The Hall effect comes about due to the nature of the current flow in the conductor. Current consists of many small charge-carrying "particles" (typically electrons) which see a force due to the magnetic field. Some of these charge elements end up forced to the sides of the conductors, where they create a pool of net charge. This is only notable in larger conductors where the separation between the two sides is large enough.
One important feature of the Hall effect is that it differentiates between positive charges moving in one direction and negative charges moving in the opposite. The Hall effect offered the first real proof that electric currents in metals are carried by moving electrons, not by protons. Interestingly enough, the Hall effect also showed that in some substances (especially semiconductors), it is more appropriate to think of the current as positive "holes" moving rather than negative electrons.
By measuring the Hall voltage across the element, one can determine the strength of the magnetic field applied. So called Hall effect sensors are readily available from a number of different manufacturers. The most common types are analog (or Linear) Hall effect sensors, which output a voltage that is proportional to the applied magnetic field, and digital Hall effect sensors, which are often used as magnetically controlled switches -- they turn on or off when the applied magnetic field reaches a certain level. These Hall effect switches generally consist of a Hall effect sensor]], one or more logic gates and a transistor used to switch the electric current on or off.
Alternately, by applying a known magnetic field (typically from a permanent magnet) one can use the Hall voltage to instead measure the current through the element. This can be particularly useful as it allows one to measure the current in a conductor remotely through induction. This is widely used commercially in "live wire detectors", which allow you to quickly identify which wires are carrying current without plugging into them.
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