Magnetic moment

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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.
 
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