Up: PHY531 Classical Electrodynamics Links
Using a Transfer Matrix
K.E. Schmidt
Department of Physics and Astronomy
Arizona State University
Tempe, AZ U.S.A.
Jackson problems 4.8, 5.8, and many similar problems
can be solved using a transfer matrix.
These problems are the shielding effect of a circular cross section dielectric
or of a permeable sheath. For the magnetic case, there are no currents,
so the curl of and the divergence of are both
zero, and
Except at the boundaries between media with different ,

= 0,
 (2)

and tangential and normal
are continuous at boundaries. Continuity of tangential
implies that the potentials on either side of a surface can differ
by at most a constant. We choose the constant to be zero so that
that boundary condition is replaced by continuity of the potential.
The dielectric case is identical to the magnetic case
with the replacements
For the twodimensional case, where the fields go to a constant along
far from the shield, only solutions with this cos()
dependence will be produced. That is the solution in region i
bounded by two circular cross section surfaces will have the form
The boundary conditions between and at = a_{i}
are then
A_{i}a_{i} +

=

A_{i + 1}a_{i} +
 

=

 (5) 
with the solution
  (6)

We can now use this transfer matrix to solve for a multilayered sheath.
At each boundary we apply Eq. 6 to get the coefficients
of the potential in the next region.
For the simple case of a sheath of permeability from = a to
= b , we have the additional boundary conditions,
C_{1} = 0 and A_{3} =  B_{0} , and two boundaries.
Identifying
A_{1} =  B_{in} , we can write
  (7)

which gives immediately the result

B_{0} = B_{in}
 (8)

or

B_{in} = B_{0}
 (9)

The coeffients of the potential inside the sheath are
A_{2} =  B_{in} =  B_{0}


 
C_{2} =  B_{in} =  B_{0}


 (10) 
and outside

C_{3} = B_{in} = B_{0}
 (11)

Jackson's example in three dimensions can also be written in terms
of a transfer matrix. The potential within each spherical shell
is
The boundary conditions are
A_{i}a_{i} +

=

A_{i + 1}a_{i} +
 

=

 (13) 
Solving for the transfer matrix as in the twodimensional case gives
  (14)

For the single shell from a to b ,
  (15)

with the immediate result

B_{0} = B_{in}
 (16)

as given in Jackson.
These same techniques can be applied to higher l values too. For
the spherical case, the equations are
The boundary conditions are
A_{i}a_{i}^{l} +

=

A_{i + 1}a_{i}^{l} +
 

=

 (18) 
or
  (19)

For the single shell from a to b ,
  (20)

with the result

B_{0} = B_{in}.
 (21)

For
,
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