Explain how the Hall effect arises when a current-carrying conductor is placed in a magnetic field perpendicular to the current, and state what determines the polarity of the Hall voltage produced.

The magnetic Lorentz force
F=q(v\times B) acts on the charge carriers moving with velocity v in the conductor. With v along +x and B along +z, the force on positive charges is along +y and on negative charges along –y, so the carriers are deflected to one side of the conductor.
This deflection causes an accumulation of charge on that face, producing an electric field (the Hall field) across the width of the conductor. The Hall field exerts an electric force qE on the carriers.
When the electric force balances the magnetic force (qE=qvB), equilibrium is reached and the carriers no longer deflect; the Hall voltage is the potential difference between the two faces.
The polarity of the Hall voltage is determined by the sign of the charge carriers: if the carriers are positive (conventional current direction) the Hall voltage is positive on the side to which they are deflected; if the carriers are negative (electrons moving opposite to conventional current) the polarity is reversed.