Physics Asked on February 7, 2021
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 $defB{mathbf B} B$ 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,
$$
nablatimes B=mu_0left(mathbf J+epsilon_0frac{partial mathbf E}{partial t}right).
$$
Since $B$ is uniform, the LHS is zero, leaving us with
$$
mathbf J=-epsilon_0frac{partial mathbf E}{partial t}.
$$
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?
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.
Answered by ProfRob on February 7, 2021
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.
Answered by Lost on February 7, 2021
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