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(90θ)
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 


