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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?

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Your solution is not self-consistent. Whilst you have assumed that the B-field is uniform, if there is a current flowing through the probe, then it cannot be - you have forgotten about the B-field attributable to the current.

What may be true is that the current in the direction of the electric field is zero and hence the component of the curl of the B-field in that direction is zero.

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  • $\begingroup$ Why the current in the direction of E field is zero? The electrons might move in a curved path. $\endgroup$
    – Ma Joad
    Oct 31 '19 at 9:29
  • $\begingroup$ @MaJoad In which direction is the electric field? Why do you think there is a current in that direction? If there were then the electric field would be increasing... $\endgroup$
    – ProfRob
    Oct 31 '19 at 9:33
  • $\begingroup$ 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? $\endgroup$
    – Ma Joad
    Oct 31 '19 at 9:39
  • $\begingroup$ @MaJoad I don't understand what you mean. $\endgroup$
    – ProfRob
    Oct 31 '19 at 12:00
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You have used the equation wrongly. The uniform magnetic field that we want to measure is not produced by the current present in the probe layer hence on the RHS of your equation you cannot use the current in the probe. The LHS magnetic field is produced by other currents etc but measured with the help of voltage produced by the deflection of current in the probe.

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