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 ∫ d^{3}**x** (**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 v^{2}

Of course, v = r ω and we know from equating

m v^{2} / 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 v^{2}

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 = I*A = current x area = e ω/(2 π)*π r^{2} = 1/2 e ω r^{2}

or using the integral over current density **J**

μ = 1/2∫ d^{3}**x** [**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 ρ**v** where **v** is the velocity of drift for the charge distribution ρ.

🙂 Very Nice

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