- #1
Sky Walker
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Hello everybody, I have a question about the magnetic induction equation. I am interested in the non ideal case, i.e. when there is a non negligible diffusivity, however as regards my question I think that this is not relevant.
I first ask the question and then give additional explanations:
How do I write the magnetic induction equation when the magnetic field is rotating (of if you prefer, when the reference system where the magnetic field is at rest is not a non inertial one)?
In particular, I am interested in the case of a rotating star, which is endowed with some magnetic field. The field is anchored to the surface of the star and corotates with it. Then I want to use the indcution equation to describe the interaction between the stellar magnetic field and some plasma which is orbiting around the central object.
Now, in general the magnetic induction equation is written as:
d_t B = nabla x ( v x B - eta nabla x B )
where:
d_t = partial derivative with respect to time
B = magnetic field
v = velocity field of the plasma
eta = total diffusivity
x = cross product and
nabla = del operator
The problem is that "in general" the electromagnetic field are not considered to be moving by themselves. "Not so bad, why don't you just go in a reference where the electromagnetic is at rest?" Yes I can do that, but the point is that this reference would be non inertial (it is rotating), therefore I would expect to find some additional terms in the equation. Otherwise I think that the only difference would be that of changing the velocity of the plasma v, with the relative velocity between the plasma and the magnetic field.
Can someone help me in finding which are these additional terms?
It might be useful to shift the problem to Maxwell's equations and Ohm's law, since the induction equation is derived from them (in the approximation of non-relativistic motion and neglecting the displacement current).
I apologise if posting in the wrong forum section.
Thanks for any help.
I first ask the question and then give additional explanations:
How do I write the magnetic induction equation when the magnetic field is rotating (of if you prefer, when the reference system where the magnetic field is at rest is not a non inertial one)?
In particular, I am interested in the case of a rotating star, which is endowed with some magnetic field. The field is anchored to the surface of the star and corotates with it. Then I want to use the indcution equation to describe the interaction between the stellar magnetic field and some plasma which is orbiting around the central object.
Now, in general the magnetic induction equation is written as:
d_t B = nabla x ( v x B - eta nabla x B )
where:
d_t = partial derivative with respect to time
B = magnetic field
v = velocity field of the plasma
eta = total diffusivity
x = cross product and
nabla = del operator
The problem is that "in general" the electromagnetic field are not considered to be moving by themselves. "Not so bad, why don't you just go in a reference where the electromagnetic is at rest?" Yes I can do that, but the point is that this reference would be non inertial (it is rotating), therefore I would expect to find some additional terms in the equation. Otherwise I think that the only difference would be that of changing the velocity of the plasma v, with the relative velocity between the plasma and the magnetic field.
Can someone help me in finding which are these additional terms?
It might be useful to shift the problem to Maxwell's equations and Ohm's law, since the induction equation is derived from them (in the approximation of non-relativistic motion and neglecting the displacement current).
I apologise if posting in the wrong forum section.
Thanks for any help.