Suppose you have a conductor. We’ve already seen that it contains individual charges. It is easy to measure the current flowing through the conductor. But, it is not necessarily easy to tell if a positive current corresponds to positive charges moving forwards, or negative charges moving backwards. It might be possible to find out using magnetism.
Magnetism exerts a force on particles proportional to their charge. The total force on a particle of charge and velocity from an electromagnetic field is:
So supposing you look at a current moving in the direction, with a magnetic field in the y direction. Since no charges are piling up just yet, . So, the force on a free particle of charge is .
We’ve been writing as positive, but there’s no need for it to be anything other than zero. Let’s look at the positive and negative cases separately, and see what kind of behavior we find.
When is negative, the velocity of the particles is in the same direction as the current. So, the velocity coefficient is also positive. Since we’re assuming that is positive (and we will get the opposite result if it is pointing in the opposite direction), we can conclude that a force will push the particle in the direction.
When is positive, the velocity of the particles is in the opposite direction as current, making the velocity coefficient negative. Since the charge is also negative, a force direction will also push a negative charge towards the direction.
No matter what sign charge we use, charges will pile up on the wall of the conductor. There, they will act like a plate with surface charge density, which we will denote with . Since the overall material is electrically neutral, there will be a corresponding pileup of not-electrons on the other side of the conductor, corresponding to a surface charge density . We know that the electric field from one such plate is , so from two plates we expect a total electric field of magnitude .
We could run the current and let the charge build up until the charges reach a steady state. In this configuration, the electric force from the charged walls of the conductor exactly cancels out the magnetic force from the external magnetic field.
should be zero, so we would expect . We don’t know , however. If we know the density of charged particles , and we know the current , then we can find the velocity of each particle, .
The potential across the plates can then be calculated.