Faraday's Unipolar Induction Equation

[Jay21]

Hello all,

I have a question regarding Maxwell's Equations and Faraday's unipolar induction equation.
If we study the case of a cylindrical magnet with a radius of r which is rotating about its axis
with angular velocity w. The electrons within the magnet collide with the moving atoms, causing
a net drift velocity, v = w x r. The electrons experience a force:
-e*(v x B) towards the center

This results in the negative charge being centralized in the magnet and a positive
charge on the outer surface, thus creating an equilibrium electrostatics field with
Lorentz force:
F = -eE - e*(v x B) = 0

This leads to the unipolar induction equation of
E = - v x B

My question is that while studying plasma physics I came across a very similar equation such that
H = v x D
How would one derive this analogous equation of the unipolar induction equation?
I read somewhere that you could derive this equation by using a thin, charged rotating ring,
but I am unsure as to how to accomplish this.

Thanks so much.
Jay

Citations I have investigated trying to derive this analgous equation:

Unipolar Induction via a Rotating, Conducting, Magnetized Cylinder by Kirk T. McDonald
for in a comoving inertia frame
H* = H - ((v/c) x D)

Basic Plasma Physics Principles by Gordon Emslie for
E' = gamma*(E + (v/c) x B)
B' = gamma*(B - (v/c) x E)

The Unipolar Induction by P. Hrasko for
B = (u_0*gamma)*H + (1/c^2)(v x E)

 

[eljose79]

Dear Jay,

Thank you for your question regarding the derivation of the analogous equation for unipolar induction in plasma physics. The equation you are referring to, H = v x D, is known as the "magnetohydrodynamic (MHD) Ohm's law" and is a fundamental equation in plasma physics.

To understand how this equation is derived, it is important to first understand the concept of MHD. MHD is a theory that combines the principles of electromagnetism and fluid dynamics to describe the behavior of a plasma, which is a gas-like state of matter consisting of charged particles (ions and electrons). In MHD, the plasma is treated as a continuous fluid with the properties of conductivity and magnetization.

The MHD Ohm's law is derived from the more general form of Ohm's law, which relates the electric field (E) to the current density (J) and the conductivity (σ) of a material: J = σE. In the case of a plasma, the current density is given by the sum of the conduction current density (Jc) and the displacement current density (Jd): J = Jc + Jd. The displacement current density is related to the rate of change of the electric flux density (D) through Faraday's law: Jd = ∂D/∂t. Combining these equations, we get J = σE + ∂D/∂t.

Now, in the case of a plasma, the electric flux density (D) is related to the magnetic field (B) through the equation D = εE + ε0B, where ε is the permittivity of the plasma and ε0 is the permittivity of free space. Substituting this into the previous equation, we get J = σE + (∂εE/∂t) + (∂ε0B/∂t). Using the fact that ε is a function of the plasma density (n) and temperature (T), and taking the time derivative, we get (∂εE/∂t) = ε(∂E/∂t) + (∂ε/∂n)(∂n/∂t)E + (∂ε/∂T)(∂T/∂t)E. Similarly, (∂ε0B/∂t) = ε0(∂B/∂t)