In this piece of conductor there are   free  electrons. Suppose that all these free electrons pass through the end x, in time, t. This means that the current is given by

where e represents the charge on one electron.

If there is a magnetic field, B, at 90° to the current, the conductor will experience a force of magnitude

Now, this is the sum of the forces acting on all the free electrons as they move through the piece of conductor.

Therefore, force per electron, F, is given by

It is clear that, on average, the electrons must be moving with speed v = /t.

Therefore, the force per electron is given by

If the direction of the velocity is not at 90° to the flux lines, we use the above result but instead of "v" we have the component of the velocity which acts at 90° to the field.

Therefore, in general, when a charged particle moves through a magnetic field it experiences a force given by

and the direction of the force is at 90° to both the velocity and the magnetic field.

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