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19 Elliptic IntegralsApplications

§19.33 Triaxial Ellipsoids


§19.33(i) Surface Area

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

19.33.1 S=3VRG(a-2,b-2,c-2),

or equivalently,

19.33.2 S2π=c2+absinϕ(E(ϕ,k)sin2ϕ+F(ϕ,k)cos2ϕ),


19.33.3 cosϕ =ca,
k2 =a2(b2-c2)b2(a2-c2).

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 x2a2+λ+y2b2+λ+z2c2+λ=1,

The potential is

19.33.5 V(λ)=QRF(a2+λ,b2+λ,c2+λ),

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

19.33.6 1/C=RF(a2,b2,c2).

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=4πabc/3, and susceptibility χ 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+Lcχ), where Lc is the demagnetizing factor, given in cgs units by

19.33.7 Lc=2πabc0dλ(a2+λ)(b2+λ)(c2+λ)3=VRD(a2,b2,c2).

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

19.33.8 La+Lb+Lc=4π,

where La and Lb are obtained from Lc 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 ρ(x,y,z)=f((x2/α2)+(y2/β2)+(z2/γ2)),

where α,β,γ 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=126ρ(x,y,z)ρ(x,y,z)dxdydzdxdydz(x-x)2+(y-y)2+(z-z)2.

Subject to mild conditions on f this becomes

19.33.11 U=12(αβγ)2RF(α2,β2,γ2)0(g(r))2dr,


19.33.12 g(r)=4πrf(t)tdt.