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Current density of a moving conductor or conducting fluid

  1. Jan 11, 2010 #1
    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,
    \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
    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?
  2. jcsd
  3. Jan 13, 2010 #2
    I couldn't understand all your doubts

    i will post some comments which may help

    the current density vector [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
  4. Jan 13, 2010 #3
    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
  5. Jan 14, 2010 #4
    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]?
  6. Jan 14, 2010 #5
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