Capacitance matrix and integral equation method

[Jack the Stri]

Dear all,

I'm having some trouble calculating the capacitance matrix, as outlined below.

So first of all, the lecture notes I'm using use an integral equation method (method of moments) to determine the capacitance of two infinitely long and thin parallel strips in vacuum, at a distance d of each other. The z-axis is taken along one of both strips, and image theory teaches us that this problem is equivalent to one strip at a distance d/2 of a PEC plane.

The notes start from the potential of a line charge, and use the superposition principle to express the potential of the surface charge density, yielding

$$\phi(\rho) = \int^{}_{c}\rho_{l}(\rho')G(\rho,\rho')dc'$$

With the position vectors in bold (rho and rho' being each other's image), $$\rho_{l}$$ the surface charge density, c the strip width and G the Green's function for a two-dimensional Laplace equation in a homogeneous half-space on top of a PEC plane.

The next step is to impose the boundary condition that the strip surface is equipotential, e.g. at 1. Hence

$$lim_{\rho\rightarrow c} \phi(\rho) = lim_{\rho\rightarrow c} \int^{}_{c}\rho_{l}(\rho')G(\rho,\rho')dc'=1$$

This is solved by subdividing the strip into N segments of length $$\Delta$$, allowing the above expression to be rewritten as
$$lim_{\rho\rightarrow c}\sum^{N}_{i=1}\rho_{li}\int^{}_{c_{i}}G(\rho,\rho')dc'_{i}=1$$

The text then goes on to explain that, when one chooses the centres $$\rho_{j}$$ as the limit c to approach, the sum forms a set of N linear equations with N unknown coefficients. In matrix form this is written:

$$I_{N}=C^{-1}Q$$

With I_N an Nx1 column matrix with elements 1, Q the Nx1 column matrix with the elementary line charge densities (or at least, the average of the segment, the way I see it), and C^-1 the inverted capacitance matrix.

The values of C_ij are then approximated to fill in the matrix.

However.

I understand the steps leading up to this, but in this particular case, the only output for the capacitance is one number, not an NxN matrix (which makes sense otherwise the matrix size would be dependent upon the accuracy of your calculation). The question then is, how do you get the capacitance C from this matrix, and, in the more general case of a capacitance matrix, how do you extract each individual C_ij (since these are not the same as the C_ij mentioned above)?

I can post some more elaboration on the maths followed if necessary.

Edit: LaTeX doesn't seem to be co-operating with me as far as formatting certain formula parts in bold, sorry...

 

[Jack the Stri]

Bumpity bump. From what I can make out of it each element i,j of the matrix represents the capacitance between two segments $$\Delta_{i}$$ and $$\Delta_{j}$$, or the self-capacitance. The capacitance value would then be the sum of the capacitances for each individual segment, i.e. $$C = \sum C_{i} = \sum (C_{ii} - \sum_{j(j \neq i)} |C_{ij}|)$$, or the diagonal minus the absolute values of the rest.

This doesn't match the textbook though. Any expertise around here?