Finding equations for electric field lines

[js2020]

TL;DR Summary

I would like to find an equation for E-field potentiality lines

Hi,

I am interested in finding the equation for electric field equipotential lines. Ideally, it would be nice to have one equation that worked to find it for different geometries. Unfortunately, I don't think that exists. Assuming it does not exist, I think I would probably have to either solve for the electric field intensity at a certain point...or arbitrarily pick a value that I know exists (using Maxwell's equations?). Assuming I'm in 2D, I would then pick a different X (or Y) and set the equation equal to the previous defined electric field intensity. I would do this for a lot of different X,Y combinations and start to see a line. Is this correct? If so, would I then have to plot those points and just use something like Matlab's curve fitting function to find the equation? Or how would I approach this?

 

[Baluncore]

Summary:: I would like to find an equation for E-field potentiality lines
Or how would I approach this?

Mapping electric fields mathematically is only possible for very simple geometry. The boundary conditions must be defined, but some can be infinite.
https://en.wikipedia.org/wiki/Electrostatics#Electric_field

Mapping electric fields arithmetically using FEM is easiest in two dimensions, and possible in 3 dimensions. Point charges have infinite gradients so must be represented by small areas. Again, the boundary conditions must be defined. But they cannot really be infinite.

In both processes, the shape of the conductors are defined as lines of equipotential.
https://en.wikipedia.org/wiki/Field_line#Construction

 

[Baluncore]

This book is well worth finding;
Analysis and Computation of Electric and Magnetic Field Problems. Second Edition. 1973.
By K. J. Binns and P. J. Lawrenson. Publisher; Pergamon Press.
ISBN 0-08-016638-5

Also;
Title; Electric Field Analysis. 2015.
By; Sivaji Chakravorti. Publisher; CRC Press.
ISBN-13: 978-1-4822-3337-7 (eBook - PDF)

 

[js2020]

Thanks for the book suggestions. I requested the first one and was able to download the second one through my library.

I understand how to do it using FEM. I was hoping to find the equation describe those lines somehow. I understand it would change based on the geometry though. Again, thanks for the book suggestions. I'll see if I can find a way there.

 

[Baluncore]

You are getting into the field of Orthogonal Transforms and Mapping.
They are useful for EM fields and fluid flow such as airfoils.

 

[js2020]

I'll see if I can read on orthogonal transforms and mapping in general then.
Just thinking about what I'm trying to do, I can see it being useful for airfoils too. I'm interested in finding out how to design high voltage electrodes. I've found some formulas for different profiles like rogowski, earnst, cheng, etc but I'm wondering how they got the equations in the first place.

I looked at their equations and I'm sure it's just because I'm new to this all together but I'm not sure how to even use those variables.

 

[Baluncore]

The geometry of orthogonal EM fields is fundamental.

For example, the evolution of Paul Neill's “N-type connector” into the BNC connector, occurred once an applied mathematician, Carl Concelman, understood the problem.
BNC = Bayonet Neill–Concelman. TNC = Threaded Neill–Concelman.
https://en.wikipedia.org/wiki/BNC_connector

But that is all in the past. Now for the future ...