# What's wrong about Maxwell's equations for a Hall probe?

I am using Maxwell's equation to analyse the current and electric field of a Hall Probe. A Hall probe is basically a thin sheet of metal with a current through it. When a uniform magnetic field $$\def\B{\mathbf B} \B$$ is applied perpendicular to the plane of this sheet, charges in the sheet will be displaced, and an electric field across the sheet will be generated. From Maxwell's equations, $$\nabla\times \B=\mu_0\left(\mathbf J+\epsilon_0\frac{\partial \mathbf E}{\partial t}\right).$$ Since $$\B$$ is uniform, the LHS is zero, leaving us with $$\mathbf J=-\epsilon_0\frac{\partial \mathbf E}{\partial t}.$$ Since $$\mathbf E$$ is what we want to measure in a Hall probe, (we measure the p.d. across the sheet of metal, which is essentially to measure $$\mathbf E$$) $$\mathbf E$$ is time-independent. So we have $$\mathbf J=0$$. But that is wrong - we must have some current flowing through the sheet of metal. What is going wrong?

• I can see what you mean: "If there were then the electric field would be increasing". I know that if ALL the current flows in the direction of $E$ field, then the charge will be redistributed. But what if you have a CURVED current that goes up and down so that the net effect on E field is zero? Oct 31, 2019 at 9:39