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In the Magnetohydrodynamics the electric current is given by the Ampere's law is:

$$\nabla \times \mathbf{B}=\mu_0\mathbf{J}$$

where $\mathbf{B}$ is the magnetic field and $\mathbf{J}$ is the electric current.

Now, consider Ohm's law:

$$\mathbf{E}+\mathbf{V}\times\mathbf{B}=\mathbf{J}/\sigma$$

where $\mathbf{E}$ is electric field and $\mathbf{V}$ is the fluid velocity.

The question is if there is a uniform external magnetic exerting in the absence of an electric field, Ampere's law says $\mathbf{J}=0$ while the Ohm's law says $\mathbf{J}\neq 0$, why?

Kyle Kanos
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Danial Rezai
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2 Answers2

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If the external magnetic field is uniform, total field $\mathbf B$ need not be uniform; there is contribution to total field due to current in the medium. If total field $\mathbf B$ is somehow made uniform, then current density vanishes. Then $\mathbf E + \mathbf V \times \mathbf B = 0$, so electric field is determined by the magnetic field.

Ján Lalinský
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Contrary to what you've asserted, assuming $\mathbf{E}=0$ and $\mathbf{B}=\text{const}$, Ohm's law does not make the claim that $\mathbf{J}\neq0$, it makes no claim on the value of $\mathbf{J}$ as there is an additional component you've neglected: the velocity (i.e., you have two unknowns & 1 equation). So by itself, it can only make the claim that the current density is orthogonal to the velocity & magnetic field, $$ \mathbf{v}\times\mathbf{B}=\mathbf{J} $$

It is only by using additional information, in the form of Ampere's law wherein we find that $\mathbf{J}=0$, that Ohm's law reduces to, $$\mathbf{v}\times\mathbf{B}=0$$ which means that the velocity $\mathbf{v}$ must be parallel to the magnetic field.

Kyle Kanos
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  • Consider a pipe carrying water, I place two magnets on the pipe's diameter and far away enough to make a uniform magnetic field which is prependicular to the velocity field, assume there is no electric field, in this configuration $\vec{V}\times\vec{B}=\vec{J}\neq 0$ according to the Ohm's law,but $\vec{J}$ is zero based on Ampere's law, how do you justify this? – Danial Rezai Aug 12 '22 at 09:02
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    I'd justify it by saying one of your assumptions must be wrong. For instance, the electric field may not actually be zero due to the moving particles (typically one assumes charged particles in the analysis, rather than neutral ones like water) via the Lorentz force. Or that Ohm's law, being an approximation, doesn't hold (cf. this PSE post or the linked question here). – Kyle Kanos Aug 12 '22 at 12:27