Combined resistance of plates made of different materials

[cuallito]

Homework Statement

This problem propped in my head yesterday related to a project I'm working on (not homework, I'm not in school, but I thought this was the best place for it) and I just can't figure out how to crack it! If we have a large plate made up of nxn smaller plates, what is the resistance of the whole plate? For simplicity, let's say each sub-plate is either a perfect conductor, regulator conductor (resistivity=1 in some units), or a perfect insulator.

Would it be possible to write a closed form solution so that if we knew the arrangement of the smaller plates, we can calculate the total resistance of the larger plate?

What I'd like to have is a formula, that given an arbitrary matrix rho representing the arrangement of the sub-plates, it gives me the top-to-bottom resistance of the whole plate.

$$
\rho=
\begin{pmatrix}
0 & 1 & \infty & 1 & 1 \\
\infty & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & \infty & \infty \\
\infty & 1 & 0 & 1 & \infty \\
1 & 0 & \infty & \infty & \infty
\end{pmatrix}
\longrightarrow
\Omega\ ?
$$

2. Homework Equations

Kind of lost on where to start, but maybe differential forms of maxwell's equations?

The Attempt at a Solution

I can figure out some simple cases:

If it's all one material, the resistance would just be the height (top to bottom) times the resistivity of the material.

Likewise, if we limit ourselves to horizontal "stripes" of different materials, the total resistance would just be the height of the first stripe times the resistivity of its material, plus the height of the second times it's reistivity, etc.

Finally if we had a big plate that was one sub-plate high vertically by n plates horizontally, it seems like it'd reduce to a parallel resistors problem? But I'm not sure...

It seems that for more complicated configurations, you'd almost have to find the path of least resistance in each case?

 

[Simon Bridge]

It does sort of depend on the exact arrangement of the connecting wires and the scale, even assuming ideal materials.
I guess you could model the sheet as if each plate were connected by thin wires to the ones adjacent (how do you handle connection through the corners?) ... the equivalent circuit would be a grid. But I very much doubt there is a simple equation for real world situations.

 

[Nidum]

But I very much doubt there is a simple equation for real world situations.

If this was a real world problem to be solved I would either write a simple computer program or find some existing software that could be made use of .

 

[Simon Bridge]

Some solutions are probably quite straight forward though ... like if there is a path of zero resistance through the plate then the resistance will be zero.
You could simplify the situation by looking at the patter of resistivity 0 plates ... those just short out every path they attach to.
i.e. the resistance of the example array is zero if we interpret it as left-to-right, and "1" read top to bottom.

 

[cuallito]

OK, I've been trying to compute this numerically in Mathematica. Please check if I'm doing it right; I somehow got thru two semesters of E&M in college, but not really much of it stuck :)

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:
$$
\nabla\ \cdot\ \sigma \nabla \Phi\ = 0
$$

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 "effective" resistance with R=I/V...

How does that sound?