0.8 Motion of a charged particle in electric and magnetic fields

Motion of a charged particle in the simultaneous presence of both electric and magnetic fields has variety of manifestations ranging from straight line motion to the cycloid and other complex motion. Both electric and magnetic fields impart acceleration to the charged particle. But, there is a qualification for magnetic field as acceleration due to magnetic field relates only to the change of direction of motion. Magnetic force being always normal to the velocity of the particle tends to move the particle about a circular trajectory. On the other hand, electric force is along electric field and is capable to bring about change in both direction and magnitude depending upon the initial direction of velocity of the charged particle with respect to electric field. If velocity and electric vectors are at an angle then the particle follows a parabolic path.

One of the important orientations of electric and magnetic fields is referred as “crossed fields”. We use the term “crossed fields” to mean simultaneous presence of electric and magnetic fields at right angle. The behavior of charged particles such as electrons under crossed fields has important significance in the study of electromagnetic measurement and application (determination of specific charge of electron, cyclotron etc.).

Before we proceed, we should understand that elementary charged particles have mass of the order of ${10}^{-28}$ kg or less. Therefore, even small electric or magnetic force is capable to generate very high acceleration of the order of ${10}^{12}$ $m/s^2$ or more. Under proper set up, these particles achieve velocity comparable to speed of light. In order to keep our discussion in the simple classical context, however, we shall confine our discussion limited to the cases which are less complicated and which neglect relativistic effects.

Some of the important applications or phenomena associated with simultaneous presence of two fields include :

• Motion of a charged particle in electric and magnetic fields
• Measurement of specific charge of an electron (J.J.Thomson experiment)
• Acceleration of charged particles (cyclotron)

In this module, we shall study first two of the listed application or phenomena. The third one i.e. cyclotron will be discussed in a separate module.

Motion of a charged particle in electric and magnetic fields

We have already studied motion of charged particle in individual fields. Here, we shall combine the effects of two fields. Few of the interesting cases are discussed here.

Charged particle is moving along parallel electric and magnetic field

The velocity, electric and magnetic vectors are in in the same direction. Let they are aligned along x-axis. Since magnetic field and velocity vectors are parallel, there is no magnetic force.

$$F_M=v_{0}qB\sin 0°=0$$

where $v_0$ is initial speed of the particle. The charged particle is, however, acted upon by electric field. It is accelerated or decelerated depending on the polarity of charge and direction of electric field. Considering positive charge, the electric force on the charge is given as :