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C:\User\Teaching\505506\F10\HW07sol.doc
PHY 505
Classical Electrodynamics I
Fall 2010
Homework 7 with solutions
Problem 7.1
(to be graded of 15 points). Calculate resistance between
two large conductors separated with a very thin, plane, insulating
partition, with a circular hole of radius
R
in it
– see Fig. on the right.
Hint
: You may like to use the degenerate ellipsoidal coordinates which
have been used in class to find the selfcapacitance of a round disk in
vacuum – see Sec. 2.3 of the lecture notes
Solution
: Selecting parameter
R
of the degenerate ellipsoidal coordinates given by Eq. (2.59) of the
lecture notes, to be equal to the hole radius, and placing the origin into its center, we see that the intact
part of the partition corresponds to
=
/2 = const, and at the conductor points slightly above the
partition,
z
≈
R
sinh
(
/2 
), so that
sinh
1
0
0
R
z
n
R
r
z
R
r
z
.
This means that if the potential is a function of
alone, the boundary condition of having no current
flow through the partition is automatically satisfied. (Due to the problem symmetry, the same conclusion
is valid for the lower semispace as well.) From the disk capacitance problem (Sec. 2.3 of the lecture
notes), we already know that with this assumption, the Laplace equation may be also satisfied if
.
)
sinh
arctan(
2
1
c
c
For the sake of notation simplicity, let us accept (anti)symmetric boundary conditions at infinity:
when
i.e.
,
when
),
sgn(
2
)
sgn(
2
z
V
z
V
(with applied voltage driving current up along axis
z
); in this case the solution takes the form
).
sinh
arctan(
V
Note that according to this formula, the contribution of distant parts of the conductor (with
r
>>
R
) into
the total voltage drop is negligible; this is why the shape of external electrodes is not important,
provided that they are not too small or too close to the hole.
What remains now is to find total current
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 Fall '08
 Stephens,P
 Resistance, Work, Laplace, Ohmic, Electric charge, 2k

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