Motion of a Charged Particle in Magnetic Field

The motion of a charged particle in electric and magnetic fields behaves differently. So, what is the motion of a charged particle in a uniform magnetic field? Today, we will study the motion of a charged particle in a uniform magnetic field. To understand this concept in-depth, we must first understand how does magnetic field lines behave?. We know that both the magnetic and electric forces somehow influence the motion in a magnetic field. Although there will be a change in the trajectory of the particles in both the forces, the charged particle can be an ion or an atom with an electric charge. Below we will learn about the effects of the electric and magnetic force on a charged particle.

Electric Field Vs Magnetic Field Lines

We will use field lines to describe the motion of a charged particle in electric and magnetic fields. So let us start by understanding what these field lines are? There are many conceptual differences between the electric and magnetic field lines. The positively charged particle has an evenly distributed and outward-pointing electric field. For the negative charge, the electric field has a similar structure, but the direction of the field lines is inwards or reverse to that of the positive charge. The field lines create a direct tangent electric field. Also, if the charge density is higher, then the lines are more tightly packed to each other. We can notice that the electric field has no curl. In case both the charges are involved, then positive charges generate field lines, and negative charges terminate them. 

As we know, magnets consist of two poles north and south. So the field lines generate from the north pole and terminate at the south pole in the case of magnets. The charges in magnets are always bipolar, i.e. they always come with two poles (north and south) and never exist in a single-pole(monopole). It generates a non-zero curl for the ordinary magnets.  If the field lines do not have a perpendicular velocity component, then charged particles move in a spiral fashion around the lines. This concept is widely used to determine the motion of a charged particle in an electric and magnetic field. We can determine the magnetic force exerted by using the right-hand rule. Let us discuss the motion of a charged particle in a magnetic field and motion of a charged particle in a uniform magnetic field. 

The Motion of Charged Particle in Electric and Magnetic Field

So, you must be wondering how do we define the motion of a charged particle in a magnetic field and motion of a charged particle in a uniform magnetic field? Suppose if a charged particle is in motion, then the directional component of the force towards motion and the force on the particle performs some amount of work.  Let us consider this particle has a charge q and it moves in the direction of magnetic field B (motion in a magnetic field). And the velocity of the particle experiences a perpendicular magnetic force. In this case, the magnetic force does not perform any work on the particle, and hence there is no change in the velocity of the charged particle. Considering the velocity to be v and representing the mathematical equation of this particle perpendicular to the magnetic field where the magnetic force acting on a charged particle of charge q is

F = q(v x B).

Here, the magnetic force becomes centripetal force due to its direction towards the circular motion of the particle. Hence, if the field and velocity are perpendicular to each other, then the particle takes a circular path. Sometimes, the magnetic field and a velocity component are in the same direction. As a result of that, the particle does not experience any effect of the magnetic field, and its magnitude remains the same in the entire motion. The motion resulting from both of these components takes a helical path, as described in the diagram below. 

(image will be uploaded soon)

We can determine the centripetal force perpendicular towards the center while the given radius of the circular path by the particle is r. 

\[F = \frac{mv^{2}}{r}\]   --(1)

Both magnetic field and velocity experiences perpendicular magnetic force and its magnitude can be determined as follows.

\[F = qvB\]           --(2)

Using equation (1) and (2)

\[F = \frac{mv^{2}}{r} = qvB\]

Simplifying the equation above

\[r = \frac{mv}{qB}\]

We know that the angular frequency of the particle is

\[v = \omega r\]

Substituting the value from the above equation in this one,

\[\omega = \frac{qB}{m} = 2\pi v\]

Let us find the time for one revolution(T),

\[T = \frac{2\pi}{\omega} = \frac{1}{v}\]

Total distance moved by the particle in one rotation or pitch can be given as,

\[p = v_{p}T = \frac{2\pi m v_{p}}{qB}\]

Where \[v_{p}\] is the parallel velocity. This process describes how the motion of a charged particle in a magnetic field takes place.