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1 2 show that fz sin z may be regarded as the complex

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Unformatted text preview: = const, we see that the lines § = constant are at right angle to the equipotentials. Put another way: The lines §= const are parallel to Ô∞ and are therefore lines of force (showing the direction of the force) Looking at the example we just did, the lines of force are given by §(z) = A[Arg(z-c) - Arg(z+c)] = Const In the homework, you will see that these too are circles (except for the one degenerate case when the constant is zero), looking something like magnetic lines of force: (In fact the are the same thing...) Using the Complex Potential to get the Electrostatic Field We know that we can recover the electrostatic field by just taking the gradient of ∞: E = Ô∞ However, ∂∞ ∂∞ ∂∞ ∂§ Ô∞ = “ , ‘=“ ,‘ ∂x ∂y ∂x ∂x ∂∞ ∂§ = -i In complex notation ∂x ∂x = F'(z) where F is the complex potential. Conclusion Conservative Vector Fields and Complex Potentials (Not in Kreyzsig) If E is a conservative field independent of the Ω-coordinate (or in the c...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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