# Effective resistance of a plate with varying conductance

I'm trying to find the resistance across a plate made up of different materials. Do my formulas and method make sense - will they yield the correct answer?

Reference I'm using: http://web.mit.edu/6.013_book/www/chapter7/7.2.html

I'm starting with equation (3) for the voltage/potential in a material with varying conductance, setting Φ = 1 at one edge, and to 0 at the other for the boundary conditions: $$\nabla\ \cdot\ \sigma \nabla \Phi\ = 0$$

Right now, I'm solving this numerically with Mathematica. Once I get the solution for Φ, I just use $$J = \sigma \nabla\Phi$$ to get the current density.

Then I just integrate along one of the edges of the platter to find the total current:

$$I = \int_{edge} J dx$$

And finally find the resistance with R=I/V.....

Should that work? Thanks for any advice/pointers/solutions, thanks!

• The quantity you're calculating is not just effective. It is the resistance. Commented Apr 12, 2016 at 17:32
• Indeed to solve this your main difficulty is solving the PDE in the first place. Calculating the total current is then sort of a trivial afterthought. To numerically solve I would recommend finite volume method solvers such as FiPy. Commented Apr 12, 2016 at 17:33
• Well, like I said, I'm a little rusty on the terminology ;p Thanks for the advice though. I'm using Mathematica, it seems to be working OK. Updated question to be a little clearer. Commented Apr 12, 2016 at 18:36
• Well, I can put an official Answer but it would simply be "Yes." :-) Your approach should indeed work. Commented Apr 12, 2016 at 18:41
• Restance? Do you want to edit the title? Commented Apr 12, 2016 at 19:05