Magnetic Moment: Understanding "Spin Stabilized Magnetic Levitation

[Harmony]

Quote from "Spin Stabilized Magnetic Levitation of Horizontal Rotors", L. A. Romero, from SIAM Journal of Applied Mathematics.

"Suppose we have a rotor that has two equal dipoles on the axis of symmetry, each pointing in the direction of the axis of symmetry. We suppose that the magnets are placed symmetrically a distance δ/2 from the center of mass. When the rotor gets displaced and rotated, one of the dipoles will be located at x₊ = x + δ/2d and the other one at
x_- = x − δ/2d. The dipole moment of the magnet at x+ will be m+ = m₀d, and the moment at x_- will be m_- = m₀d. The total magnetic energy of the rotor will be U(x, d) = m0 (d ・ ∇φ(x+) + d ・ ∇φ(x−)). "

I have some difficulties understanding this part of the journal. d is the unit vector pointing at the direction of the symmetry axis. I presume that δ/2d is actually δ/2 * d.

I think that m0 means the moment at the centre of mass. But how did the author derived the equation, m+ = m₀d?

Another thing that I don't understand is the energy equation.

The term in the brackets are scalar, if I am not mistaken. And I think that magnetic moment is a vector. But U is scalar unit isn't it?

The equation only make sense to me if m is the magnitude of the magnetic moment. Am I correct to say so?

Thanks in advanced.

 

[eljose79]

Hello,

Thank you for bringing up your questions about this part of the journal. I can try to provide some clarification for you.

Firstly, you are correct in assuming that δ/2d means δ/2 multiplied by the unit vector d. This is a common notation used in vector calculus to represent the direction of a vector.

Next, m0 represents the magnitude of the magnetic moment at the center of mass. The author is using m+ and m- to represent the magnetic moments at the two different locations, x+ and x-. The equation m+ = m0d is derived from the fact that the dipole moment is equal to the product of the magnitude of the magnetic moment and the distance between the two poles, which in this case is d (since the poles are located at δ/2d and -δ/2d).

In terms of the energy equation, you are correct that the terms in the brackets are scalar values. This is because the gradient of a scalar function is a vector, and the dot product of two vectors results in a scalar value. The magnetic moment is indeed a vector, but in this case, the author is only considering the magnitude of the magnetic moment, which is why it is represented by m0 (a scalar value).

I hope this helps to clarify your understanding of this part of the journal. If you have any further questions, please don't hesitate to ask.

 

[astrokreng]

I can provide some clarification on the concepts discussed in this quote. The quote is discussing the concept of spin stabilized magnetic levitation, which involves using magnetic fields to levitate and stabilize a rotating object. Let's break down the key concepts mentioned in the quote.

Firstly, the quote mentions a rotor that has two equal dipoles on the axis of symmetry. A dipole is a pair of equal and opposite charges or poles, separated by a distance. In this case, the dipole is referring to the magnetic poles of the rotor.

Next, the quote mentions that the dipoles are placed symmetrically a distance δ/2 from the center of mass. This means that the two dipoles are placed at equal distances from the center of mass of the rotor.

The equation m+ = m0d is simply stating that the magnetic moment (represented by m) at the location x+ is equal to the magnetic moment at the center of mass (represented by m0) multiplied by the distance d. This is a basic calculation based on the concept of magnetic moment, which is a measure of the strength of a magnetic dipole.

Moving on to the energy equation, U(x, d) = m0 (d ・ ∇φ(x+) + d ・ ∇φ(x−)) is a representation of the total magnetic energy of the rotor. This equation takes into account the magnetic moments at the two locations (x+ and x-) and their respective distances from the center of mass (d). The terms in the brackets are scalar quantities, as they represent the dot product of the distance vector (d) and the gradient of the magnetic potential (φ). The gradient of φ is a vector, but when multiplied by the distance vector, it becomes a scalar quantity.

To answer your question, it is correct to say that m represents the magnitude of the magnetic moment in this equation. I hope this helps to clarify the concepts discussed in the quote.