0.7 Motion of a charged particle in magnetic field  (Page 7/5)

Motion of a charged particle in magnetic field is characterized by the change in the direction of motion. It is expected also as magnetic field is capable of only changing direction of motion. In order to keep the context of study simplified, we assume magnetic field to be uniform. This assumption greatly simplifies the description and lets us easily visualize the motion of a charged particle in magnetic field.

Lorentz magnetic force law is the basic consideration here. Hence, we shall first take a look at the Lorentz magnetic force expression :

$$\mathbf{F}=q\left(\mathbf{v}X\mathbf{B}\right)$$

We briefly describe following important points about this expression :

1: There is no magnetic force on a stationary charge (v=0). As such, our study here refers to situations in which charge is moving with certain velocity in the magnetic field. This condition is met when the charge is released with certian velocity in the magnetic field.

2: The magnetic field ( B ) is an uniform stationary magnetic field for our consideration in the module. It means that the magnitude and direction of magnetic field do not change during motion. The charged particle, however, is subjected to magnetic force acting side way. The direction of motion of charged particle, therefore, changes. In turn, the direction of magnetic force being perpendicular to velocity also changes. Important point to underline here is that this loop of changing directions of velocity and magnetic force is continuous. In other words, the directions of both velocity and magnetic force keeps changing continuously with the progress of motion.

This aspect of continuous change is shown in the figure below. Note that direction of magnetic field is fixed in y-direction. Initially, the charged particle is at the origin of coordinate reference with a velocity v in x-direction. Applying right hand rule for vector cross product and considering a point positive charge, we see that magnetic force is directed in z-direction. As a result, the particle is drawn to move along a curved path with velocity (having same speed) directed tangential to it. The magnetic force vector also changes sign being perpendicular to velocity vector. In this manner, we see that the directions of both velocity and magnetic force keeps changing continuously as pointed out.

Motions of a charged particle in magnetic field

3: The nature of motion depends on the initial directions of both velocity and magnetic field. The initial angle between velocity and magnetic field ultimately determines the outcome i.e. nature of motion.

We shall, therefore, discuss motion of charged particular on the basis of the enclosed angle (θ) between velocity and magnetic field vectors. There are following three cases :

• The motion of the charged particle is along the direction of magnetic field.

• The motion of the charged particle is perpendicular to the direction of magnetic field.

• The motion of the charged particle is neither along nor perpendicular to the direction of magnetic field.