- #1
omyojj
- 37
- 0
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,
[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?
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,
[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
This is an empirical law which, for stationary conductors, takes the form [tex] \mathbf{J} = \sigma \mathbf{E} [/tex], where [tex] \mathbf{E} [/tex] is the electric field and [tex] \mathbf{J} [/tex] the current density. We interpret this as [tex] \mathbf{J} [/tex] being proportional to the Coulomb force [tex]\mathbf{f} = q\mathbf{E}[/tex] which acts on the free charge carriers, [tex] q[/tex] being their charge. If, however, the conductor is moving in a magnetic field with velocity [tex]\mathbf{u}[/tex], the free charges will experience an additional force, [tex] q\mathbf{u} \times \mathbf{B} [/tex] and Ohm's law becomes
[tex] \mathbf{J} = \sigma ( \mathbf{E} + \mathbf{u} \times \mathbf{B} ) [/tex]
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?