Circular motion in the magnetic field

The charged particle with velocity v has uniform circular motion when it is in a field B

If we were to take a particle carrying charge q and mass m moving in a magnetic field B, we would realize that such a particle would be subject to a force of magnetic intensity. This force, at the same time, is perpendicular to the direction of velocity and to the magnetic field. Observing the movement of this charge, we see that we can calculate its angular velocity as well as the frequency of the movement knowing that the magnetic force is the centripetal force of this movement, as shown in the figure above. We can then write that:

The magnetic force acting on the charge is the centripetal force, that is, it is a force directed towards the center of the circle. This force is always perpendicular to the displacement of the load. Thus, we say that the magnetic field does not influence the variation of the kinetic energy of the charge. In other words, the magnitude of the tangential velocity is not modified by the action of the magnetic field.

The orbit radius is obtained through the following equation:

The frequency of circular motion is given by:

Combining the two previous equations, we have:

And the angular velocity is given by:

The frequency of motion is independent of the ion’s velocity, but the radius of motion is proportional to mass and tangential velocity. Lighter ions rotate in smaller orbits than heaviest ones.

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