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Given a material with electrical conductivity $\sigma$ and an electric field $E$, the current density in the material is $$ J = \sigma E. $$ Now assume that in addition there is a magnetic field $B$, and that the material is liquid and moves around with the flux $u$. I'm reading that the induced current contains the term $u\times B$, but I'm not sure about the coefficient.

What's the current density induced by the magnetic field?

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The general Ohm's law (valid for homogeneous and isotropic material) is: $$ \mathbf{J} = \sigma \mathbf{f}, $$ where $\mathbf{f}$ is the force per unit charge and $\sigma$ a scalar. The force $\mathbf{f}$ is usually an electromagnetic force, so $$ \mathbf{f}=\mathbf{E}+\mathbf{v}\times\mathbf{B}. $$ The current induced by the magnetic field is $J_{B} =\sigma \mathbf{v}\times\mathbf{B}$. Ordinarily, the velocity of the charges is sufficiently small that the second term can be ignored.

Notice that Ohm's law is a vector equation (equivalent to three scalar equation, if you write it by coordinates).

Choose the best coordinate system according to the geometry of the problem and use $ \mathbf{J} = \sigma(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ to calculate the total current density.

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