19 Elliptic IntegralsApplications19.32 Conformal Map onto a Rectangle19.34 Mutual Inductance of Coaxial Circles

- §19.33(i) Surface Area
- §19.33(ii) Potential of a Charged Conducting Ellipsoid
- §19.33(iii) Depolarization Factors
- §19.33(iv) Self-Energy of an Ellipsoidal Distribution

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

19.33.1 | $$S=3V{R}_{G}({a}^{-2},{b}^{-2},{c}^{-2}),$$ | ||

or equivalently,

19.33.2 | $$\frac{S}{2\pi}={c}^{2}+\frac{ab}{\mathrm{sin}\varphi}\left(E(\varphi ,k){\mathrm{sin}}^{2}\varphi +F(\varphi ,k){\mathrm{cos}}^{2}\varphi \right),$$ | ||

$a\ge b\ge c$, | |||

where

19.33.3 | $\mathrm{cos}\varphi $ | $={\displaystyle \frac{c}{a}},$ | ||

${k}^{2}$ | $={\displaystyle \frac{{a}^{2}({b}^{2}-{c}^{2})}{{b}^{2}({a}^{2}-{c}^{2})}}.$ | |||

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).

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 | $$\frac{{x}^{2}}{{a}^{2}+\lambda}+\frac{{y}^{2}}{{b}^{2}+\lambda}+\frac{{z}^{2}}{{c}^{2}+\lambda}=1,$$ | ||

$\lambda \ge 0$. | |||

The potential is

19.33.5 | $$V(\lambda )=Q{R}_{F}({a}^{2}+\lambda ,{b}^{2}+\lambda ,{c}^{2}+\lambda ),$$ | ||

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

19.33.6 | $$1/C={R}_{F}({a}^{2},{b}^{2},{c}^{2}).$$ | ||

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

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 | $${L}_{c}=2\pi abc{\int}_{0}^{\mathrm{\infty}}\frac{d\lambda}{\sqrt{({a}^{2}+\lambda )({b}^{2}+\lambda ){({c}^{2}+\lambda )}^{3}}}=V{R}_{D}({a}^{2},{b}^{2},{c}^{2}).$$ | ||

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

19.33.8 | $${L}_{a}+{L}_{b}+{L}_{c}=4\pi ,$$ | ||

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).

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 | $$U=\frac{1}{2}{\int}_{{\mathbb{R}}^{6}}\frac{\rho (x,y,z)\rho ({x}^{\prime},{y}^{\prime},{z}^{\prime})dxdydzd{x}^{\prime}d{y}^{\prime}d{z}^{\prime}}{\sqrt{{(x-{x}^{\prime})}^{2}+{(y-{y}^{\prime})}^{2}+{(z-{z}^{\prime})}^{2}}}.$$ | ||

Subject to mild conditions on $f$ this becomes

19.33.11 | $$U=\frac{1}{2}{(\alpha \beta \gamma )}^{2}{R}_{F}({\alpha}^{2},{\beta}^{2},{\gamma}^{2}){\int}_{0}^{\mathrm{\infty}}{(g(r))}^{2}dr,$$ | ||

where

19.33.12 | $$g(r)=4\pi {\int}_{r}^{\mathrm{\infty}}f(t)tdt.$$ | ||