# potential of hollow ball

Let  $(\xi,\,\eta,\,\zeta)$  be a point bearing a mass  $m$  and  $(x,\,y,\,z)$  a point. If the distance of these points is $r$, we can define the potential of  $(\xi,\,\eta,\,\zeta)$  in  $(x,\,y,\,z)$  as

 $\frac{m}{r}=\frac{m}{\sqrt{(x-\xi)^{2}+(y-\eta)^{2}+(z-\zeta)^{2}}}.$

The relevance of this concept appears from the fact that its partial derivatives  $\frac{\partial}{\partial x}\!\left(\frac{m}{r}\right)=-\frac{m(x-\xi)}{r^{3}},% \quad\frac{\partial}{\partial y}\!\left(\frac{m}{r}\right)=-\frac{m(y-\eta)}{r% ^{3}},\quad\frac{\partial}{\partial z}\!\left(\frac{m}{r}\right)=-\frac{m(z-% \zeta)}{r^{3}}$

are the components of the gravitational with which the material point  $(\xi,\,\eta,\,\zeta)$  acts on one mass unit in the point  $(x,\,y,\,z)$  (provided that the are chosen suitably).

The potential of a set of points  $(\xi,\,\eta,\,\zeta)$  is the sum of the potentials of individual points, i.e. it may lead to an integral.

We determine the potential of all points  $(\xi,\,\eta,\,\zeta)$  of a hollow ball, where the matter is located between two concentric spheres with radii $R_{0}$ and $R\,(>R_{0})$. Here the of mass is assumed to be presented by a continuous function  $\varrho=\varrho(r)$  at the distance $r$ from the centre $O$. Let $a$ be the distance from $O$ of the point $A$, where the potential is to be determined. We chose $O$ the origin and the ray $OA$ the positive $z$-axis.

For obtaining the potential in $A$ we must integrate over the ball shell where $R_{0}\leq r\leq R$. We use the spherical coordinates  $r$, $\varphi$ and $\psi$ which are tied to the Cartesian coordinates  via

 $x=r\cos\varphi\cos\psi,\quad y=r\cos\varphi\sin\psi,\quad z=r\sin\varphi;$

for attaining all points we set

 $R_{0}\leq r\leq R,\quad-\frac{\pi}{2}\leq\varphi\leq\frac{\pi}{2},\quad 0\leq% \psi<2\pi.$

The cosines law implies that  $PA=\sqrt{r^{2}-2ar\sin\varphi+a^{2}}$. Thus the potential is the triple integral

 $\displaystyle V(a)=\int_{R_{0}}^{R}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{% 0}^{2\pi}\!\!\frac{\varrho(r)\,r^{2}\cos\varphi}{\sqrt{r^{2}-2ar\sin\varphi+a^% {2}}}\,dr\,d\varphi\,d\psi=2\pi\int_{R_{0}}^{R}\varrho(r)\,r\,dr\int_{-\frac{% \pi}{2}}^{\frac{\pi}{2}}\frac{r\cos\varphi\,d\varphi}{\sqrt{r^{2}-2ar\sin% \varphi+a^{2}}},$ (1)

where the factor  $r^{2}\cos\varphi$  is the coefficient for the coordinate changing

 $\left|\frac{\partial(x,\,y,\,z)}{\partial(r,\,\varphi,\,\psi)}\right|=\!\mod\!% \left|\begin{matrix}\cos\varphi\cos\psi&\cos\varphi\sin\psi&\sin\varphi\\ -r\sin\varphi\cos\psi&-r\sin\varphi\sin\psi&r\cos\varphi\\ -r\cos\varphi\sin\psi&r\cos\varphi\cos\psi&0\end{matrix}\right|.$

We get from the latter integral

 $\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{r\cos\varphi\,d\varphi% }{\sqrt{r^{2}-2ar\sin\varphi+a^{2}}}=-\frac{1}{a}\operatornamewithlimits{\Big{% /}}_{\!\!\!\varphi=-\frac{\pi}{2}}^{\,\quad\frac{\pi}{2}}\sqrt{r^{2}-2ar\sin% \varphi+a^{2}}=\frac{1}{a}[(r+a)-|r-a|].$ (2)

Accordingly we have the two cases:

$1^{\circ}$.  The point $A$ is outwards the hollow ball, i.e. $a>R$.  Then we have  $|r-a|=a-r$  for all  $r\in[R_{0},\,R]$.  The value of the integral (2) is $\frac{2r}{a}$, and (1) gets the form

 $V(a)=\frac{4\pi}{a}\int_{R_{0}}^{R}\varrho(r)\,r^{2}\,dr=\frac{M}{a},$

where $M$ is the mass of the hollow ball. Thus the potential outwards the hollow ball is exactly the same as in the case that all mass were concentrated to the centre. A correspondent statement concerns the attractive

 $V^{\prime}(a)=-\frac{M}{a^{2}}.$

$2^{\circ}$.  The point $A$ is in the cavity of the hollow ball, i.e. $a .  Then  $|r-a|=r-a$  on the interval of integration of (2). The value of (2) is equal to 2, and (1) yields

 $V(a)=4\pi\int_{R_{0}}^{R}\varrho(r)\,r\,dr,$

which is on $a$. That is, the potential of the hollow ball, when the of mass depends only on the distance from the centre, has in the cavity a constant value, and the hollow ball influences in no way on a mass inside it.

## References

• 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset II.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).
 Title potential of hollow ball Canonical name PotentialOfHollowBall Date of creation 2013-03-22 17:16:46 Last modified on 2013-03-22 17:16:46 Owner pahio (2872) Last modified by pahio (2872) Numerical id 8 Author pahio (2872) Entry type Example Classification msc 28A25 Classification msc 26B10 Classification msc 26B15 Related topic JacobiDeterminant Related topic ChangeOfVariablesInIntegralOnMathbbRn Related topic SubstitutionNotation Related topic ModulusOfComplexNumber