Page:SahaElectrodynamics.djvu/34

1. From the second of equations A); it follows that an electrical force Y, and a magnetic force N produce equal deflexions of an electron moving with the velocity v, when $Y={\frac {Nv}{c}}$. Therefore we see that according to our theory, it is possible to obtain the velocity of an electron from the ratio of the magnetic deflexion A_m, and the electric deflexion A_e, by applying the law :—

${\frac {A_{m}}{A_{e}}}={\frac {v}{c}}$.

This relation can be tested by means of experiments because the velocity of the electron can be directly measured by means of rapidly oscillating electric and magnetic fields.

2. From the value which is deduced for the kinetic energy of the electron, it follows that when the electron falls through a potential difference of P, the velocity v which is acquired is given by the following relation :—

$P=\int \ Xdx={\frac {m}{e}}c^{2}\left[{\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right]$.

3. We calculate the radius of curvature R of the path, where the only deflecting force is a magnetic force N acting perpendicular to the velocity of projection. From the second of equations A) we obtain :

$-{\frac {d^{2}y}{dt^{2}}}={\frac {v^{2}}{R}}={\frac {e}{m}}{\frac {v}{c}}N{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$,

or

$R={\frac {mv\beta c}{eN}}$

These three relations are complete expressions for the law of motion of the electron according to the above theory.