moving freely with a velocity u. If u is small compared with the velocity of light, the ion in motion corresponds to a current element of magnitude eu. If the ion moves in an external magnetic field of strength H, it is acted on by a force at right angles both to the direction of motion, and to that of the magnetic force and equal in magnitude to Heu sin [Greek: theta], where [Greek: theta] is the angle between the direction of the magnetic force and the direction of motion. Since the force due to the magnetic field is always perpendicular to the direction of motion, it has no effect upon the velocity of the particle, but can only alter the direction of its path.
If [Greek: rho] is the radius of curvature of the path of the ion, the force along the normal is equal to mu^2/ρ, and this is balanced by the force Heu sin [Greek: theta].
If [Greek: theta] = [Greek: pi]/2, i.e. if the ion is moving at right angles to the direction of the magnetic field Heu = mu^2/[Greek: rho] or H[Greek: rho] = (m/e)u. Since u is constant, [Greek: rho] is also constant, i.e. the particle describes a circular orbit of radius [Greek: rho]. The radius of the circular orbit is thus directly proportional to u, and inversely proportional to H.
If the ion is moving at an angle [Greek: theta] with the direction of the magnetic field, it describes a curve which is compounded of a motion of a particle of velocity u sin [Greek: theta] perpendicular to the field and u cos [Greek: theta] in the direction of the field. The former describes a circular orbit of radius [Greek: rho], given by H[Greek: rho] = (m/e)u sin [Greek: theta]; the latter is unaffected by the magnetic field and moves uniformly in the direction of the magnetic field with a velocity u cos [Greek: theta]. The motion of the particle is in consequence a helix, traced on a cylinder of radius [Greek: rho] = mu sin [Greek: theta]/(eH), whose axis is in the direction of the magnetic field. Thus an ion projected obliquely to the direction of a uniform magnetic field always moves in a helix whose axis is parallel to the lines of magnetic force[1].
- ↑ A full account of the path described by a moving ion under various conditions is given by J. J. Thomson, Conduction of Electricity in Gases (Camb. Univ. Press, 1903), pp. 79-90.
