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 electromagnetic 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). 
