0.7 Motion of a charged particle in magnetic field  (Page 2/5)

Motion of the charged particle along magnetic field

There are two possibilities. The enclosed angle (θ) is either 0° or 180°. In either case, sine of the angle is zero. Therefore, magnetic force is zero and the motion of particle remains unaffected (of course here we assume that there is no other force field present).

Motion of the charged particle perpendicular to magnetic field

This is the case in which charged particle experiences maximum magnetic force. It is given by :

$$F=qBv\sin 90°=qvB$$

If the span of magnetic field is sufficient around the charged particle, then it will describe a circular path as magnetic force is always perpendicular to the motion. The magnetic force provides the centripetal force required for circular motion. The span of magnetic field around charged particle is important. Here, we consider some interesting cases as shown in the figure. For all cases, we assume that motion of charged particle is in the plane of the drawing and magnetic field is perpendicular and into the plane of drawing. Magnetic field is shown by evenly distributed X sign indicating that it is an uniform magnetic field directed into the plane of drawing.

Motions of a charged particle in magnetic field

In the first case, there is sufficient span of magnetic field around charged particle and it is able to describe circular path. In second case, the charged particle enters the region of magnetic field and never completes the circular trajectory. Similarly, the charged particle in third case also does not complete the circular path as it comes out of the region of magnetic field even before completing half circle.

Now, we consider the first case in which the charged particle is able to complete circular path. Let the mass of the particle carrying charge is m. Then, magnetic force is equal to centripetal force,

$$\frac{m{v}^2}{R}=qvB$$ $$\frac{mv}{R}=qB$$

The radius of circular path, R, is given as :

$$R=\frac{mv}{qB}$$

We can easily interpret the effects of various parameters in determining the radius of circular path. Greater charge and magnetic field result in smaller radius. On the other hand, greater mass and speed result in greater radius. Now, the time period of revolution is :

$$T=\frac{2\pi R}{v}=\frac{2\pi m}{qB}$$

Frequency of revolution is :

$$\nu =\frac{1}{T}=\frac{qB}{2\pi m}$$

Angular speed is :

$$\omega =2\pi \nu =\frac{2\pi qB}{2\pi m}=\frac{qB}{m}$$

Important aspect of these results is that properties related to periodicity of revolutions i.e. time period, frequency and angular velocity all are independent of the speed of the particle. It is a very important result which is used in cyclotron (to accelerate charged particle) to synchronize with the frequency of application of electric field. We shall learn about this in another module.

Specific charge

The ratio of charge and mass of the particle is known as specific charge and is denoted by α. Evidently, its unit is Coulomb/kg. This quantity is important in describing motion of charged particle in magnetic field. We observe that magnetic force is proportional to charge q, whereas acceleration of the particle carrying charge is inversely proportional to mass m. Clearly, the effects of these two quantities are opposite and hence they appear as the ratio q/m in most of the formula describing motion. Recasting formulas with specific charge, we have :