Help to solve electric field lines in a dielectric

[sairajan]

Hi all,

I am trying to solve numerically as well as model the electric field lines coming out of a electrode (not a point charge) in a dielectric medium. ANy suggestions would be greatly beneficial

_________________________________________________
_ separated by d + separated by d _

---> Di electric medium + positive charge
_ Negative charge

_ separated by d + separated by d _ Field lines are from + to - and are elliptical
________________________________ Please help

 

[Redbelly98]

I would solve for the voltage with eg. LaPlace's equation (or was it Poison's? Sorry, it's been a while for me), then use
E = -grad V
perhaps with a Runge-Kutta algorithm.

 

[sairajan]

Hi,
thanks. How do you use Laplace for solving. Thanks. Any suggestions are greatly appreciated. Do you have any links online which tells that. Thanks

 

[Redbelly98]

Don't know of any online links, you can always try Google.

A good reference book is Numerical Recipes in C, or Numerical Recipes in Fortran, by Press, teukolsky, Vetterling, and Flannery. You can look in the section on Relaxation Methods in the chapter on Partial Differential Equations.

But I will warn you that you can spend a lot of time writing code, perhaps more than you want to. If this is a work-related project, buying some commercial software that models E-M fields is a better way to go. For example:
http://www.ansys.com/Products/emag.asp

 

[sairajan]

Hi,
very many thanks for your reply. I am actually struck on how to solve the laplace equations. I want to have a numerical solution for this problem. Should I start with electric field due to a line charge. Where can i find that material ?
Thanks

 

[Redbelly98]

Should I start with electric field due to a line charge.

I don't see how that would help. It looks like flat electrodes in your figure.

Solving Laplace's equation numerically:
http://physics.exeter.ac.uk/teaching/CDHW/EM/CW960313-2.pdf
which is the first hit (out of 55,000) listed at
http://www.google.com/search?source...lace's+equation"+numerical&btnG=Google+Search

I'll reiterate, this gives the potential, not the electric field. Also, the dielectric boundary complicates things somewhat and I imagine you need to set up the equation differently along the boundary.

You'd still need to get the E-field once you have the potential.

If others in here have better suggestions, feel free to jump in ...

 

[marcusl]

I'm afraid if you haven't studied electricity and magnetism, you could be lost in this problem. You need at least solutions to the Laplace equation, potentials and fields, and orthogonal funcitons. Complex variables would be helpful, as noted below. You might start with a standard E&M book like Griffith's, or the beautiful but not so well known one by Mel Schwartz.

There is a nice technique that probably can give you an analytic solution to this problem. You transform this problem to a parallel line (e.g., parallel plate) geometry with a conformal transformation, solve it analytically there, and transform the solution back to the circular geometry. I can't remember the transformation off the top of my head--I think it's the bilinear transform. Smythe's book Static and Dynamic Electricity covers the techniques you need, although it's not a beginner's book.