Motion of a charged particle in a magnetic field:
Suppose a charged particle of charge 'q' travels with velocity v in magnetic field B, then that particle experiences a force F is given by
For the above equation we consider the three cases
1. Suppose if the charged particle is at rest in a magnetic field, i.e., v=0, then the charged paticle doesn't experience any force.
2. If the charged particle is moved along the direction of the magnetic field, then also the particle doesn't experience any force because velocity 'v' and magnetic field 'B' are parallel.
3. If the charged particle is moving perpendicular to the magnetic field, then the charged particle experiences maximum force.
Now we have to consider the motion of the charged particle in a magnetic field as shown below figure.
At that instant, consider the case of a charged particle goes in a magnetic field at a right angle to the magnetic field. As the charged particle is moved perpendicular to the magnetic field, then the force acted upon the charged particle is perpendicular to the direction of motion of particle and direction of field such that there is no work done by the magnetic field on the charged particle which shows that the particle does not get in any kinetic energy. As a result, the velocity of the charged particle remains unchanged. But the direction of the charged particle is changed. It comes about in practical. Here three parameters v, B and F remain constant. At that instant we treat F as Fmax or Fm. This force provides necessary centripetal force for circular path of charged particle.
Now we get Fmax or Fm=qvB-------2
Let the acceleration come from the above force is 'a', then
From equation 3, we get
Then according to the angular velocity definition, angular velocity w=qB/m.
The frequency here we found that is indenpendent of v. This frequency can be known as cyclotron frequency.
Some particles are faster and some particles are slower. Therefore, faster particles can move in bigger circle, on other hand slower particles can move in smaller circle.
Now we consider a case, in which a charged particle makes angle with direction of magnetic field as shown in the below figure.
Then we can analyse that velocity can be resolved in two components as Vx and Vy. Vx is taken along B and Vy is taken along normal to B. For figure(1), we can know that the angle between Vx and B is zero such that Vx is uneffected by the field. Therefore, the particle will move in the direction of magnetic field with constant velocity. Where as the component Vy is perpendicular to the magnetic field B, in this case the particle takes circular path in Y-Z plane. The radius of the circle is given as
Altogether, we can result for a particle which is moving by making an angle with magnetic field B as helix. This is the superposition of two motions in which one is in the direction of magnetic field with constant velocity and second is circular motion which is perpendicular to magnetic field B. We can find the pitch of the helix as the distance between any two successive circular paths. Therefore
Pitch p=velocity*time to describe a circular path
The component of Lorentz force in the direction always becomes zero because
As Vx remains constant, therefore
The angle between Vy and B becomes 90 degrees because Fy is maximum.
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Thanks for this article. Will this motion be valid if the velocity of the particle is variable?