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For axisymmetric solutions, that is solution independent of
Laplace's equation is
 |
(5.27) |
As in the case of cylindrical coordinates there are many particular
solutions. There are a few basic important solutions and the rest are
given in terms of powers of
and Legendre polynomials in
.
- An isolated point source (point charge, magnetic monopole,
point source or sink) at the origin has
- If the point source is located at
then the
potential at a general point
is
- A uniform field at large
in the
-direction is
- A dipole of moment
, orientated in the
direction has a potential
Be careful of the difference in forms for the point sources in spherical
coordinates and the line sources in cylindrical coordinates.
To obtain the general solutions, we look for seperable solutions along
the lines of the cylindrical case. Hence, to solve (5.27) we set
Thus, we obtain
Rearranging by dividing by
gives,
The separation constant is taken as
where
is an integer.
Thus, the radial variations satisfy
It is straightforward to show that solutions to this are given by
The
variations are given by solving
This equation may be simplified by changing the independent variable
from
to
Thus,
Hence our equation for
becomes
and the solution is given in terms of Legendre polynomials of order
,
Finally, the potential may be expressed as either
 |
(5.28) |
Like the trigonometric functions in the cylindrical case, the
Legendre polynomials are orthogonal since,
The first two Legendre polynomials give
so that
and
Example 5. .6Find the incompressible, irrotational flow of an ideal fluid or
plasma, uniform at infinity, past a sphere of radius
. The general
solution that is even in
and uniform at infinity is
On the surface of the sphere we require that the radial velocity
component vanishes. Thus,
This means that
From the orthogonality condition on the Legendre polynomials every
coefficient must be zero, giving,
Thus,
Next: Waves
Up: Basics Potential Solutions
Previous: Cylindrical Coordinates,
Prof. Alan Hood
2000-11-06