TITLE: magnetically induced current density
QUESTION [0 upvotes]: 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?

REPLY [1 votes]: 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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