An introduction to the mathematical theory of attraction ...

Examp les. 91
2. Find the potential of a homogeneous thin circular plate at a point P on
the perpendicular to its plane through its centre.
If 'z be the distance of any point of the plate from its centre C, and c the
distance PC, taking P for origin, we have z dr = zJ d; whence
r
Y_ 27r~ 1 dr = 27ro (Va2 + c2 _ c),
ro
where a is the radius of the plate. If C be taken for origin, V assumes the
form
27(ro (a2 + r2 - r), or 2ra (Va2 + z2 - z),
according as CP is regarded as a radius vector or a coordinate. In the latter
case when z is negative V becomes 27ro (/a2 + z2 + z).
3. Show that equation 22, Art. 24, follows immediately from Poisson's
Equation.
4. Find the potential at any point of a field of force throughout which the
resultant force is constant in direction.
Take a line parallel to this direction for axis of z. Since the lines of force
are perpendicular to the equipotential surfaces, these latter are parallel planes.
Hence when z is constant V is constant, that is, Y is a function of z. Laplace's
Equation therefore becomes
d2 V
d = 0, whence V = Ciz + Cz.
dZy
5. Find the potential at any point between two infinite parallel planes, each
of which is at a constant potential.
Take as the plane of xy the plane whose potential is A, let b be the distance
of the other, and B its potential; then Vis plainly a function of z; and therefore,
by Laplace's equation, we get
A-B
V= A- z.
b
6. Two concentric spherical surfaces are each at a constant potential: find
the potential at any point between them.
Let a denote the radius of the inner sphere, and b that of the outer, the
potentials at their surfaces being A and B; then, if the centre of the spheres
be taken for origin, it is plain that the potential at any point of the space
between their surfaces is a function of r; and as this space is unoccupied, we
have, by Laplace's equation,
d( 2dV)o
dr ~gdr=
Bb -Aa (A- B) ab 1
whence V = - +
b-a b-a r
7. In the last example, if the spherical surfaces be the boundaries of two
charged conductors in electric equilibrium, find the surface density and charge
on each conductor.