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In polar coordinates equation (B.6) becomes
|
(B.15) |
Substituting in equation (2.48) for and
performing the integration with respect to gives:
|
(B.16) |
This is in the form of the standard integral from
equation (A.1) of appendix A. Substituting in
the variables: , ,
and
gives
When the evaluation is carried out the contribution when is zero giving:
|
|
|
|
(B.17) |
Now noticing from equation (A.2) that
and using the fact that is a zero of
the second term vanishes:
|
(B.18) |
Expressing this in terms of the ratio of the radii,
we
get
|
(B.19) |
hence we have proved equation (2.85).
The integration used to obtain the analytical expression for is
identical to that for except that the labels are interchanged for
surface 1 and surface 2. Interchanging and means that
will be replaced with
giving
.
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Jonathan Kemp
2003-03-24