An introduction to the mathematical theory of attraction ...

220 Electric Images.
and at any internal point U = e (log AQ - logf) - E log a, where.f is the
distance of A from the centre C.
4. In Ex. 1, if the charge on the circle be - e; find the potential V at its
circumference.
If A be outside the circle, V = e log -; if A be inside, V = O. This case
f'
of a uniplanar distribution of mass corresponds to that of a sphere put to earth
under the influence of an electrified point. For the sphere, whether A be an
internal or an external point, the potential at the surface is zero, but the total
mass on the surface is different in the two cases. For the circle, the total mass
on the circumference is the same for either position of A, but the potential is
different.
5. If uniplanar mass be distributed on the circumference of a circle, so that
the density at any point varies inversely as the square of its distance from
a fixed point, show that the distribution is centrobaric, and find the baric
centre.
f2 - a 2
6. Find the value of the integral i —f - ds, taken round the circumference
s of a circle whose radius is a, where r andf are the distances of ds, and the
centre of the circle from an external point A. Ans. 2ira.
7. If A be inside the circle, what is the value of the integral in the last
Example? Ans. - 27ra.
8. Find the force which an insulated circle, having a uniplanar charge E,
exerts on a quantity e of uniplanar mass concentrated at an external point A.
If F denote the force, and î' the distance of A from the centre of the
circle
F=e tE+e er
' r2 -- 2J a
9. Uniplanar mass is distributed on the boundary of the larger segments of
two circles cutting orthogonally so as to produce a uniform potential L; find
the potential at any point in external space, and the distribution of the mass
(see fig., Art. 115).
Let A and B be the centres of the two circles, and G the point of intersection of their common chord with AB. If we suppose equal quantities - of
uniplanar mass placed at A and B, and a quantity - -q placed at G, the potential
at the boundary of one of the circles is v log -, where c is the distance between
their centres. Hence, if v be determined by the equation - log - = L, the
c
potential V at any point Q in external space is given by the equation
V-= (log GQ - log AQ - log BQ),