I beg you to understand my poor Eng..
If there is any poor grammar or spelling..plz correct me..

While studying MHD with "An Introduction to Magnetohydrodynamics" written by Davidson,
I encountered the term 'current density'..
As you know well, empirically,
[tex]
\mathbf{J} = \sigma \mathbf{E}
[/tex]
with electric field being measured in a frame of reference in which the charged test particle is at rest.

It says

I can't understand this "empirical" Ohm's law for moving conductor(or conducting fluid) because, to my knowledge, [tex] \mathbf{J}(\mathbf{r},t) = \rho_e(\mathbf{r},t)\mathbf{v}(\mathbf{r},t) [/tex] is thought to be the more fundamental definition of current density. It is basically a vector having the (net) direction of charged particles drift velocity..
But [tex] \mathbf{u} \times \mathbf{B} [/tex] clearly does not coincide in direction with [tex] \mathbf{u} [/tex]..

Also, I'd like to raise a question about the e.m.f. generated by a relative movemnet of the imposed magnetic field and the moving fluid. Why is it of order [tex] |\mathbf{u} \times \mathbf{B}| [/tex]? Does it come from Faraday's law?

the current density vector [tex]
\mathbf{J}(\mathbf{r},t)
[/tex]
need not be in the direction of u, it can be found in any direction. so u x B may not necessarily coincide with u.

may be in a wire the J is maximum in direction of u and it is of interest

the magnetic force component u x B is also responsible for genrating an emf. the equation used to arrive to this result should be faraday and maxwells equation.

i would like to read some material and give you a concrete explanation

suppose that the prescribed magnetic field [tex] \mathbf{B} = B_0 \hat{\mathbf{z}} [/tex] is present..and suppose that ,at time t, at the origin of the inertial frame, a particle with charge q moves along the y-direction with velocity u..then the Lorentz force due to magnetic field is in the x-direction..and there is e.m.f generated around the origin..

then what is the current density at the origin at that time? is it not just [tex]q\mathbf{u}[/tex]?