$
thus B is quite independent of the charge, mass, or velocity of the particle, and depends merely on the distribution of the electric field and the distance from the point of projection at which the deflection is measured.
A very convenient method of producing the electric field is to have two parallel plates perpendicular to the axis of z; in this case the electric field is approximately constant between the plates and vanishes outside them. If b is the length of the plates measured parallel to the axis of x, and if one end of the plates just comes up to the point from which the particle is projected, putting Z=Z from x=0 to x=b, and Z=0 from
x= b to x=l, we find that
    $B=Zb{\Bigl-{\frac {b}{2}}{\Big )}$
so that If z is the deflection when x=l
    $z={\frac {e}{mv^{2}}}\,Zb\,{\Bigl-{\frac {b}{2}}{\Big )}.$
The electric field Is not absolutely constant between the plates, It is greater close to the edges than in other parts of the field, nor it absolutely vanish at all places outside the plates; when great accuracy is required these points have to be taken Into account in the calculation of B. A method by which this may be done was given by the author in the "Phil. Mag.," VI, vol. xx, p, 752.
If the particle is simultaneously acted on by magnetic and electric forces parallel to the axis of z, we may, If the deflections are small, superpose the effects due to the magnetic and electric forces, so that the y, z deflections of the particle parallel to the axis of y and z respectively are given by the equation
    $y={\frac {e}{mv}}\,A\,\qquad (1)$
    $z={\frac {e}{mv^{2}}}\,B\,\quad (2)$
Thus if we had a stream of charged particles of different kinds (i.e. with different values of e/m) projected from the origin with different velocities parallel to the axis of x, in the absence of electric and magnetic forces they would all strike a at x = l at the same point. When, however, they are submitted to the action of electric and magnetic forces they get sorted out, and no two particles strike the same point on the screen unless they are moving at the same speed and also have the same value of e/m. If we know the deflected of the particle we can by equations (1) and (2) calculate both the values of v and also the value of e/m; we have from these equations
    $v={\frac {y}{z}}\,{\frac {B}{A}}\qquad (3)$
    ${\frac {e}{m}}={\frac {y^{2}}{z}}\,{\frac {B}{A^{2}}}\quad (4)$
Thus y/z will be constant for all particles moving with a given speed whatever may be their charge or mass, hence all such particles will strike the screen in a straight line passing through the undeflected position of the particles.
Again for the same kind of particle y$^2$/z is constant whatever may be the velocity of the particles, hence particles of the same kind will all strike the screen in a parabola with its vertex at the undeflected position of the particles, and there will be as many of these parabolas as there are different kinds of particles.