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

mr=m(x-ξ)2+(y-η)2+(z-ζ)2.

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

x(mr)=-m(x-ξ)r3,y(mr)=-m(y-η)r3,z(mr)=-m(z-ζ)r3

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

x=rcosφcosψ,y=rcosφsinψ,z=rsinφ;

for attaining all points we set

R0rR,-π2φπ2,0ψ<2π.

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

|(x,y,z)(r,φ,ψ)|=mod|cosφcosψcosφsinψsinφ-rsinφcosψ-rsinφsinψrcosφ-rcosφsinψrcosφcosψ0|.

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

V(a)=4πaR0Rϱ(r)r2𝑑r=Ma,

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(a)=-Ma2.

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

V(a)=4πR0Rϱ(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.

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
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Related topic SubstitutionNotation
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