An introduction to the mathematical theory of attraction ...

70 The Potential.
3. Mass acting inversely as the square of the distance is uniformly distributed on the circumference of a circle; prove that the chord of contact of
tangents drawn from an external point P divides the mass into two parts having
equal potentials at P.
The normals to the circle at the two points at which it is met by a line
drawn through P make equal angles with this line. Hence, the potentials of the
two portions of mass are composed of elements which are equal respectively.
4. Find the uniplanar potential of a homogeneous circle at any point P, the
force varying inversely as the distance.
Let v denote the uniplanar density of the circle, a its radius, c the distance
of P from its centre O; then if Q be any point on its circumference, and if
r and p denote the distance PQ and the angle POQ, the potential Vis given by
the equation
27r / _ \
V= v log (-) do.
Now r2= a2 + c2-2ac cos = c2 ( 1 - - e (1 - e ),
whence, if P be external to the circle,
log r = log c - (ei4 + e-i4) + ~ a (e2 + e-2i) &c
=log c - cos cp + - cos 2p + &c., and therefore log r d< = 2ir log c.
Substituting in the expression for V, we get V = 2rva log -. If P be internal,
we obtain, in a similar manner,
27r
log r dp = 27r log a,
whence V = 27rva log-.
5. Find an expression for the potential V of a plane lamina, of uniform density o, at any point O, the force varying inversely as the square of the distance.
Take O for origin, let c be the perpendicular distance of O from the plane of
the lamina, N the foot of this perpendicular, and un the distance of any point in
the lamina from N; then the element of mass is expressed by o- w d d<, and,
as r2 = -2 + c2, we have
= orf f =drd{ - (r-c) dc,
where the integral is to be taken round the curve bounding the lamina. If N
be inside this curve, V= = { rd) - 27rc}, and if outside, V = Jr dc.
The original expression for V may be transformed in another manner, as
follows:
_r r/r2 -\ 2 + C-2 r_.2 do r-c
j (r - o)d - jd c = r do-= -- - do.