Understanding the magnetic dipole moment

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.

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