Mathematical Methods for Physicists: A concise introduction - Site Map

PARTIAL DIFFERENTIAL EQUATIONS
We now illustrate the usefulness of the above result for an electrostatic problem
having spherical symmetry. Consider a conducting spherical shell of radius a and
charge per unit area. The problem is to ®nd the potential …r;;'† at a point P a
distance r > a from the center of shell. Take the origin of coordinates to be at the
center of the shell. As the surface of the shell is an equipotential, we have the ®rst
boundary condition
…r† ˆconstant ˆ …a† for r ˆ a and all and ': …10:38†
The second boundary condition is that
! 0 for r !1and all and ': …10:39†
Of the three types of solutions (10.37) only the last can satisfy the boundary
conditions. Thus
…r;;'†ˆ…cr 1 ‡ d†P 0 …cos †:
Now P 0 …cos † ˆ1, and from Eq. (10.38) we have
…10:40†
…a† ˆca 1 ‡ d:
But the boundary condition (10.39) requires that d ˆ 0. Thus …a† ˆca 1 ,or
c ˆ a…a†, and Eq. (10.40) reduces to
Now
…r† ˆa…a† : …10:41†
r
…a†=a ˆ E…a† ˆQ=4a 2 ";
where " is the permittivity of the dielectric in which the shell is embedded,
Q ˆ 4a 2 . Thus …a† ˆa=", and Eq. (10.41) becomes
…r† ˆa2
"r : …10:42†
Solutions of the wave equation: separation of variables
We now use the method of separation of variables to solve the wave equation
@ 2 u…x; t†
@x 2 ˆ v 2 @ 2 u…x; t†
@t 2 ; …10:43†
subject to the following boundary conditions:
u…0; t† ˆu…l; t† ˆ0; t 0; …10:44†
u…x; 0† ˆf …t†; 0 x l; …10:45†
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