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Force Acting on a Charged Particle Moving Through a Magnetic Field
Consider a conductor of length L, having n free electrons per unit volume.  
A current I (shown in the conventional current sense) is flowing through it.  
   
This piece of the conductor has volume V given by  
 
so the number of electrons in this piece of the conductor is  
 
Let us suppose that all these free electrons pass through the end x in time t.  
As electric current is defined as rate of flow of electric charge, this means that the current flowing through this conductor is given by  
 
where e represents the charge on one electron.  
Now, if there is a magnetic field of flux density B at 90 to the current, then we know that the conductor will experience a force of magnitude ILB (as discussed here).  
   
This force is the sum of all the forces acting on the free electrons as they move through the conductor.  
Therefore the force on each electron is given by  
 
which, using the expression for I from above reduces to   
 
If all these electrons are going to pass though end x in time t as stated above then their average drift velocity must be  
   
and so we conclude that the force F acting on each electron is simply given by   
   
   
  Consider now a situation in which a charged particle moves at some other angle to the magnetic field lines.  

We can still use the result above but the velocity will be the component of velocity which acts at 90 to the flux lines.

From the diagram we can see that this component has magnitude vcos(90-θ) which equals vsinθ.
 
Therefore, the more general expression for the force F acting on a particle of charge q moving at angle θ to a magnetic field of flux density B  
   
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