An introduction to the mathematical theory of attraction ...

60 Lines of Force.
EXAMPLES.
1. Determine the lines of force and the equipotential surfaces in the interior
of a homogeneous ellipsoid.
The differential equations of a line of force are
dx dy dz
Ax y Cz'
where A,, Care the same as in (15), Art. 21.
Integrating these equations we have
zu = K1x = - 2y
where K1 and K2 are arbitrary constants.
The differential equation of an equipotential surface is
Ax dx + By dy + Cz dz = O,
which integrated becomes
Ax2 + By2 + Cz2 = E,
where K is an arbitrary constant.
2. The attraction of a homogeneous spherical shell at an internal point is
zero, and at an external point is the same as if its entire mass were concentrated at its centre.
It is obvious from symmetry that the shell is centrobaric both for an internal
and an external point, the baric centre being the centre of the sphere. Hence
by Art. 38 we obtain the above results.
3. If the resultant force in unoccupied space be uniform in direction it must
be uniform in magnitude.
In this case the lines of force are parallel straight lines; a tube of force is
therefore a cylinder whose section is constant.
Hence by Art. 28 the force is constant along any one line of force; and by
a method similar to that employed in Art. 38, we can show that the force does
not vary in going from one line of force to another.
In this case the equipotential surfaces are planes.
4. A hollow closed conductor is charged with electricity in equilibrium. If
there be no mass in the interior hollow, show that at every point in it the
resultant force is zero, and that there is no charge at any point on the inner
surface of the conductor.
Since the interior hollow @ is unoccupied, no tube of force can begin or end
there; and therefore, since a tube of force cannot be a closed curve, if there be
a tnbe of force in @, it must begin and end on the interior surface of the conductor, which by Art. 31 is impossible.
Again, since there is no force on either side of the interior surface of the
conductor, by Art. 29 there can be no charge.