Let
be a point bearing a mass
and
a variable point. If the distance of these points is
, we can define the potential of
in
as
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 density 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 obtaining the potential in we must integrate over the ball shell where
. We use the spherical coordinates
,
and
which are tied to the Cartesian coordinates via
![]() |
(1) |
We get from the latter integral
![]() |
(2) |
. 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
. 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
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