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Unformatted text preview: he vector components are then the real and imaginary parts of this.
(B) Potential between two non-coaxial cylinders.
Find the potential between two cylinders C 1: |z| = 1 being grounded (potential 0) and
C2: |z - 0.4| = 0.4 having a potential of 110 volts.
This is hard to solve without some trick: First, consider the general FLT
z - z0
where c = z0– and |z0| < 1.
cz - 1
Then I claim that r maps the unit disc onto itself, but takes z0 to 0. The latter claim is
obvious. Let us check that first claim: Mapping the unit disc onto itself:
|z| = 1ﬁ |z - z0| = |z– - z0–|
Since |w| = |w–| for every w
= |z| |z– - z–0–|
Since |z| = 1
= |zz– - z z0–|
= |1 - z z–0–|
Again, since |z| = 1
Therefore, |r(z)| = 1, as claimed.
Notice one further thing about this strange map: If we choose z0 to be real; z0 = b, say,
bz - 1
= -1, and also r(-1) =
= 1 so that r flips the unit circle over.
Notice that, since this is an FLT, circles inside the unit disc must...
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