Cauchy Integral Formula and Electrodynamics

AI Thread Summary
The discussion centers on the potential application of Cauchy Integral Formulas from complex variables to solve for an electric field (E field) derived from a charge density function. The Cauchy Integral Formula, which involves closed loops in the complex plane, is noted for its utility in evaluating integrals involving derivatives of functions. However, participants express concerns about the complexity of the integral and the need for divergence-type formulas, as opposed to curl-type formulas. The conversation draws parallels between the Cauchy residue theorem and Gauss's theorem, suggesting that both can identify singularities related to charge distributions. The main inquiry is whether this mathematical similarity can be leveraged to address electrodynamics problems specifically in two dimensions, indicating a search for innovative methods in theoretical physics.
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Is it possible to solve for an E field from a charge density function using the Cauchy Integral Formulas from complex variables?

Cauchy Integral Formula about a closed loop in the complex plane
(Integral[f[z]/ (z-z0)^(n+1)dz = 2 pi i /n! d^n f(z0)/dz ])

that is the n derivative of f with respect to z evaluated at z0
 
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If I remember that was a curl type formula and you need a divergence type formula. It may not require math that complicated. It is just that doing the integral might be awkward. In the formula E=Integral (rho/r^2) dr.
 
I was thinking that just like the Gauss's theorem (the surface integral version of the Div[E] = rho/ epsilon) picks out charges which are in effect mathematical singularities, so to the cauchy residue theorem picks out every 1/z of a function.

Can this similarity be used to solve electrodynamics problems in two dimensions?
 
Thread 'Motional EMF in Faraday disc, co-rotating magnet axial mean flux'

Thread 'Motional EMF in Faraday disc, co-rotating magnet axial mean flux'

So here is the motional EMF formula. Now I understand the standard Faraday paradox that an axis symmetric field source (like a speaker motor ring magnet) has a magnetic field that is frame invariant under rotation around axis of symmetry. The field is static whether you rotate the magnet or not. So far so good. What puzzles me is this , there is a term average magnetic flux or "azimuthal mean" , this term describes the average magnetic field through the area swept by the rotating Faraday...
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