Mathematical Methods for Physicists: A concise introduction - Site Map

PARTIAL DIFFERENTIAL EQUATIONS
Let us now apply the solutions of Laplace's equation in cylindrical coordinates
to an in®nitely long cylindrical conductor with radius l and charge per unit length
. We want to ®nd the potential at a point P a distance >l from the axis of the
cylindrical. Take the origin of the coordinates on the axis of the cylinder that is
taken to be the z-axis. The surface of the cylinder is an equipotential:
…l† ˆconst: for r ˆ l and all ' and z:
The secondary boundary condition is that
E ˆ@=@ ˆ =2l" for r ˆ l and all ' and z:
Of the four types of solutions to Laplace's equation in cylindrical coordinates
listed above only the ®rst can satisfy these two boundary conditions. Thus
…† ˆb… f ln ‡ g† ˆ
2" ln l ‡ …a†:
(4) Laplace's equation in spherical coordinates …r;;'†: The spherical coordinates
are shown in Fig. 10.2, where
x ˆ r sin cos ';
y ˆ r sin sin ';
z ˆ r cos ':
Laplace's equation now reads
r 2 @
…r;;'†ˆ1
r 2 @
‡ 1
@ @
r @r @r r 2 sin
sin @ @
‡ 1 @ 2
r 2 sin 2 @' 2 ˆ 0:
…10:28†
Figure 10.2.
Spherical coordinates.
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