S2_Magnetic permeability

S2_Magnetic permeability - An educative approach on the...

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An educative approach on the problem of defining magnetic permeability S.Koutandos [email protected] Abstract : In this article we take a short path for studying Maxwell’s equations in the view that isotropic materials and space obey this system of equations and therefore their wave equation as well, by defining the equivalent of magnetic permeability ) ( r r μ as self-induction factor per unit length of a circuit in space. The advantage of our method lies in the fact that we define the magnetic permeability through Ampere’s law alone and that way we leave room for the magnetization M not to be necessarily parallel to H , as could be the case for quantum systems. First of all we shall take new coordinates as the x axis will lie tangent on the lines of the current flux, that is J (the current density). The magnetic field we shall take to always coincide with the z axis as it will of course be perpendicular to the current. Then , together with a y axis an orthogonal coordinate system will be formed. The only difference with our normal Cartesian reference system is going to be that dx=(dx(s)/ds)ds where ds is the length of the curve of constant J, so that way we will have parameterized everything with the length of the curve s. This is displayed in picture 1.
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Picture 1: Now the first relation as it is natural is: = = 0 0 dz dB B z r (1a) B z = B z (x,y) (1b) Ampere’s law in the differential operator form states that: 0 = dx dB z (2) Equations 1 and 2 combine to give: B z =B z (y) (3) If we were to talk about the flux Φ =BS having a perfect differential then it would be as we are about to prove: SdB BdS d d d + = Φ + Φ = Φ 2 1 (4) First we will prove that the first part of the magnetic flux is a perfect differential: ∫∫ = = Φ dydx y B dS B d z z z ) ( 1 (5) Equation 5 is else written as: dx dy y B d z z 0 1 ) ( = = Φ (6)
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This clearly means that this part of the flux changes along with x since the function inside the integral is a function of x, the only variable left and 1 Φ d is a perfect differential. Now we are about to study the second part of the flux: dy y dy dB S dB S d z z z z ) ( 2 = = Φ (7) We shall express Ampere’s law in the usual differential form: ) ( y J J dy dB x x z μ = = (8) The second part of the flux changes with y only for the following reason:
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S2_Magnetic permeability - An educative approach on the...

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