# Motion of Charged Particles in an Electromagnetic Field

The motion of charged particles in an electromagnetic field is of great practical importance. It is used in observation instruments (oscilloscopes, electron microscopes etc.), accelerators, mass spectroscopy, the investigation of nuclear and particle reactions, etc. It is also important in some other fields of physics: plasma physics, astrophysics, cosmic ray physics, electronics, etc. In this chapter we analyze the simplest problems of motion in uniform electric and magnetic fields both in Newtonian and relativistic mechanics. We also consider some simple applications.

## 14.1. Motion of a charged particle in an electric field

In this section we consider the motion of a charged particle in a uniform electric field E. The relevant effect is its acceleration and deviation, which is widely used in observation and measurement instruments (oscilloscopes, television sets, etc.).

A) Non-relativistic analysis

An electric field E exerts a force F = qE on a particle of mass m and charge q. Thus, its equation of motion is

If the field E is uniform, the acceleration a is constant. The equation of motion [14.1] may be integrated to give the velocity and the position

where vo and ro are the initial velocity and the initial position, ...

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