MOTION OF CHARGES IN CROSSED MAGNETIC AND ELECTRIC FIELDS

Let us consider the motion of a charge q in case of simultaneous presence of homogeneous and constant electric E and magnetic H fields, which are perpendicular to initial direction of charge motion and to each other (as shown in animation). We shall limit our consideration by not relativistic case, when speed of a charge V << c. For fulfillment of this condition the intensity of electric field E must be much less than intensity of magnetic field H. In this approximation the trajectory of a particle is a trochoid which can be presented as a sum of two motions: in the direction perpendicular to both fields the charge moves with constant drift speed V_d = cE/H and in the plane perpendicular to magnetic field it moves in a circle path with cyclotron frequency w = qH/mc and radius R = | (V₀-cE/H)/w |, where V₀ is the initial speed of a charge.

Depending the initial velocity the trajectory of the particle motion in the crossed electric and magnetic fields can be trochoid, cycloid or even a straight line. In the animation above the initial velocity of the particle equals to zero. It moves in a cycloid curve.

In the next animation the initial velocity of the particle V₀ > cE/H. The particle moves in a trochoid curve.

If there is no electric field, the particle will move in a spiral curve.