potential of hollow ball

Let  (ξ,η,ζ)  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  (ξ,η,ζ)  in  (x,y,z)  as


The relevance of this concept appears from the fact that its partial derivativesMathworldPlanetmath


are the components of the gravitational with which the material point  (ξ,η,ζ)  acts on one mass unit in the point  (x,y,z)  (provided that the are chosen suitably).

The potential of a set of points  (ξ,η,ζ)  is the sum of the potentials of individual points, i.e. it may lead to an integral.

We determine the potential of all points  (ξ,η,ζ)  of a hollow ball, where the matter is located between two concentric spheres with radii R0 and R(>R0). Here the of mass is assumed to be presented by a continuous functionMathworldPlanetmathϱ=ϱ(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 R0rR. We use the spherical coordinatesMathworldPlanetmath r, φ and ψ which are tied to the Cartesian coordinatesMathworldPlanetmath via


for attaining all points we set


The cosines law implies that  PA=r2-2arsinφ+a2. Thus the potential is the triple integral

V(a)=R0R-π2π202πϱ(r)r2cosφr2-2arsinφ+a2𝑑r𝑑φ𝑑ψ=2πR0Rϱ(r)r𝑑r-π2π2rcosφdφr2-2arsinφ+a2, (1)

where the factor  r2cosφ  is the coefficient for the coordinate changing


We get from the latter integral

-π2π2rcosφdφr2-2arsinφ+a2=-1a/φ=-π2π2r2-2arsinφ+a2=1a[(r+a)-|r-a|]. (2)

Accordingly we have the two cases:

1.  The point A is outwards the hollow ball, i.e. a>R.  Then we have  |r-a|=a-r  for all  r[R0,R].  The value of the integral (2) is 2ra, and (1) gets the form


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


2.  The point A is in the cavity of the hollow ball, i.e. a<R0 .  Then  |r-a|=r-a  on the interval of integration of (2). The value of (2) is equal to 2, and (1) yields


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.


  • 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
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