We have to show that, in terms of the Hall electric field E and the current density j, the number of charge carriers per unit volume is given by

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The Hall electric field arises so that the net Lorentz force on charge carriers inside a conductor is zero. Let be the drift velocity of charge carriers. The magnetic field is so arranged as to have

This condition implies that

From this equation, we note that

As and are orthogonal, we have

The current density j in terms of volume density of charge carriers, n, drift speed,, and charge, e, is

We thus have