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

$$\frac{\partial }{\partial x}\left(\frac{m}{r}\right)=-\frac{m(x-\xi )}{r^3},\frac{\partial }{\partial y}\left(\frac{m}{r}\right)=-\frac{m(y-\eta )}{r^3},\frac{\partial }{\partial z}\left(\frac{m}{r}\right)=-\frac{m(z-\zeta )}{r^3}$$

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 $R_0\le r\le R$. We use the spherical coordinates $r$, $\phi$ and $\psi$ which are tied to the Cartesian coordinates via

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

for attaining all points we set

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

$V(a)={\displaystyle {\int }_{R_0}^R}{\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}}{\displaystyle {\int }_0^{2\pi }}{\displaystyle \frac{\varrho (r)r^2\cos \phi }{\sqrt{r^2-2ar\sin \phi +a^2}}}𝑑r𝑑\phi 𝑑\psi =2\pi {\displaystyle {\int }_{R_0}^R}\varrho (r)r𝑑r{\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}}{\displaystyle \frac{r\cos \phi d\phi }{\sqrt{r^2-2ar\sin \phi +a^2}}},$ (1)

where the factor  $r^2\cos \phi$  is the coefficient for the coordinate changing

We get from the latter integral

${\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}}{\displaystyle \frac{r\cos \phi d\phi }{\sqrt{r^2-2ar\sin \phi +a^2}}}=-{\displaystyle \frac{1}{a}}\underset{\phi =-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{/}}\sqrt{r^2-2ar\sin \phi +a^2}={\displaystyle \frac{1}{a}}[(r+a)-|r-a|].$ (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

$$V(a)=\frac{4\pi }{a}{\int }_{R_0}^R\varrho (r)r^2𝑑r=\frac{M}{a},$$

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. .  Then  $|r-a|=r-a$  on the interval of integration of (2). The value of (2) is equal to 2, and (1) yields

$$V(a)=4\pi {\int }_{R_0}^R\varrho (r)r𝑑r,$$

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.

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