Current density of a moving conductor or conducting fluid

[omyojj]

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

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

It says

This is an empirical law which, for stationary conductors, takes the form $$\mathbf{J} = \sigma \mathbf{E}$$, where $$\mathbf{E}$$ is the electric field and $$\mathbf{J}$$ the current density. We interpret this as $$\mathbf{J}$$ being proportional to the Coulomb force $$\mathbf{f} = q\mathbf{E}$$ which acts on the free charge carriers, $$q$$ being their charge. If, however, the conductor is moving in a magnetic field with velocity $$\mathbf{u}$$, the free charges will experience an additional force, $$q\mathbf{u} \times \mathbf{B}$$ and Ohm's law becomes
$$\mathbf{J} = \sigma ( \mathbf{E} + \mathbf{u} \times \mathbf{B} )$$

I can't understand this "empirical" Ohm's law for moving conductor(or conducting fluid) because, to my knowledge, $$\mathbf{J}(\mathbf{r},t) = \rho_e(\mathbf{r},t)\mathbf{v}(\mathbf{r},t)$$ 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 $$\mathbf{u} \times \mathbf{B}$$ clearly does not coincide in direction with $$\mathbf{u}$$..

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 $$|\mathbf{u} \times \mathbf{B}|$$? Does it come from Faraday's law?

 

[payumooli]

I couldn't understand all your doubts

i will post some comments which may help

the current density vector $$\mathbf{J}(\mathbf{r},t)$$
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

 

[Bob S]

In J = σ·E, σ is electrical conductance (ohm-m)^-1.

In F = σ·V(E + v x B), σ is charge density (Coulombs per unit volume).

I think.

Bob S

 

[omyojj]

suppose that the prescribed magnetic field $$\mathbf{B} = B_0 \hat{\mathbf{z}}$$ 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 $$q\mathbf{u}$$?

 

[Bob S]

I think the original post refers to the operation of a homopolar generator, also called a Faraday disc. See
http://en.wikipedia.org/wiki/Homopolar_generator
The equations for the Faraday disc are presented in
http://www.physics.brown.edu/physics/demopages/Demo/em/demo/5k1080.htm
Bob S
.