Determine the constraints to be able to observe (measure) the Hall voltage in a conductor. We would like to use a PCB as a prototype Hall element. Specific constraints to determine are:
- Magnetic field limitations; how big can we go??? (constrained by $$$)
- Electric current limitations
- Dimensions of hall element; in this case I would like to use a thin conductor, like PCB or metal tape (dimensions constrained by 2)
- Instrumentation & Error Budget (what levels of hall voltage can be attained, how do we plan to measure the hall voltage, and do it somewhat accurately?)
Once we figure out the above, we'll get the materials and test.
Hall Voltage TL;DR
VH = -(1 ⁄ nq) * I * B ⁄ t
The 3 factors which influence Hall voltage magnitude:
- The longitudinal current in a material (conductor or semi-conductor).
- The magnetic field component cutting through the material, perpendicular to the direction of current
- The material itself where (1 / nq) is known as the Hall coefficient for conductors and t is the thickness
Note: The Hall coefficient (1 / nq from above) becomes a bit more complex when you start using materials that have more than one type of charge carrier (e.g. semi-conductors). However, the concept is the same.
Design of Experiment
Now to the fun part: if we are intending to see what magnitude of current and magnetic field we will need to measure Hall voltage, we know we can't apply an infinite amount of either...we obviously have some constraints :).
What Is the Hall Effect?
The hall effect is the phenomena of a voltage (known as Hall voltage) formed due to the presence of:
- current flowing in a material
- magnetic field which is perpendicular to the current
where the Hall voltage:
- appears in the plane which is perpendicular to both the current and magnetic fields, and
- whose polarity is determined by the polarity of charge carriers and the directions of both the current and magnetic field.
The Hall Voltage: A Closer Look
When we talk about electric current, we're talking about moving charges. When these moving charges cut through a perpendicular external magnetic field, a force is exerted upon these charges: the Lorentz force. The same phenomena that accelerates projectiles beyond the speed of sound (aka the Railgun) is the same force responsible for Hall Effect.
Notice how I said "moving" charges above. This means that if I take an otherwise non-magnetic (or weak magnetic) material (say copper), put an electric current through it, then place it in a magnetic field, the moving charges will start responding to the magnetic field. Again, this response depends upon the magnitude of the current (moving charges) and magnetic field (in addition to some material properties). Pretty sweet, right?
Here's the mathematical formula (don't be scared, use the google or the references):
FL = q(v x B + E)
Let's break it down:
FL: Lorentz Force
q: charge per charge carrier (scalar; include sign!!)
v: velocity of charge carriers (vector)
B: external magnetic field (vector)
E: resulting electric field due to dipole charge distribution formed in conductor (vector)
Where Did the E Come From?
Realize that we are dealing with an isolated system here. We're only concerned with the electric current in the material and the external magnetic field applied through that material. There are no other external fields coming from out of nowhere.
The electric field in this case (E) is a result of the interaction of the charge carriers and magnetic field, where the majority of the charge carriers are "compressed" to one side of the material. This makes one side of the material more negatively charged and the other side more positively charged. This phenomena forms an effective dipole, where the separation of positive and negative charges causes an E field to polarize in a direction opposite to the polarity of the Hall voltage.
<image to come>
Putting It All Together
From physics, we know that electric field lines (basically the strength and direction of the field) align going from the positive to the negative side of the dipole. Within the conductor the strength (force) keeping the charge distribution in equilibrium is related to the separation distance between the electrons and the more positively charged space which is unoccupied by electrons. And as the electrons are compressed to one side, the force trying to bring them back to equilibrium increases linearly with distance (like a spring).
So what does this mean?...this means the Hall Field (and hence Hall Voltage) will have a limit because there will be a point where increasing current or magnetic field strength will yield very little gains in Hall voltage.
Mathematically, when there is no Lorentz force (FL = 0):
-qE = qvB
Using this concept, along with some other concepts in physics, we get the following equation for the Hall voltage in terms of current and external magnetic field (see reference material):
VH = -(1 ⁄ nq) * I * B ⁄ t where;
nq: charge density of material
I: current in the conductor
B: magnetic field cutting through the conductor
t: thickness of the conductor material
Hall Effect Animation (Thanks Wikipedia!)
Note: The above video clip uses the electron current model which assumes negative charge carriers. Please understand that when calculating the Lorentz force, the sign of the charge carrier matters, so choose your model and stick to it. Please google "right-hand rule" for more info (also see the references down below).