#### Hall Effect

The Hall Effect can distinguish between positive charges flowing in one direction and negative charges flowing in the other. It is also a common way of measuring the strength of a magnetic field.

Start by picturing a wire of square cross-section, carrying a current to the right. There is a uniform magnetic field pointing into the page.

If the current is made up of positive charges flowing right, in what direction is the magnetic force on the charges?

1. Up
2. Down
3. Left
4. Right
5. Into the page
6. Out of the page

The force is up. The positive charges are deflected up, leaving a deficit of positive charge (i.e., a net negative charge) at the bottom. This looks like a set of charged parallel plates with an electric field pointing down. The electric field builds up until the force experienced by the charges in this electric field is equal and opposite to the force applied on the charges by the magnetic field.

The electric field is associated with a potential difference across the wire that can be measured with a voltmeter. This is known as the Hall voltage VH. If we assume the electric field is uniform, the Hall voltage is:

VH = -Ed, where d is the width of the wire.

The force on the charges due to the electric field is balanced by the magnetic force, so:

qE = qvdB, and E = vdB, so the Hall voltage is:

VH = -vdBd, where vd is the drift velocity of the charges.

The Hall voltage is proportional to the magnetic field, so a voltage measurement can easily be turned into a measurement of B. Or, in a known magnetic field the Hall voltage can be used to measure the drift velocity.

Now, what if the charges flowing through the wire are really negative, flowing left? In what direction is the magnetic force on the charges?

1. Up
2. Down
3. Left
4. Right
5. Into the page
6. Out of the page

The magnetic field again deflects the charges to the top of the wire. This creates an electric field that is opposite in direction to the field obtained with the positive charges, flipping the sign of the Hall voltage. So, figuring out whether the current is made up of positive charges or negative charges is as easy as checking the sign of the Hall voltage.