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(zc)  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 Ωcoordinate (or in the c...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.
 Fall '03
 StefanWaner
 Math, Algebra, Geometry, Complex Numbers

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