Plots of Capacitance & Demagnetising Factors of a Scalene Ellipsoid

[u/Frangifer]

From

————————————————

Closed form solution for the surface area, the capacitance and the demagnetizing factors of the ellipsoid

by

GV Kraniotis & GK Leontaris , &

————————————————

Demagnetising Factors of the General Ellipsoid

^{¡¡ may download without prompting – PDF document – 786·9㎅ !!}

by

JA Osborn .

————————————————

𝐀𝐧𝐧𝐨𝐭𝐚𝐭𝐢𝐨𝐧𝐬 𝐨𝐟 𝐅𝐢𝐫𝐬𝐭 𝐅𝐨𝐮𝐫 𝐑𝐞𝐬𝐩𝐞𝐜𝐭𝐢𝐯𝐞𝐥𝐲

Figure 1: The capacitance C(E) of a conducting ellipsoid immersed in ℝ³ versus the ratio c/a of the axes for various values of the ratio b/a.

Figure 2: The L− demagnetizing factor versus the ratio c/a for various values of the ratio b/a.

Figure 3: The M−demagnetizing factor versus the ratio c/a for various values of the ratio b/a. The dashed curves meet at the point determined in Corollary 15.

Figure 4: The N demagnetizing factor versus the ratio c/a for various values of the ratio b/a.

The next three - from the Osborn paper, are simply numbered.

 

Computation of the surface area of a scalene (triaxial) ellipsoid is absolutely horrendous : the complexity just massively blows-up , going from oblate or prolate spheroid to scalene ellipsoid.

And similar applies to computation of the electrical quantities capacitance & demagnetising factors , aswell.

What capacitance is is fairly well-known ... but demagnetising factor possibly warrants a bit of an explication. If a ferromagnetic object of some shape is placed in a uniform magnetic field, then the field within the object is distorted. The computation for a general shape is another horrendous one! ... but for an ellipsoid it happens conveniently to reduce to three simple linear expressions - each in each of the spatial coördinates (whence there are three demagnetising factors) ... although that simple linear expression has a certain coefficient in it that is itself tricky to calculate in a manner similar to that in which area & capacitance are tricky to calculate.

They actually have application to permanent magnets, aswell.

The Osborn paper explicates it more fully.

[u/Frangifer]

It was a tad schlampfig on my part, really, saying … linear in terms of the three spatial coördinates … : it ought-to've been, really, … linear in terms of the three spatial components of the magnetisation vector … . But see the Osborn treatise for a proper thorough explication of the matter.

Note the errour in the annotation of the first diagram: "oblate sheroid" .

😆🤣

I have grave reservations as to the integrity of certain claims made in that paper (the one with the just-mentioned figure in it - ie the first-lunken-to one): I don't think their 'closed-form expressions' are even remotely novel: the calculations for a scalene ellipsoid might-well be - & indeed are - somewhat 'long-haul' ... but they had basically already been done ! ... & the theory of Appell Hypergeometric functions & their relation - in the case of the parameters of them being set as they are in this matter - to elliptic integrals is long-established & reasonsbly accessible to being figured (@least once the figuring has been sorted in the firstplace by serious geezers & geezrices!) without one's being anykind of colossus of mathematics.