91. Determination of the constants of the rays. If the deviation of the rays in both an electric and magnetic field is known, the values of the velocity of the rays, and the ratio e/m of the charge of the particle to its mass can be determined by the method, first used by J. J. Thomson for the cathode rays, which is described in section 50. From the equations of a moving charged body, the radius of curvature [Greek: rho] of the path of the rays in a magnetic field of strength H perpendicular to the path of the rays is given by
H[Greek: rho] = (m/e)V.
If the particle, after passing through a uniform magnetic field for a distance l_{1}, is deviated through a small distance d_{1} from its original direction,
2[Greek: rho]d_{1} = l_{1}^2
or d_{1} = (l_{1}^2/2) (e/m) (H/V) (1).
If the rays pass through a uniform electric field of strength X and length l_{2} with a deviation d_{2},
d_{2} = (1/2) (Xel_{2}^2})/(mV^2) (2),
since Xe/m is the acceleration of the particle, at right angles to its direction, and l_{2}/V is the time required to travel through the electric field.
From equations (1) and (2)
V = (d_{1}/d_{2}) (l_{2}^2/l_{1}^2) (X/H),
and e/m = (2d_{1}/l_{1}^2) (V/H).
The values of V and e/m are thus completely determined from the combined results of the electric and magnetic deviation. It was found that
V = 2·5 × 10^9 cms. per sec.
e/m = 6 × 10^3.
On account of the difficulty of obtaining a large electrostatic deviation, these values are only approximate in character.
