§19.33 Triaxial Ellipsoids

§19.33(i) Surface Area

The surface area of an ellipsoid with semiaxes $a,b,c$, and volume $V=4\pi abc/3$ is given by

 19.33.1 $S=3VR_{G}\left(a^{-2},b^{-2},c^{-2}\right),$ ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind, $V$: volume and $S$: area Referenced by: §19.33(i) Permalink: http://dlmf.nist.gov/19.33.E1 Encodings: TeX, pMML, png See also: Annotations for §19.33(i), §19.33 and Ch.19

or equivalently,

 19.33.2 $\frac{S}{2\pi}=c^{2}+\frac{ab}{\sin\phi}\left(E\left(\phi,k\right){\sin^{2}}% \phi+F\left(\phi,k\right){\cos^{2}}\phi\right),$ $a\geq b\geq c$,

where

 19.33.3 $\displaystyle\cos\phi$ $\displaystyle=\frac{c}{a},$ $\displaystyle k^{2}$ $\displaystyle=\frac{a^{2}(b^{2}-c^{2})}{b^{2}(a^{2}-c^{2})}.$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\phi$: real or complex argument and $k$: real or complex modulus Permalink: http://dlmf.nist.gov/19.33.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.33(i), §19.33 and Ch.19

Application of (19.16.23) transforms the last quantity in (19.30.5) into a two-dimensional analog of (19.33.1).

For additional geometrical properties of ellipsoids (and ellipses), see Carlson (1964, p. 417).

§19.33(ii) Potential of a Charged Conducting Ellipsoid

If a conducting ellipsoid with semiaxes $a,b,c$ bears an electric charge $Q$, then the equipotential surfaces in the exterior region are confocal ellipsoids:

 19.33.4 $\frac{x^{2}}{a^{2}+\lambda}+\frac{y^{2}}{b^{2}+\lambda}+\frac{z^{2}}{c^{2}+% \lambda}=1,$ $\lambda\geq 0$. ⓘ Permalink: http://dlmf.nist.gov/19.33.E4 Encodings: TeX, pMML, png See also: Annotations for §19.33(ii), §19.33 and Ch.19

The potential is

 19.33.5 $V(\lambda)=QR_{F}\left(a^{2}+\lambda,b^{2}+\lambda,c^{2}+\lambda\right),$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $Q$: charge and $V(\lambda)$: potential Permalink: http://dlmf.nist.gov/19.33.E5 Encodings: TeX, pMML, png See also: Annotations for §19.33(ii), §19.33 and Ch.19

and the electric capacity $C=Q/V(0)$ is given by

 19.33.6 $1/C=R_{F}\left(a^{2},b^{2},c^{2}\right).$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind and $C$: electric capacity Permalink: http://dlmf.nist.gov/19.33.E6 Encodings: TeX, pMML, png See also: Annotations for §19.33(ii), §19.33 and Ch.19

A conducting elliptic disk is included as the case $c=0$.

§19.33(iii) Depolarization Factors

Let a homogeneous magnetic ellipsoid with semiaxes $a,b,c$, volume $V=4\pi abc/3$, and susceptibility $\chi$ be placed in a previously uniform magnetic field $H$ parallel to the principal axis with semiaxis $c$. The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength $H/(1+L_{c}\chi)$, where $L_{c}$ is the demagnetizing factor, given in cgs units by

 19.33.7 $L_{c}=2\pi abc\int_{0}^{\infty}\frac{\mathrm{d}\lambda}{\sqrt{(a^{2}+\lambda)(% b^{2}+\lambda)(c^{2}+\lambda)^{3}}}=VR_{D}\left(a^{2},b^{2},c^{2}\right).$

The same result holds for a homogeneous dielectric ellipsoid in an electric field. By (19.21.8),

 19.33.8 $L_{a}+L_{b}+L_{c}=4\pi,$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $L_{a}$, $L_{b}$: factor Permalink: http://dlmf.nist.gov/19.33.E8 Encodings: TeX, pMML, png See also: Annotations for §19.33(iii), §19.33 and Ch.19

where $L_{a}$ and $L_{b}$ are obtained from $L_{c}$ by permutation of $a$, $b$, and $c$. Expressions in terms of Legendre’s integrals, numerical tables, and further references are given by Cronemeyer (1991).

§19.33(iv) Self-Energy of an Ellipsoidal Distribution

Ellipsoidal distributions of charge or mass are used to model certain atomic nuclei and some elliptical galaxies. Let the density of charge or mass be

 19.33.9 $\rho(x,y,z)=f\left(\sqrt{(x^{2}/\alpha^{2})+(y^{2}/\beta^{2})+(z^{2}/\gamma^{2% })}\right),$

where $\alpha,\beta,\gamma$ are dimensionless positive constants. The contours of constant density are a family of similar, rather than confocal, ellipsoids. In suitable units the self-energy of the distribution is given by

 19.33.10 $U=\frac{1}{2}\int_{{\mathbb{R}^{6}}}\frac{\rho(x,y,z)\rho(x^{\prime},y^{\prime% },z^{\prime})\mathrm{d}x\mathrm{d}y\mathrm{d}z\mathrm{d}x^{\prime}\mathrm{d}y^% {\prime}\mathrm{d}z^{\prime}}{\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z% ^{\prime})^{2}}}.$

Subject to mild conditions on $f$ this becomes

 19.33.11 $U=\tfrac{1}{2}(\alpha\beta\gamma)^{2}R_{F}\left(\alpha^{2},\beta^{2},\gamma^{2% }\right)\int_{0}^{\infty}(g(r))^{2}\mathrm{d}r,$

where

 19.33.12 $g(r)=4\pi\int_{r}^{\infty}f(t)t\mathrm{d}t.$