FOCUSING OF THE PARAXIAL BEAM OF THE CHARGED PARTICLES IN A MAGNETIC FIELD.

When a charged particle moves in a magnetic field then the Lorenz force F_L = (q/c)[vB] is acting upon it perpendicularly the velocity. As a result the particle will move in a spiral curve. Angle frequency of rotation equals ω = V/R = qB/mc. Radius of the rotation equals R = mcV/qB. Along the direction of the magnetic field the particle will move without acceleration.

Let us consider a paraxial beam of the charged particles in a magnetic field. The direction of the magnetic field coincides with direction of the beam. The velocity of a particle in the direction of the field equals V_B=V·cosα, where α is a small angle between the initial velocity of the particle and the direction of the magnetic field. For one cycle of rotation T = 2π/ω = 2πmc/qB the particle moves at the distance L = V_BT = 2πc·Vcosα/(q/m)B along magnetic field. Angle α is a small and for this reason  cosα ≈ 1 č L ≈ 2πcV/(q/m)B. So we can see that distance L does not depend upon the angle α and after one cycle of rotation all the particles will come to the same point which is the focus of the beam. This phenomena can be used for many applications, particularly to determine the ratio of the charge of a particle to the mass of this particle.