• Jay21
In summary: B + (∂ε0/∂T)(∂T/∂t)B. Substituting these into the previous equation and rearranging, we get J = σE + ε(∂E/∂t) + ε0(∂B/∂t) + (∂ε/∂n)(∂n/∂t)E + (∂ε/∂T)(∂T/∂t)E + (∂ε0/∂n)(∂n/∂t)B + (∂ε0/∂T)(∂T/
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)

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)

## 1. What is Faraday's Unipolar Induction Equation?

Faraday's Unipolar Induction Equation, also known as Faraday's Law of Induction, is a fundamental law in electromagnetism that describes the relationship between a changing magnetic field and an induced electric field. It was discovered by the British scientist Michael Faraday in the 1830s.

## 2. How does Faraday's Unipolar Induction Equation work?

Faraday's Unipolar Induction Equation states that an electric field is induced in a conductor when it is exposed to a changing magnetic field. This induced electric field is proportional to the rate of change of the magnetic field and the length of the conductor. The direction of the induced electric field is perpendicular to both the direction of the magnetic field and the direction of the conductor.

## 3. What is the significance of Faraday's Unipolar Induction Equation?

Faraday's Unipolar Induction Equation is significant because it provides the basis for understanding the relationship between electricity and magnetism. It is also the underlying principle behind important technologies such as generators, transformers, and electric motors.

## 4. Can Faraday's Unipolar Induction Equation be applied to all types of conductors?

Yes, Faraday's Unipolar Induction Equation can be applied to all types of conductors, including metals, liquids, and even living cells. As long as the conductor is exposed to a changing magnetic field, an induced electric field will be created.

## 5. How is Faraday's Unipolar Induction Equation related to the laws of conservation of energy and charge?

Faraday's Unipolar Induction Equation is related to the laws of conservation of energy and charge because it states that the induced electric field is a result of the changing magnetic field, and thus energy is conserved. It also follows the law of charge conservation as the induced electric field is created by the movement of charges within the conductor.

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