potential of hollow ball
The relevance of this concept appears from the fact that its partial derivatives
are the components of the gravitational with which the material point acts on one mass unit in the point (provided that the are chosen suitably).
The potential of a set of points is the sum of the potentials of individual points, i.e. it may lead to an integral.
We determine the potential of all points of a hollow ball, where the matter is located between two concentric spheres with radii and . Here the of mass is assumed to be presented by a continuous function at the distance from the centre . Let be the distance from of the point , where the potential is to be determined. We chose the origin and the ray the positive -axis.
for attaining all points we set
The cosines law implies that . Thus the potential is the triple integral
We get from the latter integral
Accordingly we have the two cases:
. The point is outwards the hollow ball, i.e. . Then we have for all . The value of the integral (2) is , and (1) gets the form
where 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
. The point is in the cavity of the hollow ball, i.e. . Then on the interval of integration of (2). The value of (2) is equal to 2, and (1) yields
which is on . 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|
|Date of creation||2013-03-22 17:16:46|
|Last modified on||2013-03-22 17:16:46|
|Last modified by||pahio (2872)|