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The Motion of Charged Particles in Magnetic Fields
A stationary charged particle in a magnetic field experiences no force.
A charged particle moving parallel to the lines of magnetic field experiences no force.

A charged particle moving so as to cross lines of magnetic field experiences a force which acts at 90° to both the field lines and the direction of motion of the particle.
We can try to guess at an equation to calculate the magnitude of the force.
We will suggest that magnitude of the force depends on
1. the magnitude of the charge, q
2. the strength of the magnetic field (the magnetic flux density), B
3. the magnitude (and direction) of the velocity of the particle, v

Let us make the simplest suggestion, that the magnitude of the force is directly proportional to these factors

so, to get our equation, we need a constant of proportionality

Forgetting the constant for a minute... how about looking at the units of the quantities in our invented equation?
On the left, we obviously have: Newtons
On the right we have: Coulombs × metres per second × Teslas.
Looking at this more closely (and using symbols for simplicity), we find, on the right:
C ms-1 N A-1 m-1 which equals C ms-1 N s C-1 m-1 which equals... wait for it... yes, N!
Now this analysis of the units does not prove that we have the correct equation but it does tell us that the equation is at least possible.
We can now do some experiments and what we find is that this equation is correct and, what's more, the constant of proportionality is 1.

As has been shown in more detail here, if the angle between the velocity and the field lines is θ then the complete equation is

Consider an electron projected into a region where there is a uniform magnetic field.

To represent field lines into or out of the plane of a diagram we use the same system as with currents into or out of the plane of the diagram (see here)
The region of uniform magnetic field is indicated by the regularly spaced crosses.
The field is into the plane of the diagram.
The electron, initially moving in a straight line, enters the field, crossing the field lines at 90°.
It will experience a force at 90° to its direction of motion and to the field.

This means that it will follow a circular path.
Once it leaves the field it will continue in a straight line.
Fleming's left hand rule can be used to predict the direction of the force.
However, if we want to assign a sense to the conventional current in this situation, remember that it will be the opposite sense to the motion of the electron!

Time Period of the Circular Motion of a Charged Particle in a Magnetic Field
As stated above, a charged particle moving so as to cut across the lines of a magnetic field follows a circular path.
It is interesting to note that the time period of the circular motion is independent of the speed of the particle.
(If necessary, see here for information about circular motion, centripetal force etc)

If we assume that the particle crosses the field lines (flux lines) at 90° then the electro-magnetic force acting on the particle has magnitude given by

This force provides a centripetal acceleration to the particle and so can also be written as

therefore, we can write

and since

this simplifies to

Also, we have

which means that the time period is given by

This shows that the time period depends only on the charge to mass ratio of the particle and the flux density of the field.
If this sounds surprising, consider it as follows:
- throw particle into field, it follows a circular path of a certain radius
- throw particle in faster, it follows a path of (proportionally) bigger radius, so taking the same time to get round once.

Now consider a particle projected into a field as shown here.

Its component of velocity which is perpendicular to the flux lines will cause it to go in a circular path.
However, its component of velocity which is parallel to the flux lines will not be affected by the field.
These two effects result in a helical path (the path of the particle is a helix).
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