Math144Notes

# 7 ln z 2 ln05 the real potential is 2z 1 1587

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 r(z) = where c = z0– and |z0| &lt; 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, then z-b r(z) = bz - 1 1-b -2 and r(1) = = -1, and also r(-1) = = 1 so that r flips the unit circle over. b-1 -2 Notice that, since this is an FLT, circles inside the unit disc must...
View Full Document

## This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

Ask a homework question - tutors are online