The force acting on the particle is given by the familiar Lorentz law:

(194) |

It turns out that we can eliminate the electric field from the above equation by
transforming to a different inertial frame. Thus, writing

(196) |

where we have made use of a standard vector identity (see Section A.10), as well as the fact that . Hence, we conclude that the addition of an electric field perpendicular to a given magnetic field simply causes the particle to drift perpendicular to both the electric and magnetic field with the fixed velocity

irrespective of its charge or mass. It follows that the electric field has no effect on the particle's motion in a frame of reference which is co-moving with the so-called

Let us suppose that the magnetic field is directed along the -axis.
As we have just seen, in the
frame, the particle's equation of motion reduces to Equation (197), which can be written:

Here,

(202) |

where we have judiciously chosen the origin of time so as to eliminate any phase offset in the arguments of the above trigonometrical functions. According to Equations (203)-(205), in the frame, our charged particle gyrates at the cyclotron frequency in the plane perpendicular to the magnetic field with some fixed speed , and drifts parallel to the magnetic field with some fixed speed . The fact that the cyclotron frequency is positive for positively charged particles, and negative for negatively charged particles, just means that oppositely charged particles gyrate in opposite directions in the plane perpendicular to the magnetic field.

Equations (203)-(205) can be integrated to give

where we have judiciously chosen the origin of our coordinate system so as to eliminate any constant offsets in the above equations. Here,

(209) |

We conclude that the general motion of a charged particle in crossed electric and magnetic field is a combination of drift [see Equation (198)] and spiral motion aligned along the direction of the magnetic field--see Figure 12. Particles drift parallel to the magnetic field with constant speeds, and gyrate at the cyclotron frequency in the plane perpendicular to the magnetic field with constant speeds. Oppositely charged particles gyrate in opposite directions.