If you want to understand more about the parameter μ, try this.
We know (or at least accept) the usual equation E = – μ • B and perhaps one or two expressions for μ, like I*A or maybe even one using the current density J
μ = 1/2 ∫ d3x (r x J)
But this mysterious parameter can be found using only elementary physics, simply proposing that E is proportional to B, and finding that constant of proportionality.
Start with the same example that we used to find the gyromagnetic ratio of the electron, the charged particle motion in a circle due to uniform B. Now write down the kinetic energy of the particle – that’s right, just plain old
E = 1/2 m v2
Of course, v = r ω and we know from equating
m v2 / r = e v B (the force due to B)
that m ω = e B. Propose there exists a constant of proportionality μ obeying
E = μ B = 1/2 m v2
where E is your expression you wrote down for kinetic energy. Solve for μ until you are satisfied that you have the same expression as you’d get by doing the integral above or using the current loop model I*A. You will obtain the same result as if you were to calculate
μ = IA = current x area = e ω/(2 π)π r2 = 1/2 e ω r2
or using the integral over current density J
μ = 1/2∫ d3x [r x J] = 1/2 r ρ*(Volume) v = 1/2 r e (rω) !!!
Note that here, ρ is charge density, so ρ (Volume) = total charge. The current density J is defined as ρv where v is the velocity of drift for the charge distribution ρ.